Thermodynamics Skeleton Guide

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Thermodynamics

Skeleton Guide

Joseph E. Shepherd

Division of Engineering and Applied Science California Institute of Technology Pasadena, CA 91125

December 7, 2019

1. You can’t win, you can only break even.

2. You can only break even at the absolute zero.

3. You cannot reach absolute zero.

Conclusion: You can neither win nor break even.

Quoted by Dugdale on the last page of Entropy and its physical meaning.

Warning This is not a textbook on thermodynamics or even a set of notes that you can learn from. There are no ﬁgures or derivations, nothing about applications, and no examples. Fur- thermore, it is not logically organized. It is a collection of facts and formulas that represent the essential mathematical framework or “skeleton” that I use when teaching thermodynamics. Some of formulas may be wrong and the notation may be inconsistent. The main purpose is to typeset the equations and symbols so that when the students can’t read my terrible handwriting on the chalk board they have something to fall back on. If you need a text, some suggestions are discussed in SectionA. JES, April 1, 2007. Contents

1 Fundamentals1 1.1 Thermodynamic Systems...... 1 1.2 Equilibrium...... 1 1.3 State Variables...... 2 1.4 Energy...... 2 1.5 Zeroth Law of Thermodynamics...... 2

2 First Law of Thermodynamics3 2.1 Point vs. Path Function...... 3 2.2 Work...... 3 2.3 Enthalpy...... 3 2.4 Speciﬁc Properties...... 4 2.5 Heat Capacity and Speciﬁc Heat...... 4

3 Cycles 5 3.1 Thermodynamic Efﬁciency...... 5 3.2 Carnot Cycle...... 5 3.2.1 Carnot’s Theorem...... 6 3.2.2 Thermodynamic Temperature...... 6

4 Second Law of Thermodynamics - Cycle Version6 4.1 Kelvin-Planck...... 6 4.2 Clausius...... 6

5 Entropy 7 5.1 Entropy as a State Function...... 7

6 Second Law of Thermodynamics - Entropy Version7

7 Open Systems8 7.1 Mass...... 8 7.2 Energy...... 8 7.3 Entropy...... 8 7.4 Steady State, Steady Flow...... 9 7.5 Ideal Stagnation Properties...... 9 7.6 Sound speed and Mach number...... 10 7.7 Efﬁciency...... 10

8 Availability, Maximum Work and Reversibility 11 8.1 Availability of a Closed System...... 11 8.2 Second Law analysis of ﬂowing systems...... 11 i 9 Ideal Gas 13 9.1 Ideal Gas Law...... 13 9.2 Internal Energy...... 13 9.3 Adiabatic Reversible Compression...... 14 9.4 Entropy...... 15 9.5 Gas Kinetics...... 15 9.5.1 Mean Free Path...... 16 9.5.2 Distribution of Molecular Velocities...... 17 9.5.3 Distribution of Molecular Energies...... 20 9.5.4 Ideal Gas Law...... 21 9.5.5 Speciﬁc Heat Capacity and Partition of Energy...... 21 9.5.6 Molecular Flux...... 23 9.6 Mixtures...... 23 9.6.1 Partial Molar Properties...... 24

10 Relationships 27 10.1 Thermodynamic potentials and fundamental relations...... 27 10.2 Maxwell relations...... 27 10.3 Calculus identities...... 27 10.4 Various defined quantities...... 28 10.5 Specific heat relationships...... 28 10.6 Gruneisen¨ Coefficient...... 28 10.7 Thermal Pressure Coefficient...... 29 10.8 Enthalpy, Energy and Entropy...... 29

11 Equation of State 30 11.1 Constructing Equations of State...... 30 11.2 Critical Points...... 31 11.2.1 Law of corresponding states...... 31 11.3 Compressibility Factor...... 32 11.4 van der Waals Equation...... 32 11.5 Throttling and the Joule-Thompson Coefﬁcient...... 32 11.6 Virial Equation...... 32 11.7 Ideal Solid...... 33

12 Equilibrium Liquid-Vapor Mixtures 34 12.1 Lever Rule...... 34 12.2 Humidity...... 34

13 Phase Equilibrium 36 13.1 Clapeyron Equation...... 36 13.2 Maxwell’s Construction...... 36 13.3 Multiple Phases...... 37 ii 14 Chemical Transformations 38 14.1 Heat of Reaction...... 38 14.1.1 Enthalpy...... 38 14.1.2 Heat of formation...... 39 14.2 Combustion Reactions...... 39 14.3 Heat of Combustion...... 40 14.4 Flame Temperature...... 40 14.5 Explosion Pressure and Temperature...... 41

15 Chemical Equilibrium 41 15.1 Le Chatelier’s Rule...... 43 15.1.1 Pressure...... 43 15.1.2 Temperature...... 44

A Textbooks and references 45 A.1 Introductory...... 45 A.2 Many others...... 46 A.3 Biography and History...... 46

B Famous Numbers 47

C Fundamental Dimensions 49

D Ideal Gas Properties up to 4000K 50

iii 1 1 Fundamentals

At the most basic level, thermodynamics is concerned with energy and the transformation of energy into different forms such as internal energy, work, and heat. Thermodynamics also provides fundamental limitations on the efficiency of energy transformation processes. On a theoretical level, thermodynamics introduces the concept of entropy, a measure of disorder in physical matter. On a more mathematical level, thermodynamics provides a methodology for describing the properties of matter and the relationship between properties such as internal energy, entropy, pressure, temperature and volume. Thermodynamics is a conceptual framework that is based on empirical observations and the- ories about natural processes. It is not a complete theory of nature or method for computing properties of matter but rather a set of ideas for relating properties of matter and governing energy transformation. Thermodynamics can be applied to a fixed or variable quantity of matter that contains a macroscopic amount of material that can be considered to be an equilibrium state. Macroscopic means that sufficiently large enough numbers of molecules and atoms are in the system that average properties of matter exist and the fluctuations from the average are negligibly small. Equilibrium means that the system has reached a state that is stable, and free from external influences, will not evolve further.

1.1 Thermodynamic Systems The essential unit that we study in thermodynamics is a system.

System A quantity of matter or a region in space that we select to study. Systems can be: closed – no exchange of matter with the surroundings open – exchanges matter with the surroundings isolated – closed and no exchange of energy with the surroundings

Control boundary or surface – The envelope surrounding and deﬁning the system. Surroundings – Everything other than the system. State – A well-deﬁned condition of a system described by certain observable macroscopic properties. Process – A sequence of states that describe the evolution of a system.

1.2 Equilibrium Condition of a system that has ceased to evolve after a period of time. Isolated systems reach a state of internal equilibrium. Systems that interact with the surroundings or each other reach a state of external or mutual equilibrium. A system in equilibrium is uniquely described by its equilibrium state. 2 1 FUNDAMENTALS

1.3 State Variables A system in internal equilibrium is uniquely defined by a set of thermodynamic parameters or coordinates such as temperature, pressure, volume, composition, etc. Specifying the values of an independent and complete set of these state variables defines a unique equilibrium state of a system. There are two types of state variables: Extensive Properties that are proportional to the amount of substance in the system. Examples are: M mass, V volume, U internal energy, S entropy, quantity – number of molecules N or moles n Intensive Properties that are independent of the amount of substance in the system. Examples are: pressure P , temperature T , specific energy u, composition measured in mole y or mass fraction x of a component or phase.

1.4 Energy The energy of a system consists of the sum of kinetic energy (KE), potential energy (PE) and the internal energy E = KE + PE + U. (1) The KE and PE are due to the motion of the system as a whole. For a mechanical system of mass M moving with velocity v in a gravitational ﬁeld with acceleration g and height z above a reference level of zero potential, the total energy can be written 1 E = M|v|2 + Mgz + U. (2) 2 The internal energy is a state variable, for example U = U(P,T ) . (3)

1.5 Zeroth Law of Thermodynamics Two systems are in thermal equilibrium if there is no net transfer of energy as heat when they are placed in thermal contact. Systems in thermal equilibrium are characterized by having the same temperature. T1 = T2 (4) Empirical temperature is a measure of the relative “hotness” or “coldness” of a system – heat ﬂows from a hot to a cold system; degrees Celsius (◦C) is an example of an empirical temperature: 0◦C is the freezing point of water, 100◦C is the boiling point of water. Thermodynamic temperature or absolute temperature, measured in Kelvin (K), is the appropriate temperature to use in thermodynamic computations in SI units T (K) = T (◦C) + 273.15 . (5) Systems are in mechanical equilibrium if they have the same pressure.

P1 = P2 (6) Pressure can be deﬁned from a purely mechanical viewpoint as force/area. The SI unit of pressure is the Pascal (Pa) equal to 1 N/m2. 3 2 First Law of Thermodynamics

Energy is conserved. For a closed system that undergoes a process between states 1 and 2, the change in energy is due only to the energy that is added to the system in the form of heat, Q, and the work, W , that is done by the system.

E2 − E1 = Q − W (7)

Where Q > 0 if heat is added to the system and W > 0 if work is done by the system on the surroundings. If the system is stationary and there are no external ﬁelds then

U2 − U1 = Q − W. (8)

For small changes in internal energy, dU, and small amounts of heat addition, δQ, and system work, δW , the ﬁrst law is dU = δQ − δW . (9)

2.1 Point vs. Path Function A point function, like internal energy, has well-deﬁned derivatives and the integral of the derivative, dU, has a well-deﬁned value that only depends on the end points. Thus, dU is an exact differential.

Z 2 U2 − U1 = dU (10) 1 Heat added to and work done by a system depend on the path or manner in which the process is carried out. Heat and work are not exact differentials, so the result of integration depends on the process path. Z 2 Z 2 Q = Q1→2 = δQ W = W1→2 = δW (11) 1 1

2.2 Work For a system with pressure P and volume V , the work done by the system is Z W = P dV . (12)

For small, slow changes in volume, the effect is reversible and work can be treated as a differential

dW = P dV . (13)

2.3 Enthalpy If the pressure is held constant, the work done by the system in changing from state 1 to 2 is

W = P (V2 − V1) . (14) 4 2 FIRST LAW OF THERMODYNAMICS

The enthalpy is deﬁned as H ≡ U + PV. (15) The amount of heat added in a constant pressure process is equal to the change in enthalpy.

U2 − U1 = Q − P (V2 − V1)

(U2 + PV2) − (U1 + PV1) = Q

∆H = H2 − H1 = QP = constant

2.4 Specific Properties Thermodynamic extensive properties and associated SI units are property unit specific property unit U internal energy J u specific internal energy J/kg V volume m3 v specific volume m3/kg M mass kg ρ mass density kg/m3 H enthalpy J h specific enthalpy J/kg S entropy J/K s specific entropy J/K-kg G Gibbs energy J g specific Gibbs energy J/kg A Helmholz energy J a specific Helmholz energy J/kg Thermodynamic potentials: energy U enthalpy HU + PV Helmholtz energy AU − TS Gibbs energy GU − TS + PV

2.5 Heat Capacity and Specific Heat The change in internal energy with the addition of a small amount of heat is known as the heat capacity. The value of the heat capacity depends on the process used to add heat. Heat capacity at constant volume is defined as dU δQ CV ≡ = lim . (16) δT →0 dT V δT V The constant volume specific heat is the constant volume heat capacity per unit mass. du 1 cv ≡ = CV (17) dT v M Heat capacity at constant pressure is defined as

dH δQ CP ≡ = lim . (18) δT →0 dT P δT P The constant pressure speciﬁc heat is the constant pressure heat capacity per unit mass. dh 1 cp ≡ = CP (19) dT P M 5 3 Cycles

In a cycle, we consider a process in which the final state (2) is identical to the initial state (1). Since energy is a state function, E2 = E1 . (20) Because the work and heat exchanged during a process depends on the path taken, the work, W , done by the system and heat, Q, transferred to the system during the cycle are in general finite and by the first law, Eqn.7, must be equal Q = W. (21) A heat engine is a cycle operated between two reservoirs of thermal energy and producing work, W > 0. An amount of heat QH is transferred from a hot reservoir at temperature TH and an amount of heat QC is transferred to a cold reservoir at temperature TC . The total amount of heat transferred to the engine is W = Q = QH − QC . (22) A refrigeration cycle is a heat engine run in reverse, putting work into the cycle W < 0 and transferring heat out of the cold reservoir and into the hot reservoir.

3.1 Thermodynamic Efficiency The thermodynamic efficiency of a heat engine is defined to be

W Q η = = 1 − C . (23) QH QH The efﬁciency of a refrigerator is measured by the coefﬁcient of performance.

ω = |QC |/|W | (24)

3.2 Carnot Cycle The upper bound to efﬁciency for a cycle is found in an ideal reversible heat engine. An engine is reversible if all the component processes are reversible. Reversibility means that there is no dissipation of any kind in the operation of the process and it operates equally effectively in either direction. The Carnot cycle can be implemented as the following sequence of events

a. 1 → 2 Isothermal expansion at temperature TH with heat transfer of QH from the hot reservoir.

b. 2 → 3 Adiabatic expansion, temperature decreases from TH to TC .

c. 3 → 4 Isothermal compression at temperature TC with heat transfer of QC to the cold reservoir.

d. 4 → 1 Adiabatic compression from temperature TC to TH . 6 4 SECOND LAW OF THERMODYNAMICS - CYCLE VERSION

The temperatures and amounts of heat transferred are related by

Q Q H = C . (25) TH TC

The efﬁciency of the Carnot cycle is TC ηc = 1 − (26) TH If a Carnot cycle is run backwards and used as a refrigerator, the coefﬁcient of performance is

1 − ηc ωc = (27) ηc

3.2.1 Carnot’s Theorem

No heat cycle or engine operating between two given reservoirs can be more efﬁcient than the Carnot cycle. Consequences:

1. All reversible engines operating between two given reservoirs have the same efﬁciency.

2. Operation of a reversible engine is independent of design, details of operation or working substance.

3.2.2 Thermodynamic Temperature

A thermodynamic or absolute temperature scale, Θ, is deﬁned by the ratio of energies transferred as heat by the Carnot cycle operating with a constant value of QC to a ﬁxed cold reservoir of temperature TC . Θ Q 2 = 2 (28) Θ1 Q1

4 Second Law of Thermodynamics - Cycle Version

4.1 Kelvin-Planck

No process is possible whose sole result is the absorption of heat from a reservoir and complete conversion of this heat into work.

4.2 Clausius

No process is possible whose sole result is the transfer of heat from a cooler to a hotter body. 7 5 Entropy

A reversible transfer of heat, dQ, to a system at temperature, T , results in an increase in entropy in the amount dQ dS = . (29) T For any real (irreversible) process dQ dS ≥ . (30) T 5.1 Entropy as a State Function Entropy S is a measure of the disorder in a system. Entropy always increases with increasing temperature. Entropy is a state function and is independent of path. Z 2 dQ S2 − S1 = = S(U2,V2) − S(U1,V1) (31) 1 T The First Law of Thermodynamics for a PVT system can be written in the form dU = T dS − P dV , (32) which is sometimes referred to as the fundamental relationship of thermodynamics. The canonical equation of state for a substance can be written as U = U(S,V ) (33) ∂U ∂U dU = dS + dV . (34) ∂S V ∂V S where temperature and pressure are deﬁned by ∂U T = , (35) ∂S V ∂U P = − . (36) ∂V S 6 Second Law of Thermodynamics - Entropy Version

Consider a system that can exchange work and heat with its surroundings but is otherwise isolated. The Second Law of Thermodynamics can be stated as:

The total entropy of the universe consisting of a system and its surroundings either remains constant or increases.

∆Suniverse = ∆Ssystem + ∆Ssurroundings ≥ 0 (37) Consequence: For an isolated system, the entropy tends towards a maximum as the system ap- proaches equilibrium. ∆Ssystem ≥ 0 (38) 8 7 OPEN SYSTEMS 7 Open Systems

An open system can have mass and energy ﬂuxes into and out of the system. There are balance statements or conservation laws which determine how the time rate of change of the energy and mass within the system control volume are related to these ﬂuxes.

7.1 Mass The mass M inside a control volume CV is Z M = ρ dV (39) CV where ρ is the mass density of the material inside the control volume. If the mass ﬂow per unit time ˙ (kg/s) into the control volume is MIN , and the mass ﬂow per unit time out of the control volume is ˙ MOUT , then dM = M˙ − M˙ . (40) dt IN OUT

7.2 Energy The energy, E, inside the control volume, CV , is

Z |v|2 E = ρ u + + gz dV . (41) CV 2

The balance equation for the energy inside the control volume is

2 2 dE ˙ ˙ ˙ |v| ˙ |v| = Q − W + MIN h + + gz − MOUT h + + gz (42) dt 2 IN 2 OUT where Q˙ is the rate of heat addition to the system control volume (J/s), W˙ is the rate that the system does work (J/s), and h is the speciﬁc enthalpy h = u + pv or h = u + p/ρ.

7.3 Entropy The entropy, S, inside the control volume, CV , is Z S = ρs dV (43) CV where s is the speciﬁc entropy of the material inside the CV. The balance inequality for the entropy inside the control volume is ˙ dS Q ˙ ˙ ≥ + MIN sIN − MOUT sOUT (44) dt TQ 7.4 Steady State, Steady Flow 9

˙ where Q is the rate of heat addition to the system control volume (J/s) at temperature TQ. For distributed heat transfer, the ﬁrst term on the RHS becomes

Z q˙ dA (45) TQ where q˙ is the heat ﬂux per unit area and the integral is taken over the entire surface of the control volume.

7.4 Steady State, Steady Flow For a steady flow situation, the mass and energy inside the control volume are constant (independent of time) and we have ˙ ˙ MOUT = MIN (46) and |v|2 |v|2 h + + gz =q ˙ − w˙ + h + + gz (47) 2 OUT 2 IN where q˙ = Q/˙ M˙ is the heat addition to the system per unit mass flowing through it and w˙ = W/˙ M˙ is the rate of work done by the system per unit mass flowing through it. The entropy inequality is

q˙ s2 − s1 ≥ (48) TQ

7.5 Ideal Stagnation Properties The ideal stagnation state (total conditions) is defined by isentropic deceleration of of a fluid to rest. The ideal stagnation enthalpy is defined by

v2 h = h + (49) t 2 and the ideal stagnation pressure is deﬁned by considering h(P, s)

h(Pt, s) = ht (50)

The ideal stagnation temperature is determined by considering h(T, s)

h(Tt, s) = ht (51) and the ideal stagnation density is given by the equation of state ρ(P,T )

ρt = ρ(Pt,Tt) (52) 10 7 OPEN SYSTEMS

7.6 Sound speed and Mach number

The sound speed is deﬁned as

s ∂P a = (53) ∂ρ s

and the Mach number is

v Ma = (54) a

For an ideal gas, the sound speed is

p a = γRT (55)

7.7 Efﬁciency

The efﬁciency of a turbine is deﬁned as the ratio of actual work w˙ a to the maximum amount of work w˙ s that can be obtained working between a given inlet and exhaust pressure.

w˙ a ηturbine = (56) w˙ s

An alternative definition for the turbine efficiency can be derived by applying the energy balance relationship to the flow.

∆ha ηturbine = (57) ∆hs where ∆ha is the actual enthalpy change across the turbine and ∆hs is the enthalpy change for a reversible (isentropic) process between the same inlet and exit pressures. The efﬁciency of an adiabatic compressor is deﬁned as the ratio of work required for ideal (isentropic) compression w˙ s to the actual amount of work w˙ a required to increase the pressure by a given amount.

w˙ s ηcompressor = (58) w˙ a An alternative definition for the compressor efficiency can be derived by applying the energy balance relationship to the flow.

∆hs ηcompressor = (59) ∆ha where ∆ha is the actual enthalpy change across the compressor and ∆hs is the enthalpy change for a reversible (isentropic) process between the same inlet and exit pressures. 11 8 Availability, Maximum Work and Reversibility

Consider a system immersed in an environment with temperature, T◦, and pressure, P◦. In the process of the system coming to equilibrium with the environment, the system can produce external work, Wext. The maximum amount of work that can be obtained is determined by applying the Second Law of Thermodynamics and requiring that there is no net entropy generation. This is equivalent to supposing that the mechanisms of bringing the system to equilibrium with the environment are optimal, i.e., reversible.

8.1 Availability of a Closed System

A = E + P◦V − T◦S (60) or alternatively as A = (E − U◦) + P◦ (V − V◦) − T◦ (S − S◦) (61) where the system initially has energy E, volume V , and entropy S. Since only the differences in availability are important for thermodynamics, subtracting a constant does not affect the ﬁnal result below for maximum work. The system and environment can exchange energy in the form of heat and P -V work due to the change in system volume. After the system reaches equilibrium with the environment, T → T◦, P → P◦, the state of the system is given by E◦ = U◦ = U(T◦,P◦), V◦ = V (T◦,P◦), and S◦ = S(T◦,P◦). The maximum amount of external work that can be obtained from the system is found by using reversible processes that bring the system to equilibrium with the environment.

Wext,max = −∆A (62) Note that the external work is utilized outside of the universe consisting of the system and the environment together. The actual external work, W , that is obtained by using real (irreversible processes) will always be less than this amount.

W ≤ Wext,max (63)

8.2 Second Law analysis of ﬂowing systems For a steady-state, steady-ﬂow process in an open system, the energy balance equation is v 2 v 2 W˙ = M˙ h + in − h − out + Q˙ (64) in 2 out 2 12 8 AVAILABILITY, MAXIMUM WORK AND REVERSIBILITY

The entropy balance equation is

˙ dS Q ˙ ˙ ˙ = + Minsin − Moutsout + I. (65) dt Ts where the entropy generation rate is non-negative

I˙ ≥ 0 . (66) 13 9 Ideal Gas

An ideal gas is a model of a gas in which the molecules have negligible interactions aside from very brief collisions that serve to randomize the distribution of energy and velocity among the molecules and atoms. It is a useful approximation for many real gases and the simplest example of an equation of state. The ideal gas relationships apply both to pure substances and to homogeneous mixtures of gases.

9.1 Ideal Gas Law An Ideal Gas is deﬁned by the thermodynamic relationship

PV = nRT˜ (67) where n is the number of moles of substance and R˜ is the universal gas constant, 8314.5 J/kmol·K.

˜ R = kBNA (68) −23 26 where kB is the Boltzmann constant (1.381×10 J/K) and NA is Avogadro’s number (6.022×10 molecules/kmol). This ideal gas law can also be written as

P v = RT (69) where R = R/˜ W . (70) The ideal gas law can also be written as

P = ρRT or P v = RT (71) where W is the molar mass of the gas and R is simply referred to as the gas constant. The speciﬁc volume v is the reciprocal of the mass density ρ. 1 v = (72) ρ

For air, W = 28.96 kg/kmol and R = 287.05 m2s−2K−1.

9.2 Internal Energy An ideal gas has an internal energy that is a function of temperature only. Z 0 0 U = U(T ) = CV (T ) dT (73)

A perfect gas has a heat capacity that is constant (independent of thermodynamic properties) du = c = constant (74) dT v 14 9 IDEAL GAS

and the energy change between two states can be computed as

u2 − u1 = cv (T2 − T1) . (75)

The speciﬁc enthalpy of a perfect gas is

h = u + P v = u + RT (76) and the constant pressure speciﬁc heat is

cp = cv + R. (77)

The speciﬁc enthalpy difference between two states for a perfect gas is

h2 − h1 = cp (T2 − T1) . (78)

The ratio of constant pressure to constant volume heat capacity of a perfect gas is a constant known as the ratio of speciﬁc heats c γ = p . (79) cv In terms of γ, the speciﬁc heat capacities are

R γR c = c = . (80) v γ − 1 p γ − 1

9.3 Adiabatic Reversible Compression A process is adiabatic if there is no heat transfer into or out of a system, Q = 0. For a perfect gas,

c dT dv du = c dT = −dW = −P dv ⇒ v = − (81) v R T v

For the case of a constant speciﬁc heat ratio, this can be integrated between states 1 and 2 to yield the relationship T v −(γ−1) 2 = 2 . (82) T1 v1 Combine this with the ideal gas law to obtain

−γ P2 v2 γ γ = or P2v2 = P1v1 . (83) P1 v1

This also means that γ P T γ−1 2 = 2 . (84) P1 T1 9.4 Entropy 15

9.4 Entropy The entropy changes in a substance can be computed from Eqn. 32 du P ds = + dv (85) T T

For an ideal gas, du = cv dT and P/T = R/v. For a perfect gas, cv = constant: T2 v2 s2 − s1 = cv ln + R ln (86) T1 v1 or cp = constant: T2 P2 s2 − s1 = cp ln − R ln . (87) T1 P1

For an ideal gas where cp and cv can vary with temperature, cp(T ) = cv(T ) + R ◦ ◦ P2 s2 − s1 = s (T2) − s (T1) − R ln (88) P1 where Z T 0 ◦ cp(T ) 0 s (T ) = 0 dT (89) T ◦ T and T ◦ is the standard state reference temperature of 25◦C or 298.15 K.

9.5 Gas Kinetics A gas is ideal if the size d of the atoms or molecules can be neglected compared to the average 1/3 spacing (V/N) . A typical molecular size is about 0.35 nm for O2 or N2. At NTP (room temperature 25◦C, and pressure 1 bar), the number density of molecules N/V = 2.46×1025 m−3 so that the average spacing is 3.4 nm. In an ideal gas, the molecules are sufﬁciently far apart that there is no contribution of potential energy of intermolecular interaction to the internal energy of a collection of molecules. This means that the internal energy is not a function of volume and is only a temperature for an ideal gas

U = U(T ) for ideal gases (90)

This is very different than in liquids or solids, for which the intermolecular interactions dominate the internal energy and are the key to deﬁning the dependence of pressure on volume.

U = U(T,V ) for liquids and solids (91)

Although the molecular size is small compared to the mean spacing in ideal gases, collisions between the molecules do occur and play an essential role in how gases behave. Collisions are the mechanism by which energy introduced into the gas is redistributed in space. For small disturbances, this occurs by the propagation of sound waves, which travel at speeds that are comparable to the average molecular speed. For large disturbances, shock waves can be generated that can 16 9 IDEAL GAS

abruptly change the speed of the molecules in one spatial direction. Compression by moving walls is transmitted into a gas ﬁrst by energy transfer from a moving wall to adjacent molecules and then subsequent collisions of those molecules in the interior of the gas. Collisions of pairs of very energetic molecules are responsible for the producing excited states, chemical reaction, and ionization. Gas kinetics is the study of gas properties derived by considering the collision mechanics of a large collection of molecules.

9.5.1 Mean Free Path Collisions in ideal gases are essentially random in character, occurring between pairs of molecules with every possible value of kinetic energy and directions. If we average over a very large number of collision pairs, we will obtain an average rate of collision of the molecules within the gas and a corresponding average distance traveled between collisions. The average distance ` between collisions is termed the mean free path and can be estimated by considering the behavior of the average molecule. Consider a representative molecule of diameter d traveling in a gas of randomly distributed molecules of the same size. On the average, this molecule will collide with another molecule when the product of the swept volume πd2` and the number density [n] = N/V is unity or 1 ` = mean free path (92) πd2[n] For air at NTP, ` = 106 nm or about 300 molecular diameters. If the average speed of the molecule is ¯v, then the average frequency of collision will be fc = ¯v/` and the average time between col- 9 −1 lisions will be tc = 1/fc. For air at NTP, fc = 4.4 ×10 s and tc = .22 ns. From our previous considerations about molecular size, time between collisions is about 1000 times longer than the collision itself. Although a collision occurs on the average after one mean free path, the random character of the collision process results is a distribution of collision distances when we make a large number of observations. Since collisions are rare, then the probability of a collision per unit distance traveled is a constant value α. This means that of a sample of molecules of size N, a number dN will have collided after traveling a distance dx and the fractional decrease in number that have not collided yet will be dN = −αdx (93) N The number of molecules N that travel a distance x without colliding decreases exponentially with distance N = No exp(−αx) (94) The probability per unit length is just the reciprocal of the mean free path since: Probability of collision between x and x + dx = (Probability of reaching x with no collision) × (Probability of collision in dx) = exp(−αx) × α. The average distance between collisions can then be computed as the average over all possible distances times the probability of colliding at that distance. Z ∞ 1 ` = xα exp(−αx)dx = (95) 0 α 9.5 Gas Kinetics 17

which means that the probability of a mean free path of length x is exp(−x/`) probability distribution of collision distance (96) This means that distance between collisions is described by the Poisson distribution of classical statistics. NOTE: The Poisson distribution occurs in many situations where there is a small chance of an event happening at random. The frequency of radioactive decay is usually very small, for example, if there is one decay per second for a milligram of material, then the probability is O(10−19) that any one atom in the sample will decay in one second. If the average time between decay events is τ, then the probability of observing no events in time t is exp(−t/τ) (97) which is exactly the analog of the probability of having no collisions in distance x. The Pois- son distribution is speciﬁcally deﬁned as the probability of observing exactly n events (decays, collisions, etc.) within a time t. You can show (Bevington p. 39) that this is t n exp(−t/τ) P (n, t) = (98) τ n! This can in turn can be considered an approximation to the binomial distribution that governs the statistics of a Bernoulli trial, epitomized by the outcome of a coin toss or game of chance in which there is a chance p of succeeding and a chance q = 1 − p of failure in each independent coin toss or play of the game. As shown in elementary probability theory, considering all the possible ways or combinations of having m successes in n trials leads to the probability of this outcome being given by the binomial distribution n! P (m, n) = pmqn−m (99) (n − m)!m! As the probability p of individual events becomes small while the average number of events pn = t/τ is held constant, it can be shown that (Bevington, p. 36) the binomial distribution can be closely approximated by the Poisson. In this fashion, the statistics of molecular collisions can be viewed as a result of rare, chance encounters and the probability distribution of collisions can be discussed in the same fashion as the outcome of gambling by considering that the underlying processes, whatever they might be, result in a truly random outcome. It is not surprising that kinetic theory is closely related to probability since one of the key assumptions of gas kinetics is that a statistical description of the molecular motion is both possible and necessary. Kinetic theory starts from physical considerations and through a chain of very detailed and subtle arguments, arrives at the statistical distributions of molecular properties. On the other hand, for our purposes it perhaps easier to just accept that the behavior is statistical and adopt the simplest probability distribution for collisions (Poisson) that is consistent with the idea of rare events.

9.5.2 Distribution of Molecular Velocities

The key concept in gas kinetics is to consider the components of molecular velocities v = (vx, vy, vz) as random variables. At ordinary temperatures, gases have sufﬁciently high energy that quantum 18 9 IDEAL GAS

effects are unimportant for the translation motion. This means that we can consider the kinetic 1 2 energy of each molecule = 2 m|v| and the velocity components to be continuous random variables. This means that the appropriate description of molecular velocity is in terms of a probability distribution function P(v) where P(v)dv (100) is the probability of ﬁnding a molecule with velocity between v and v + dv. The average properties of the gas can be found by considering by integrating over all possible values of velocities and weighting by the probability of that velocity. This average value obtained in this fashion is denoted < f > and is computed as Z < f >= f(v0)P(v0)dv0 (101)

For example, the average speed Z < v >= v0P(v0)dv0 (102)

is zero if the system is at rest since it is equally likely that the molecules have velocities in all directions. The form of the velocity distribution function for a gas in equilibrium was deduced by Maxwell and Boltzmann. In a gas at rest, there is nothing to distinguish directions in space so that the distribution function is the product of three identical functions for each component of velocity

P(v) = Fv(vx)Fv(vy)Fv(vz) (103)

where Fv is 1/2 m 2 Fv(v) = exp(−mv /2kbT ) − ∞ < v < +∞ (104) 2πkbT NOTE: The distribution of velocities is identical to the normal distribution or Gaussian distribution that describes the outcome of measurements subject to repeated and uncorrelated random errors. The normal distribution function is deﬁned as the probability of ﬁnding an observation lying within X and X + dX as " # 1 1 X − µ2 F (X) = √ exp − (105) 2πσ2 2 σ

The distribution has two parameters, the mean µ and standard deviation σ.

Z p √ µ =< X >= XF (X) dX σ = < (x − µ)2 > = < X2 > − < X >2 (106)

For gas kinetics, if the mean velocity of the gas is zero, then µ = 0. The parameter σ is the width or spread of the distribution. The larger σ, the larger the range of values of X will be found in a set of observations. The fraction of observations that will fall within a given range are given by the integral of the distribution

Z x+ P (x− < X < x+) = F (X) dX (107) x− 9.5 Gas Kinetics 19

For example, one-half of the observations are expected to fall within ±0.6745σ of the mean value. Comparing the normal distribution and the Maxwell-Boltzmann result, we see that the spread in p velocities is proportional to the square-root of temperature, σ = = kbT/m. This provides a statistical interpretation of temperature as deﬁning the width of the distribution of velocities. The width is known as the root mean square or RMS speed of the molecules. NOTE: There is a very deep connection of the MB distribution to statistical reasoning. The Central Limit Theorem of classical statistics (Lingren et al., p. 124) states that: The sum of a large number of independent observations on a random variable is ap- proximately normally distributed. The beauty of this is that it applies to almost any1 sort of independent random variables! For example, the distribution of each individual observation could be uniformly distributed but when a large number of observations is summed, the distribution of the sums will be tend to be normally distributed. The connection to gas kinetics is that the current value of a velocity component v of a molecule can be considered to be the sum of the effects ∆v of the very large number of random collisions that have taken place. After n collisions, a molecular which initially has zero velocity in the x direction will have an x velocity equal to

vx(n) = ∆v1 + ∆v2 + ··· + ∆vn (108) The central limit theorem then immediately suggests that for these molecules the velocities should be normally distributed. It is plausible that this holds for all molecules since we expect that the initial velocity is a random variable that will also follow the same normal distribution since for a system in equilibrium, samples at different times should have the same statistical properties, i.e., an equilibrium system is in a statistically stationary state. This provides an alternative view of the MB distribution that is purely statistical in nature and relies on the only random nature of the collision process. The details of the collision or even the exact character of the randomness does not have to be known! It is not a proof but if you accept the mathematical origin of the statistical reasoning, then it provides a very simple explanation of why this distribution should come about.

Distribution of molecular speeds In many situations, we are not interested in the individual velocity distributions but rather the distribution of speeds q 2 2 2 c = vx + vy + vz (109) for example, the energy distribution depends on the distribution of speed. We can find the distribution of speed by transforming to polar coordinates in velocity space 2 dvxdvydvx = 4πc dc and Fc(c)dc = Fv(vx)Fv(vy)Fv(vz)dvxdvydvx (110) and solving for the distribution function 3/2 m 2 2 Fc(c) = exp(−mc /2kbT )4πc 0 ≤ c ≤ ∞ (111) 2πkbT 1The random variables must be drawn from a common distribution with finite mean and variance. The meaning of “large” depends on the details but a value of 20 is usually sufficient and in the context of gas kinetics, we always are considering a very large number of events. 20 9 IDEAL GAS

The mean speed c¯ is Z ∞ r 8kbT c¯ = cFc(c)dc = (112) 0 πm

The most probable speed, deﬁned by the maximum of Fc is r 2k T c = b (113) mp m The root mean square or rms velocity in one direction is r q k T v = < v2 > = b . (114) i,rms i m and the rms total velocity is r q 3k T v = < v2 + v2 + v2 > = b . (115) rms x y z m

9.5.3 Distribution of Molecular Energies For a isotropic situation, the average kinetic energy in each independent direction is equal 1 1 1 < mv2 >=< mv2 >=< mv2 > (116) 2 x 2 y 2 z where m is the molecular mass. The total kinetic energy of a molecule is therefore 1 1 = m(v2 + v2 + v2) = mc2 (117) 2 x y z 2 We can transform the speed distribution to an energy distribution using the chain rule √ dc F() = Fc(c) = 3/2 exp(−/kBT ) (118) d (kBT ) The total average kinetic energy of the molecules is nonzero and for a monatomic gas, is equal to the internal energy of the gas Z ∞ 1 2 U = N < m|v| >= N < >= F() d (119) 2 0 where N is the number of molecules. We can evaluate this using the distribution of molecular speeds or the distribution of energies to obtain 3 U = Nk T (120) 2 B For multi-atom molecules, the vibrational and rotational motion of the molecules about the center mass also contributes to the internal energy as discussed subsequently. For sufﬁciently high temperatures, dissociation of molecules in atoms, electronic excitation, and ionization also has to be considered when computing internal energy. 9.5 Gas Kinetics 21

9.5.4 Ideal Gas Law The average pressure the gas exerts on a surface can be determined by computing the impulse for elastic collisions on a rigid wall and averaging the results over the distribution of velocities. For a wall perpendicular to the x direction N 2 1 P = 2 m < v2 > or PV = N < m|v|2 > . (121) V x 3 2 For an ideal monatomic gas, the internal energy is therefore 3 U = PV. (122) 2 Combining this with the ideal gas law, we have

PV = NkBT (123) which is the ideal gas law that was deduced by through experiments by Boyle (1662)

PV = constant for ﬁxed T and N (124)

Charles (1787), Dalton (1801) and Gay-Lussac (1802)

V ∝ T for ﬁxed P and N (125) and Avogadro (1811) V ∝ N for ﬁxed T and P (126) In modern terms, ¯ ¯ PV = nRT R = NakB n = N/NA (127) 23 ¯ where NA = 6.023 ×10 is Avogadro’s number, R is the universal gas constant, and n is the amount of substance measured in moles.

9.5.5 Speciﬁc Heat Capacity and Partition of Energy The constant volume speciﬁc heat of a monatomic ideal gas is apparently dU 3 Cv = = NkB . (128) dT V 2

For one mole of substance, N = NA, and the molar heat capacity of a monatomic gas is

3 c = R˜ (129) v 2 and the mass speciﬁc heat capacity is

3 c = R. (130) v 2 22 9 IDEAL GAS

Compare this with the standard engineering expression for speciﬁc heat capacity

R c = (131) v γ − 1

and we see that for an monatomic gas the ratio of speciﬁc heats γ = 5/3. This can be generalized for molecular gases using the concept of degrees of freedom, f. The average energy per independent direction of motion in space is 1 1 < mv2 >= k T. (132) 2 i 2 B For all classical forms of motion (translation, vibration, rotation), the general rule of equipartition of energy) is that every degree of freedom of a molecule contributes 1/2kBT to the internal energy and 1/2kB to the heat capacity 1 1 U = Nf k TC = Nf k (133) 2 B v 2 B For real gases, the translation and rotational motion can be treated classically but all other motions, such as vibration and electronic excitation have to be treated with quantum mechanics. As a consequence of the quantization of energy levels, i, these motions will not share equally in the partition of energy unless the temperature is sufﬁciently high kBT ∆. At temperatures of interest for engineering, this means that the effective number of degrees of freedom depend on the temperature and the molecular structure, f = f(T ). Only at very high temperatures are all of the degrees of freedom are excited and the classical limit of complete equipartition is reached. lim f(T ) = fclassical (134) T →∞

For diatomic gases such at N2 and O2, fclassical = 7; 3 for center-of-mass motion; 2 for rotation (spinning about the axis of symmetry doesn’t count); 2 for vibration (1 for kinetic energy and 1 for potential energy of vibration). For nitrogen and oxygen at room temperature, the vibrational degrees of freedom are not excited and f = 5 and γ = 1.4, giving rise to the usual approximation that 5 c (air, 300K) ≈ R. (135) v 2 In general, the classical limit for a molecule containing m atoms is:

linear fclassical = 6m − 5 (136)

nonlinear fclassical = 6m − 6 . (137)

The effective value of the ratio of speciﬁc heats will be 2 + f γ = 1 ≤ γ ≤ 5/3 . (138) f

Equipartition for a gas that consists of a mixture of different molecules or atoms means that each type of molecule or atom shares equally in the energy for the fully classical modes of motion. 9.6 Mixtures 23

At equilibrium, there is a common temperature for all species so that each species follows the MB law with the same temperature. For example, this means for translational energy 1 1 3 < m c2 >=< mc2 >= k T, (139) 2 1 1 2 2 2 B so that p< c2 > rm 1 = 2 (140) p 2 m < c2 > 1 and q r 2 3kBT < ci > = . (141) mi which is just the total rms velocity deﬁned above for a single-component gas.

9.5.6 Molecular Flux The one-way molecular flux is the rate at which molecules cross an imaginary planar surface per unit area. This quantity is key to computing processes like evaporation or condensation or the reaction of molecules at a solid or liquid surface. It is also used in simplified derivations of the transport by gradients (diffusion) of mass, momentum, or energy. Consider a gas at rest and a plane surface normal to the x direction. According to Eq. 104, there will be on the average equal numbers of molecules crossing this surface in the +x and -x direction so that there is no average net flux. However, if we concentrate on the flux in one direction, say +x, then for molecules with speed v in the +x direction, the instantaneous flux will be J = [n]v (142) where [n] = N/V is the concentration of molecules, i.e., number per unit volume. Integrating the flux over all velocities v in the positive direction, we have Z ∞ [n]¯c J+ = [n] vFv(v)dv = (143) 0 4 We have written the answer in the conventional notation using the mean speed of all the molecules but it should be clear from the derivation that we are only considering molecules with a positive velocity component normal to the plane of interest. Since a gas in equilibrium with no external forces is isotropic, the one-way molecular flux across a plane is always given by Eq. 143 independent of the orientation of the plane.

9.6 Mixtures A mixture of ideal gases is also an ideal gas. The composition of a mixture can be speciﬁed by the amounts of each species i = 1, 2, ...K. These amounts can be speciﬁed as masses Mi of each species or moles ni of each species. The mole fraction Xi = ni/n where n is the total number of moles of gas K X n = ni . (144) i=1 24 9 IDEAL GAS

The mass fraction Yi = Mi/M where M is the total mass of gas

K X M = Mi . (145) i=1 The ideal gas law applies to mixtures and the total pressure is proportional to the total number of gas molecules, all other factors being equal.

PV = nRT˜ (146)

The partial pressure, Pi, of a species i is deﬁned to be RT˜ P = n (147) i i V or Pi = XiP. (148) The average molar mass of a mixture is

K M X W = = X W (149) n i i i=1 where Wi is the molar mass of species i. The gas constant for a mixture can be written as R˜ R = . (150) W The ideal gas law can then be written in terms of the average mass density. M ρ = P = ρRT (151) V Note that the average molar mass can also be written as

K 1 X Yi = . (152) W W i=1 i

Mole (Xi) and mass (Yi) fractions are related by W Xi = Yi . (153) Wi

9.6.1 Partial Molar Properties For the thermodynamic treatment of mixtures of nonideal gases and liquid or solid mixtures, we need to consider how the properties depending on the amount of substance. For any property B, deﬁne the partial molar property as

¯ ∂B bi = (154) ∂n i nk6=i 9.6 Mixtures 25

The signiﬁcance of this to computing mixture properties is due to Euler’s theorem for homogeneous functions, which states that for a function F (n1, n2, . . . , nk) which is homogeneous

F (αn1, αn2, . . . , αnk) = αF (n1, n2, . . . , nk) (155)

then we can always write F as

k X ∂F F (n1, n2, . . . , nk) = ni (156) ∂ni i=1 nk6=i Proof: Differentiate (155) with respect to α and set α = 1 to obtain (156). Thermodynamic potentials are all extensive properties and therefore satisfy the conditions of (155) and can be written in the form of (156. For example, we can write the enthalpy of a mixture of k species as k X ¯ H = nihi (157) i=1 ¯ The distinction between partial molar properties such as hi and ideal gas mixture components hi ¯ ¯ ¯ is that in general hi = hi(P, T, n1, n2,...) and for the ideal gas hi = hi(T,Pi) only. Obviously ¯ including the pressure dependence is trivial for ideal gas enthalpy since hi = hi(T ) only but for other properties like entropy, or Gibbs energy, this is crucial. The internal energy of an ideal gas mixture is simply a weighted average of the individual species internal energies K X U = miui(T ) . (158) i=1 For a general mixture, this can be written in terms of the molar amounts by using the partial molar internal energy of each species u¯i.

K X U = nu¯ = niu¯i (159) i=1

The SI units of u¯i are J/mol. The enthalpy of a gas mixture can be expressed in the same fashion

K ¯ X H = nh = mihi(T ) (160) i=1 and the general expression is K X ¯ H = nihi (161) i=1 ¯ where hi is the partial molar enthalpy of species i. The speciﬁc internal energy of ideal gases can be written in terms of the mass fraction

K X u = Yiui(T ) (162) i=1 26 9 IDEAL GAS

as can the speciﬁc enthalpy K X h = Yihi(T ) . (163) i=1 The entropy of a mixture is calculated by

K X S = nis¯i (164) i=1

where s¯i is the partial molar entropy of species i. For an ideal gas mixture, this is equivalent to summing the entropies of each component evaluated at the partial pressure of that component

K X S = misi(T,Pi) (165) i=1 or the speciﬁc entropy is K X s = Yisi(T,Pi) (166) i=1 where the contribution of each species is

◦ ◦ si = si (T ) − Ri ln Pi/P . (167)

For a perfect gas, cp = constant, the component entropy can be estimated as

◦ ◦ si = cp ln T/T − Ri ln Pi/P . (168)

The partial molar entropies of the components are

◦ ˜ ◦ s¯i =s ¯i (T ) − R ln Pi/P (169)

and the total entropy of a mixture can be expressed as

K X S = ns¯ = nis¯i(T,Pi) . (170) i=1 The entropy of mixing is the increase in entropy due to creating a mixture by an adiabatic constant volume process K ˜ X ∆¯smix = −R Xi ln Xi > 0 . (171) i=1 27 10 Relationships

10.1 Thermodynamic potentials and fundamental relations

energy u(s, v) du = T ds − P dv (172) enthalpy h(s, P ) = u + P v dh = T ds + v dP (173) Helmholtz a(T, v) = u − T s da = −s dT − P dv (174) Gibbs g(T,P ) = u − T s + P v dg = −s dT + v dP (175)

10.2 Maxwell relations

∂T ∂P = − (176) ∂v s ∂s v ∂T ∂v = (177) ∂P s ∂s P ∂s ∂P = (178) ∂v T ∂T v ∂s ∂v = − (179) ∂P T ∂T P

10.3 Calculus identities ∂F ∂F F (x, y, . . . ) dF = dx + dy + ... (180) ∂x y,z,... ∂y x,z,...

∂f ∂x ∂y = − x (181) ∂y ∂f f ∂x y ∂x 1 = (182) ∂f ∂f y ∂x y 28 10 RELATIONSHIPS

10.4 Various deﬁned quantities

∂u ∂s specific heat at constant volume cv ≡ = T (183) ∂T v ∂T v ∂h ∂s specific heat at constant pressure cp ≡ = T (184) ∂T P ∂T P c ratio of specific heats γ ≡ p (185) cv s ∂P sound speed c ≡ (186) ∂ρ s 1 ∂v coefficient of thermal expansion α ≡ (187) v ∂T P 1 ∂v isothermal compressibility βT ≡ − (188) v ∂P T 1 ∂v 1 isentropic compressibility βs ≡ − = 2 (189) v ∂P s ρc

10.5 Speciﬁc heat relationships ∂P ∂P βT = γβs or = γ (190) ∂v s ∂v T

∂P ∂v 2 cp − cv = −T (191) ∂v T ∂T P

2 ∂Cv ∂ P = T 2 (192) ∂v T ∂T v

2 ∂Cp ∂ v = −T 2 (193) ∂P T ∂T P

10.6 Gruneisen¨ Coefﬁcient

vα G ≡ (194) cvβT ∂P = v (195) ∂u v vα = (196) cpβs v ∂T = − (197) T ∂v s 10.7 Thermal Pressure Coefﬁcient 29

10.7 Thermal Pressure Coefﬁcient ∂P α = (198) ∂T v βT

10.8 Enthalpy, Energy and Entropy Energy u(T, v) ∂P du = CvdT + T − P dv (199) ∂T v Enthalpy h(T,P ) ∂v dh = CpdT + v − T dP (200) ∂T P Entropy s(T,P ) C ∂v ds = p dT − dP (201) T ∂T P Entropy s(T, v) C ∂P ds = v dT + dv (202) T ∂T v 30 11 EQUATION OF STATE 11 Equation of State

Strictly speaking, a complete equation of state of a substance is known only when one of the Thermodynamic potential functions is given in terms of the canonical variables. For example, the Helmholtz energy A(T,V ) is such a function. The entropy S and pressure P (as well as any other properties) can be determined with the help of the fundamental relation of thermodynamics

∂A ∂A dA = SdT − P dV S ≡ P ≡ − (203) ∂T V ∂V T

11.1 Constructing Equations of State Frequently, partial or incomplete information is given about the equation of state of a substance and thermodynamic relationships have to be used to compute potential function. A typical situation is to be given P (v, T ) and CV (v, T ) information from ﬁts to experimental data and to be asked to compute internal energy, enthalpy, and entropy. Given cv(v, T ) and P (v, T ), integrate

∂P du = cv dT + T − P dv (204) ∂T v c ∂P ds = v dT + dv (205) T ∂T v along two paths: I: variable T , ﬁxed ρ and II: variable ρ, ﬁxed T . Energy: ! Z T Z ρ ∂P dρ u = u◦ + cv(T, ρ◦) dT + P − T 2 (206) T◦ ρ◦ ∂T ρ ρ | {z } | {z } I II

Ideal gas limit ρ◦ → 0,

ig lim cv(T, ρ◦) = cv (T ) (207) ρ◦→0 The ideal gas limit of I is the ideal gas internal energy

Z T ig ig u (T ) = cv (T ) dT (208) T◦ Ideal gas limit of II is the residual function

Z ρ ! r ∂P dρ u (ρ, T ) = P − T 2 (209) 0 ∂T ρ ρ and the complete expression for internal energy is

ig r u(ρ, T ) = u◦ + u (T ) + u (ρ, T ) (210) 11.2 Critical Points 31

Entropy: Z T Z ρ ! cv(T, ρ◦) ∂P dρ s = s◦ + dT + − 2 (211) T◦ T ρ◦ ∂T ρ ρ | {z } | {z } I II

The ideal gas limit ρ◦ → 0 has to be carried out slightly differently since the ideal gas entropy, unlike the internal energy, is a function of density and is singular at ρ = 0. Deﬁne

Z T cig(T ) Z ρ dρ sig = v dT − R (212) T◦ T ρ◦ ρ where the second integral on the RHS is R ln ρ◦/ρ. Then compute the residual function by subtracting the singular part before carrying out the integration ! Z ρ 1 ∂P dρ sr(ρ, T ) = R − (213) 0 ρ ∂T ρ ρ and the complete expression for entropy is

ig r s(ρ, T ) = s◦ + s (ρ, T ) + s (ρ, T ) (214)

11.2 Critical Points The critical point or CP of a pure substance is defined as the limiting thermodynamic state for the coexistence of liquid and vapor. The pressure, temperature, and specific volume have unique values (Pc,Tc,Vc) characteristic of the substance at the critical point. For temperatures T > Tc, the distinction between liquid and vapor ceases to exist and there is only a fluid or dense fluid state. Given a P (V,T ) equation of state, the critical point is defined by

∂P ∂2P = 0 2 = 0 (215) ∂V T ∂V T

11.2.1 Law of corresponding states

For a series of compounds with similar molecular structure, the thermodynamic properties can be expressed as functions of the reduced thermodynamic state. The reduced variables are deﬁned as

P T V Pr = Tr = Vc = (216) Pc Tc Vc

The Law of Corresponding States for the P -V -T relationship can be expressed as

Pr = f(Tr,Vr) (217) where f is a universal function for this series of compounds. 32 11 EQUATION OF STATE

11.3 Compressibility Factor The compressibility factor is deﬁned as P v Z = (218) RT For an ideal gas state, Z = 1. For substances in which the intermolecular forces are primarily attractive Z < 1; for substances in which the intermolecular forces are predominantly repulsive Z > 1.

11.4 van der Waals Equation The classical example of a simple analytic equation of state that has a critical point and approxi- mates real liquid-vapor behavior is the van der Waals equation RT a P = − (219) v − b v2 where a and b are substance-dependent constants. The values of a and b can be related to the critical point state. 2 2 27 R Tc RTc vc = 3b a = 2 b = (220) 64 Pc 8Pc

The compressibility factor at the critical point Zc = 3/8.

11.5 Throttling and the Joule-Thompson Coefﬁcient The change in temperature during a throttling process, i.e., an isenthalpic decrease in pressure, depends on the sign of the Joule-Thompson coefﬁcient

∂v ∂T T ∂T P − v µJ = = (221) ∂P h CP

For cooling to take place during throttling, µJ ≥ 0.

11.6 Virial Equation The virial equation of state is defined by power series in negative powers of volume or sometimes positive powers of pressure. In terms of specific volume, this is P v B(T ) C(T ) = 1 + + + ... (222) RT v v2 where B is the second virial coefficient. The temperature at which B = 0 is known as the Boyle temperature at which the compressibility correction vanishes as the specific volume tends to infinity, i.e., the ideal gas limit ∂Z lim = 0 (223) P →0 ∂P TBoyle 11.7 Ideal Solid 33

11.7 Ideal Solid

The ideal solid can be described in terms of the isothermal compressibility βT and the coefﬁcient of thermal expansion α. Considering the speciﬁc volume v(T,P ), small changes in volume can be related to corresponding changes in temperature and pressure ∆v = α∆T − β ∆P (224) v T

If we approximate βT and α as constants, then we can integrate to obtain a relationship between states 1 and 2 v2 = v1 exp (α(T2 − T1) − βT (P2 − P1)) (225) The internal energy can be determined as a function of temperature and pressure by expanding u(T,P ) and using the fundamental relations to obtain

du = (cP − P vα) dT + (P vβT − T vα) dP (226) and integrating, holding the pressure constant in the ﬁrst term and temperature in the second. Somewhat simpler is to consider u(v, T ) and to integrate

α du = cvdT + T − P dv (227) βT In order to carry out these integrations, we have to make some assumptions about the heat capacities, typically that either cv or cP is dependent on temperature only. Note that they cannot both satisfy this since, in general, α2vT cv = cp − (228) βT The entropy change can be evaluated by considering s(T,P ) to obtain

ds = cpdT − αvdP (229) 34 12 EQUILIBRIUM LIQUID-VAPOR MIXTURES 12 Equilibrium Liquid-Vapor Mixtures

A pure substance can exist in different thermodynamic phases or states such as liquid, solid, and vapor or different crystallographic conﬁgurations (bcc or fcc) at different thermodynamic conditions. For a substance that is in an equilibrium state, the phase is a unique condition exception at phase boundaries where adjacent phases coexist. At a phase boundary, the pressure and temperature are uniquely related

Psat or Pσ = Pσ(T ) (230) The pressure or temperature alone uniquely determines the thermodynamic state of an equilibrium two-phase mixture but the proportion of the two phases can be arbitrary. The pressure-temperature relation is known as the saturation or vapor pressure relationship for liquid-vapor equilibrium.

Psat = Pσ vapor-liquid equilibrium (231)

12.1 Lever Rule The quality is the mass fraction of the vapor phase in the liquid-vapor equilibrium region M x = g (232) Mf + Mg The quality can be used to determine the equilibrium mixture properties

v = xvg + (1 − x)vf (233)

u = xug + (1 − x)uf (234)

h = xhg + (1 − x)hf (235)

s = xsg + (1 − x)sf (236)

where ( )f are the ﬂuid (liquid) properties on the phase boundary and ( )g are the gas (vapor) properties on the phase boundary. The enthalpy change across the phase boundary is

∆hfg = hg − hf (237) and is known as the enthalpy of vaporization or latent heat for a vapor-liquid system.

12.2 Humidity A mixture of water vapor and air is characterized by the fraction of water vapor. This is measured in several ways. The relative humidity φ is the ratio of the partial pressure of water vapor Pv to the saturation pressure of water Psat at the ambient temperature T P φ = v (238) Psat(T ) Another measure is the humidity ratio ω deﬁned as m ω = v (239) ma 12.2 Humidity 35

where ma is the mass of air and mv is the mass of water vapor. The two indices are related by 1 P φ = ω sat (240) 0.622 Pa where Pa is the partial pressure of air. The dew point is the temperature at which the current vapor pressure is equal to the saturation pressure of water. This is temperature at which the onset of condensation is possible Pv = Psat(Tdew) (241) The adiabatic saturation temperature is equal to the exit temperature of a saturated air-water vapor mixture that has passed through an adiabatic humidiﬁer operating in steady-state, steady- ﬂow conditions. The dry bulb temperature is simply the ambient temperature. The wet bulb temperature is measured by a psychrometer and idealized as the adiabatic saturation temperature. 36 13 PHASE EQUILIBRIUM 13 Phase Equilibrium

Equilibrium between physical phases can be considered in the same fashion as equilibrium between chemical species. For example, liquid-vapor equilibrium follows from the simple constraint

M(f) ←−−→ M(g) for any pure substance M with a liquid phase f and vapor phase g. The minimization of Gibbs energy requires that

µf = µg or gg = gf (242) at equilibrium. This is true not only of liquid-vapor phase changes but also of equilibrium between any pair of coexisting phases or species.

13.1 Clapeyron Equation For a pure substance, the fundamental relationship for the Gibbs energy is

dg = −sdT + vdP (243)

Along a phase coexistence curve, the Gibbs energies must always be equal so that

dgf = dgg (244)

Substituting from the fundamental relationship, we have

(sg − sf ) dT = (vg − vf ) dP (245)

For a phase change at constant temperature and pressure, we have

∆hfg = hg − hf = T ∆sfg = T (sg − sf ) (246)

Substituting, we ﬁnd the Clapeyron equation for the slope of the phase boundary in the T − P plane. ∂P h − h ∆h = g f = fg (247) ∂T sat T (vg − vf ) T ∆sfg

13.2 Maxwell’s Construction

The relationship g1 = g2 can be used to solve for the phase boundary between two coexisting phases. Consider integrating along an isotherm from state 1 to state 2 at constant temperature

Z 2 Z P2 0 = g2 − g1 = dg = v(T,P )dP (248) 1 P1 This is known as Maxwell’s construction and can be used to implicitly determine the location of the phase boundary P (T ) given an integrable expression for v(T,P ). 13.3 Multiple Phases 37

13.3 Multiple Phases For a pure substance, more than one phase can coexist at a given time. In order for all of these phases to be at equilibrium at a common temperature and pressure, the chemical potentials of each phase must all be equal to a common value. If there are P phases, then there will be P -1 equilibrium conditions which are equivalent to

µ1 = µ2 (249)

µ1 = µ3 (250) ... (251)

µ1 = µP−1 (252)

For a pure substance, the chemical potentials are a function of only two state variables, for example, temperature and pressure, µ(T,P ). In general, it is not possible to have more constraints than variables so that there are only three cases: P = 1, one phase with arbitrary T and P ; P = 2, two phases can coexist on a curve P (T ); P = 3, a triple-point, three phases can coexist at a point. When the phases are mixtures with multiple chemical components, then more complex situations can exist since the chemical potentials are a function of the composition of that phase. Consider C components, i.e., chemically distinct species in the system, then there will be C -1 independent composition variables y1, y2, . . . , yC−1 (253) for each phase. This is one less than the number of components since the yi are not all independent - the sum is equal to one. The total number of thermodynamic state variables in an equilibrium mixture of P phases each with C components will be

2 + P (C − 1) (254)

The total number of constraints will be the total number of chemical potential equality relations, of which there are P - 1 for each of the C species for a total of

C (P − 1) (255)

The total number of degrees of freedom or possible variance V is the difference between the number of variables and the number of constraints, which is

V = C + 2 − P (256)

This is known as Gibbs phase rule. 38 14 CHEMICAL TRANSFORMATIONS 14 Chemical Transformations

The basis for thermodynamic computations with chemically reacting mixtures is a balanced reaction equation

Reactants −→ Products

Mass must be conserved and this implies that the number of atoms of each kind must be equal in the reactants and the products.

14.1 Heat of Reaction If the reaction occurs at constant pressure, the energy balance can be written in terms of the heat of reaction X ¯ X ¯ Q = HP (TP ) − HR(TR) or Q = nihi − nkhk (257) P R ¯ Where hi is the partial molar enthalpy of species i and ni is the number of moles of species i. A quantity Q of energy is transferred into the reactor as heat during the reaction process. If the reaction occurs at constant temperature, Q is known as the heat of reaction.

∆H = HP (T ) − HR(T ) (258)

This is the amount of energy that must be added in the form of heat to keep the reactor at constant temperature. We distinguish three cases:

∆H > 0 endothermic (259) ∆H = 0 neutral (260) ∆H < 0 exothermic (261)

14.1.1 Enthalpy The partial molar enthalpy for an individual species can be written

◦ ◦ h = hf + h(T ) − h(T ) (262)

◦ ◦ ◦ where hf is the heat of formation and T is standard temperature 25 C or 298.15 K. The variation with temperature is determined by integrating the speciﬁc heat

Z T ◦ 0 0 h(T ) − h(T ) = Cp(T )dT (263) T ◦

This is straightforward for ideal gases since there is no pressure dependence but for non-ideal gases or condensed phases, the variation of enthalpy (and the other energy potentials) with pressure must be considered as discussed in Section 11. 14.2 Combustion Reactions 39

14.1.2 Heat of formation

The heat of formation is deﬁned to be the heat of reaction for

Stable Elements −→ 1 mol of species of interest

at standard conditions ◦ ◦ ◦ hf = ∆HT = T P = P (264)

14.2 Combustion Reactions In combustion, the overall reaction can be written

Fuel + Oxidizer −→ Products

The most common fuels are hydrocarbons CmHn and the common oxidizer is air, usually approximated as consisting of 0.21 O2 and 0.79 N2, referred to as combustion air. The products consist of a mixture of major species such as H2O, CO2,H2, CO, and N2, and also minor species such as H, O, OH, NO, CH, N, etc. Frequently, it is sufficient to consider only major species in the energy balance at low temperatures for lean mixtures (defined subsequently). Mixtures are classified in terms of the amount of oxidizer relative to the fuel. A stoichiometric mixture has the proper proportions of fuel and oxidizer so that when dissociation is neglected, the hydrogen is completely oxidized to H2O and the carbon is oxidized to CO2. The balanced stoichiometric reaction of a hydrocarbon fuel with air can be written as

m m m CnHm + (n + 4 )(O2 + 3.76N2) −→ nCO2 + 2 H2O + 3.76(n + 4 )N2

A lean mixture is one with excess air so that O2 also appears in the products. A balanced lean reaction of a hydrocarbon fuel with air can be written

φ CnHm + (n + m/4)(O2 + 3.76N2) −→ φnCO2 + φ m/2H2O + 3.76(n + m/4)N2+ (1-φ)(n + m/4)O2

where φ is the equivalence ratio

n /n φ = fuel oxidizer (265) (nfuel/noxidizer)stoichiometric

where nfuel is the amount of fuel and noxidizer is the number of moles of oxidizer (both in moles or number of molecules). For lean combustion, φ < 1; for stoichiometric combustion, φ = 1.

A rich mixture (φ > 1) is one with excess fuel so that the products contain a mixture of CO2, H2O, H2, and CO. The combustion of the products is indeterminant without solving for chemical equilibrium composition that corresponds to the water-gas shift reaction

−→ H2 + CO2 ←− H2O + CO 40 14 CHEMICAL TRANSFORMATIONS

Mixture compositions are also speciﬁed in terms of the fuel mole or mass fractions. The ﬂamma- bility limits are usually given in terms of fuel mole fractions

nfuel Xfuel = (266) nfuel + noxidizer

A mixture is considered flammable if a propagating flame is created when a sufficiently strong ignition source is used. Experimentally, mixtures are observed to have both a lower flammability limit or LFL and an upper flammability limit UFL. The flammable region is defined as

XLF L ≤ Xfuel ≤ XUFL (267)

Another way to express mixture composition is in terms of fuel-air mass ratio deﬁned as n W f = fuel fuel (268) noxidizerWoxidizer

14.3 Heat of Combustion The heat of combustion is deﬁned to be the heat of reaction for complete combustion of one mole of fuel at standard conditions.

fuel + oxidizer −→ major products (T = T ◦)

It is more convenient to deﬁne the heat of combustion as a positive quantity

X ◦ X ◦ Qc = −∆H = nihi(T ) − nkhk(T ) (269) R P The heating value q of the fuel is deﬁned on a mass basis

Qc qc = (270) Wf

The lower heat value or LHV is deﬁned by using gaseous H2O in the products. The higher heat value or HHV is deﬁned by using liquid H2O in the products.

14.4 Flame Temperature A low-speed ﬂame can be approximated as an adiabatic, constant-pressure reactor. The ﬂame temperature is determined by carrying out an energy balance on the reactor. This is equivalent to considering the reaction

Reactants at TR −→ Products at TP at constant pressure and solving the corresponding adiabatic energy balance X X nihi(TR) = nkhk(TP ) (271) R P 14.5 Explosion Pressure and Temperature 41

This is a nonlinear equation for TP that can be solved by iteration. An approximate direct solution ◦ can be found for a special case TR = T and approximating the enthalpy of each species as

◦ ◦ hi(T ) ≈ hf,i+ < cP,i > (T − T ) (272)

◦ where < cP,i > is the average heat capacity, evaluated at a temperature intermediate to T and TP . The ﬂame temperature can be estimated as

◦ Qc X TP ≈ T + where < cP >= ni < cP,i > (273) < cP > P

14.5 Explosion Pressure and Temperature An explosion inside a vessel can be approximated as an adiabatic, constant-volume reactor for the purposes of estimating the maximum peak temperature and pressure. This is equivalent to considering the reaction

Reactants at TR −→ Products at TP at constant volume and solving the corresponding adiabatic energy equation X X niui(TR) = nkuk(TP ) (274) R P The explosion pressure is pressure in the products

nP TP PP = PR (275) nRTR ◦ with the same volume as the reactants. For the special case of TR = T , the explosion temperature can be approximated as

˜ ◦ ◦ Qc + ∆nRT X TP ≈ T + where < cV >= ni < cV,i > (276) < cV > P where X X ∆n = nk − ni (277) P R

15 Chemical Equilibrium

The basis for chemical equilibrium computations is the Second Law of thermodynamics. For a system at constant temperature and pressure, the Second Law implies that the Gibbs Energy G = H - TS is a minimum dG = 0 at constant T and P (278) The Gibbs energy is deﬁned to be

K X G = G(T, P, ni) = nig¯i(T,P,Xi) (279) i=1 42 15 CHEMICAL EQUILIBRIUM

where the partial molar Gibbs energy is deﬁned as

∂G g¯ = (280) i n i T,P,nk6=i P where Xi = ni/n, n = ni is the mole fraction of species i. This quantity is also known as the chemical potential µi ≡ g¯ − i. The chemical potential for an ideal gas can be written ¯ g¯ig = µig = hig − T sig (281)

Substances in equilibrium are related by an equilibrium constraint such as −→ H2 + CO ←− H2 + CO2 that can be written symbolically as K X νiMi = 0 (282) i=1 where the stoichiometric coefﬁcients are positive for species on the left-hand or “product” side of the constraint and negative for species on the right-hand or “reactant” side of the constraint. This implies that changes dni in the amounts of each species are related by dn dn dn 1 = 2 = ··· = k = dξ (283) ν1 ν2 νk where ξ is referred to as the progress variable for this reaction. The minimization of Gibbs energy is equivalent to K ! X dG = νiµi dξ = 0 (284) i=1 In order for this to be true for all perturbations dξ, then

K K X X νiµi = giµi = 0 (285) i=1 i=1

In the case of an ideal gas, the equilibrium condition (285) can be further simpliﬁed by taking advantage of the special form of the chemical potential

P µ (T,P ) = µ◦(T ) + RT˜ ln i (286) i i i P ◦ where the temperature-dependent portion of the chemical potential is deﬁned as

◦ ˜ ◦ µi (T ) = hi(T ) − RT s (T ) (287)

The reference pressure is 1 bar (100 kPa) for the tables in Appendix A.12 of the text. In what follows, we will use a unit of bar for the pressure and drop the P ◦ = 1 from the formulas. 15.1 Le Chatelier’s Rule 43

The equilibrium condition can be rewritten as

K ν1 ν2 νk Y νi P1 · P2 · ...Pk = Pi = KP (T ) (288) i=1

where the equilibrium constant is deﬁned to be

∆G◦(T ) KP = exp − (289) RT˜

where K ◦ X ◦ ∆G (T ) = νiµi (T ) (290) i=1

15.1 Le Chatelier’s Rule

The variation of composition with changes in state can be summarized by Le Chatelier’s Rule:

A system responds to a perturbation with changes in state that tend to counteract or minimize the effect of the perturbation.

15.1.1 Pressure

The effect of pressure changes on composition depends on the net change in moles ∆ν in the equilibrium constraint. K X ∆ν = νi (291) i=1 The effect of pressure can be seen more clearly by rewriting the equilibrium relation using mole fractions Xi and the deﬁnition of partial pressure Pi = XiP

K Y KP (T ) Xνi = (292) i P ∆ν i=1

There are three cases depending on the sign of ∆ν. The effect of increasing pressure can be summarized as follows

∆ν P ↑

< 0 shift to “products” = 0 no shift > 0 shift to “reactants” 44 15 CHEMICAL EQUILIBRIUM

15.1.2 Temperature The effect of temperature changes on composition depends on the heat of reaction that corresponds to the equilibrium constraint. K X ∆H(T ) = νihi(T ) (293) i=1 The rate of change of the equilibrium constant with temperature can be computed to be

dKP ∆H = KP (294) dT RT˜ 2 which is known as the van’t Hoff equation. The effect of increasing temperature can be summarized as follows

∆H type T ↑

< 0 exothermic shift to “reactants” = 0 neutral no shift > 0 endothermic shift to “products” 45 A Textbooks and references

If you want to teach yourself thermodynamics, a good place to start is with: Schaum’s Outline of Thermodynamics With Chemical Applications by Hendrick C. Van Ness and Michael M. Abbott, 2nd Edition, McGraw-Hill, 1990. Don’t be put off by the title, low price, and cheap binding. It is more than just an ”outline” at 384 pages and covers all the topics in more than sufﬁcient depth for our class. The presentation and problems in this book are superior to most hardbound texts selling for almost ten times this amount. Despite the inclusion of ”Chemical Applications” in the title, the majority of the text is very general in approach and most examples are appropriate for all engineers and applied scientists. If you can’t stand the book, you only spent $18 as opposed to $150 for the slicked up hardbacks in use by most engineering schools.

A.1 Introductory There are numerous introductory texts on thermodynamics, with many different points of view, writing style and levels of presentation. Instead of starting with a textbook, I suggest the the following brief introductions because they are extremely well written and the authors have thought deeply about the subject. They also have the advantage of being relatively short and inexpen- sive paperbacks. There is a lot of overlap in topics between these books but each author brings something a little different to the subject and has a different style. The first two books explain the fundamentals of thermodynamics in a very simple fashion with a minimum of mathematics. The emphasis is on physical reasoning and developing your understanding of the concepts. 1. Understanding Thermodynamics by H. C. Van Ness, Dover, 1983. 103 pages. 2. Engines, Energy and Entropy: A Thermodynamics Primer by John B. Fenn, W.H. Freeman & Company, 1982. 293 pages. Reprinted by Global View Publishing. Precisely defining and measure temperature is an essential part of thermodynamics and thermal engineering. 3. Temperatures: Very Low and Very High by Mark Waldo Zemansky, Dover, 1981. 144 pages. There is a very concise but quite clear classic by Fermi that is representative of the classical pedagogical approach to thermodynamics. 4. Thermodynamics by E. Fermi, Dover, 1937. 160 pages. If you are put off by the old-fashioned feeling of Fermi, then try the following modern and elegantly produced text. 5. Modern Thermodynamics by D. Kondepudi and I. Prigogine, Wiley, 1998. 486 pages This text is not written specifically for engineers but the first two parts contain an excellent introduction to classical thermodynamics with wonderful historical comments. The last part of the text is an introduction to Prigogine’s ideas about fluctuations, dissipative structures and self-organization. Leave that to after you have built sturdy fundamentals. There is a surprising amount of technical detail in the entries on the history of science, matter, and thermodynamics in the 46 A TEXTBOOKS AND REFERENCES

6. Encyclopaedia Brittanica (I read the 15th edition in print - you can access the most recent version on line from a Caltech IP) A thorough discussion of the various aspects of entropy is given in:

7. J.S. Dugdale Entropy and its Physical Meaning, Taylor and Francis, 1996

A.2 Many others There are probably a dozen great books on thermodynamics and hundreds more that were de- servedly pulped shortly after being printed. Some of the authors that I like include Adkins, Guggenheim, Zemansky, Denbigh, Abbott.

A.3 Biography and History Reading history and scientiﬁc biography is also a pleasant way to learn while gaining an apprecia- tion for the historical context. Popular (and in-print) biographies of historical ﬁgures include:

1. Degrees Kelvin: A Tale of Genius, Invention, and Tragedy by David Lindley

2. Boltzmanns Atom: The Great Debate That Launched A Revolution In Physics by David Lindley

3. The Man Who Changed Everything : The Life of James Clerk Maxwell by Basil Mahon.

4. Never at Rest : A Biography of Isaac Newton by Richard Westfall. More technical historical material can be found in the following:

5. Statistical Physics and the Atomic Theory of Matter by Stephen G. Brush

6. The kind of motion we call heat : a history of the kinetic theory of gases in the 19th century by Stephen G. Brush

7. The kinetic theory of gases : an anthology of classic papers with historical commentary by Stephen G Brush ; edited by Nancy S Hall

8. From Watt to Clausius;: The rise of thermodynamics in the early industrial age by D. S. L Cardwell Short (between a few paragraphs and several pages) but very informative, and often enter- taining (see Onsager) entries on both famous and not-so-famous can be found in

9. The Biographical Dictionary of Scientists Vols 1 and 2. Oxford University Press. and the history of science from 1550 to present is covered with a broad brush in

10. The Oxford Companion to the History of Modern Science by John L. Heilbron 47 B Famous Numbers

Fundamental Physical Constants

8 co speed of light in a vacuum 2.998×10 m/s −12 2 o permittivity of the vacuum 8.854×10 C /kg-m −7 µo permeability of the vacuum 4π×10 H/m h Planck constant 6.626×10−34 J-s k Boltzmann constant 1.381×10−23 J/K 23 No Avogadro number 6.022×10 molecules/mol e charge on electron 1.602×10−19 C amu atomic mass unit 1.661×10−27 kg −31 me electron mass 9.109×10 kg −27 mp proton mass 1.673×10 kg G universal gravitational constant 6.673×10−11 m3/kg-s2 σ Stefan-Boltzmann constant 5.670×10−8 W/m2K4

Astronautics

2 go gravitational acceleration at earth’s surface 9.807 m/s RE radius of earth 6378 km 24 ME mass of earth 5.976×10 kg 30 MS mass of sun 1.99×10 kg 8 au mean earth-sun distance 1.496×10 km mass of moon 7.349×1022 kg mean earth-moon distance 3.844×105 km

Ideal Gas Stuff

R˜ Universal gas constant 8314.5 J/kmol-K 8.3145 J/mol-K 82.06 cm3-atm/mol-K 1.9872 cal/mol-K mechanical equivalent of heat 4.186 J/cal volume of 1 kmol at 273.15 K and 1 atm 22.41 m3 number of molecules at 298.15 K and 1 atm 2.46×1025 m3 collision frequency at 273.15 K and 1 atm 4.3×109 s−1 mean free path in N2 at 273.15 K and 1 atm 74 nm

Consistent with the 1998 CODATA adjustment of the funadamental physical constants 48 B FAMOUS NUMBERS

Our Atmosphere

composition (mol fractions) 0.7809 N2 0.2095 O2 0.0093 Ar 0.0003 CO2

Sea level

P pressure 1.01325×105 Pa ρ density 1.225 kg/m3 T temperature 288.15 K c sound speed 340.29 m/s R gas constant 287.05 m2/s2-K W molar mass 28.96 kg/kmol µ viscosity (absolute) 1.79×10−5 kg/m-s k thermal conductivity 2.54×10−3 W/m-K cp heat capacity 1.0 kJ/kg-K

30 kft

P pressure 3.014×104 Pa ρ density 0.458 kg/m3 T temperature 228.7 K c sound speed 303.2 m/s

Based on the U.S. Standard Atmosphere, 1976 NOAA-S/T 76-1562, 1976.

Unit Conversions

1 m ≡ 3.28 ft 0.3048 ft ≡ 1 m 1 lb (force) ≡ 4.452 N 1 lb (mass) ≡ 0.454 kg 1 btu ≡ 1055 J 1 hp ≡ 745.7 W 1 hp ≡ 550 ft-lbf /s 1 mile (land) ≡ 1.609 km 1 mph ≡ 0.447 m/s 1 mile (nautical) ≡ 1.852 km 49 C Fundamental Dimensions L length meter (m) M mass kilogram (kg) T time second (s) θ temperature Kelvin (K) I current Ampere (A)

Some derived dimensional units

force Newton (N) MLT −2 pressure Pascal (Pa) ML−1T −2 bar = 105 Pa energy Joule (J) ML2T −2 frequency Hertz (Hz) T −1 power Watt (W) ML2T −3

Pi Theorem Given n dimensional variables X1, X2, ..., Xn, and f independent fundamental dimensions (at most 5) involved in the problem:

1. The number of dimensionally independent variables r is

r ≤ f

2. The number p = n - r of dimensionless variables Πi

Xi Πi = α1 α2 αr X1 X2 ··· Xr that can be formed is p ≥ n − f 50 D IDEAL GAS PROPERTIES UP TO 4000K D Ideal Gas Properties up to 4000K

The data given in the attached tables are derived from the data ﬁts of NASA which are closely related to the JANNAF compilation (Chase et al. 1985) of thermodynamic properties for ideal gases. These data are almost universally used by combustion scientists and engineers to compute energy balances and equilibrium compositions in combustion systems. The data are given in molar units. Extensive information on the computation of thermodynamic properties for ideal gases given by Gurvich et al. 1989. The speciﬁc heats are computed using basic ideas of statistical mechanics and partition functions that are as accurate as possible (Gurvich et al. 1989, McBride and Gordon 1992). The properties of the elemental compounds are documented further in McBride et al. (1993) for high temperatures and for a very large number of at lower temperatures (less than 6000 K) in Chase et al. (1985).

specific heat CP This is the specific heat for each species defined per mol of substance.

Cp(T ) = (∂H/∂T )P (J/mol-K)

o standard heat of formation ∆f H This is the heat of reaction for the formation of one mole of the species of interest from the elements in their most stable state at standard thermodynamic conditions, 298.15 K and 1 bar. It is also, by deﬁnition, the enthalpy at the standard state T = T o = 298.15 K.

o o ∆f H ≡ H(T ) = H(298.15K)

For the elements in their most stable state, the standard heat of formation is, by deﬁnition, zero.

o ∆f H (stable elements) ≡ 0

enthalpy difference H − H(T o) This is the difference in enthalpy between the temperature of interest and the standard state.

Z T o 0 0 H(T ) − H(T ) = CP (T ) dT (kJ/mol) T o

In order to ﬁnd the total enthalpy for any species i, the heat of formation and the speciﬁc enthalpy difference must be added together.

o o Hi(T ) = ∆f Hi + (H(T ) − H(T ))i

The reference temperature T o is, by convention, taken to be 298.15 K. 51 standard entropy So This is the temperature dependent portion of the molar speciﬁc entropy .

Z T 0 o CP (T ) 0 S (T ) = 0 dT (J/mol-K) 0 T This is equivalent to the entropy of a pure substance evaluated at the standard pressure P o = 1 bar. At any other pressure P , the entropy of a pure substance will be

S(T,P ) = So(T ) − R˜ ln(P/P o)

The total entropy of an ideal gas mixture is

K X o ˜ o S(T,P ) = Xi Si − R ln(Pi/P ) i=1

where Pi = XiP is the partial pressure of the ith component. An alternate formulation of this rule is

K K X o ˜ X ˜ o S = XiSi − R Xi ln Xi − R ln(P/P ) i=1 i=1 standard Gibbs energy Go This is the temperature dependent portion of the Gibbs energy.

Go(T ) ≡ H − TSo ≡ µo (kJ/mol)

This is equivalent to the Gibbs energy of a pure substance evaluated at the standard pressure P o. The actual Gibbs energy of a pure substance at any pressure can be determined by evaluating:

G(T,P ) = Go(T ) + RT˜ ln(P/P o)

The Gibbs energy of a mixture can be evaluated as

K X o ˜ o G(T,P ) = Xi Gi + RT ln(Pi/P ) i=1 The chemical potential of a species i in a mixture can be evaluated as

o ˜ o µi = µi + RT ln(Pi/P ) Using this deﬁnition, the mixture Gibbs energy can be evaluated as

K X G = Xiµi i=1 52 D IDEAL GAS PROPERTIES UP TO 4000K

K X g = niµi. i=1

References Gurvich, L. V., Veyts, I. V., and Alcock, C. B. 1989 Thermodynamic Properties of Individual Substances, Fourth Edition, Hemisphere Pubishing. Volume I, Parts 1 and 2.

McBride, B. J., and Gordon, S. 1992 Computer Program for Calculating and Fitting Thermody- namic Functions NASA Reference Publication 1271.

McBride, B. J., Gordon, S., and Reno, M. A. 1993 Thermodynamic Data for Fifty Reference Elements NASA Technical Paper 3287.

Chase, M. W. et al. 1985 JANAF Thermochemical Tables, Third Edition, Parts I and II, J. Phys. Chem. Ref. Data 14,Supplement No. 1. 53

Thermodynamic Properties of H2O Molar mass W (g/mol) 18.02 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) -241.84 Entropy So(298.15) (J/mol-K) 188.71 Gibbs energy Go(298.15) (kJ/mol) -298.11 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 33.45 .00 188.71 -298.11 300.0 33.47 .06 188.92 -298.46 400.0 34.44 3.46 198.69 -317.86 500.0 35.34 6.95 206.47 -338.13 600.0 36.29 10.53 212.99 -359.11 700.0 37.36 14.21 218.67 -380.70 800.0 38.59 18.01 223.73 -402.83 900.0 39.93 21.93 228.35 -425.43 1000.0 41.32 25.99 232.63 -448.48 1100.0 42.64 30.19 236.63 -471.95 1200.0 43.87 34.52 240.40 -495.80 1300.0 45.03 38.96 243.96 -520.02 1400.0 46.10 43.52 247.33 -544.59 1500.0 47.10 48.18 250.55 -569.48 1600.0 48.03 52.94 253.62 -594.69 1700.0 48.90 57.79 256.56 -620.20 1800.0 49.71 62.72 259.37 -646.00 1900.0 50.45 67.73 262.08 -672.07 2000.0 51.14 72.81 264.69 -698.41 2100.0 51.78 77.95 267.20 -725.01 2200.0 52.38 83.16 269.62 -751.85 2300.0 52.93 88.43 271.96 -778.93 2400.0 53.44 93.74 274.22 -806.24 2500.0 53.91 99.11 276.42 -833.77 2600.0 54.34 104.52 278.54 -861.52 2700.0 54.74 109.98 280.60 -889.48 2800.0 55.11 115.47 282.59 -917.64 2900.0 55.46 121.00 284.53 -945.99 3000.0 55.78 126.56 286.42 -974.54 3100.0 56.08 132.16 288.25 -1003.28 3200.0 56.35 137.78 290.04 -1032.19 3300.0 56.61 143.43 291.78 -1061.28 3400.0 56.85 149.10 293.47 -1090.55 3500.0 57.08 154.80 295.12 -1119.98 3600.0 57.29 160.51 296.73 -1149.57 3700.0 57.49 166.25 298.31 -1179.32 3800.0 57.68 172.01 299.84 -1209.23 3900.0 57.86 177.79 301.34 -1239.29 4000.0 58.03 183.58 302.81 -1269.50 54 D IDEAL GAS PROPERTIES UP TO 4000K

Thermodynamic Properties of CO2 Molar mass W (g/mol) 44.01 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) -393.55 Entropy So(298.15) (J/mol-K) 213.74 Gibbs energy Go(298.15) (kJ/mol) -457.27 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 37.20 .00 213.74 -457.27 300.0 37.28 .07 213.97 -457.67 400.0 41.28 4.00 225.26 -479.65 500.0 44.57 8.30 234.83 -502.66 600.0 47.31 12.90 243.21 -526.57 700.0 49.62 17.75 250.68 -551.27 800.0 51.55 22.81 257.44 -576.68 900.0 53.14 28.05 263.60 -602.74 1000.0 54.36 33.43 269.27 -629.39 1100.0 55.33 38.91 274.50 -656.58 1200.0 56.21 44.49 279.35 -684.28 1300.0 56.98 50.15 283.88 -712.44 1400.0 57.68 55.88 288.13 -741.04 1500.0 58.29 61.68 292.13 -770.06 1600.0 58.84 67.54 295.91 -799.46 1700.0 59.32 73.45 299.49 -829.23 1800.0 59.74 79.40 302.89 -859.35 1900.0 60.11 85.39 306.13 -889.80 2000.0 60.43 91.42 309.22 -920.57 2100.0 60.72 97.48 312.18 -951.64 2200.0 60.97 103.56 315.01 -983.00 2300.0 61.19 109.67 317.72 -1014.64 2400.0 61.38 115.80 320.33 -1046.55 2500.0 61.55 121.94 322.84 -1078.71 2600.0 61.70 128.11 325.26 -1111.11 2700.0 61.84 134.28 327.59 -1143.75 2800.0 61.97 140.47 329.84 -1176.63 2900.0 62.08 146.68 332.02 -1209.72 3000.0 62.19 152.89 334.12 -1243.03 3100.0 62.30 159.12 336.17 -1276.54 3200.0 62.41 165.35 338.15 -1310.26 3300.0 62.51 171.60 340.07 -1344.17 3400.0 62.61 177.85 341.93 -1378.27 3500.0 62.72 184.12 343.75 -1412.56 3600.0 62.82 190.40 345.52 -1447.02 3700.0 62.93 196.68 347.24 -1481.66 3800.0 63.04 202.98 348.92 -1516.47 3900.0 63.15 209.29 350.56 -1551.44 4000.0 63.26 215.61 352.16 -1586.58 55

Thermodynamic Properties of CO Molar mass W (g/mol) 28.01 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) -110.54 Entropy So(298.15) (J/mol-K) 197.55 Gibbs energy Go(298.15) (kJ/mol) -169.44 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 29.07 .00 197.55 -169.44 300.0 29.08 .05 197.73 -169.81 400.0 29.43 2.98 206.14 -190.02 500.0 29.86 5.94 212.75 -210.97 600.0 30.41 8.95 218.24 -232.53 700.0 31.09 12.03 222.98 -254.60 800.0 31.86 15.18 227.18 -277.11 900.0 32.63 18.40 230.98 -300.02 1000.0 33.25 21.70 234.45 -323.29 1100.0 33.72 25.05 237.64 -346.90 1200.0 34.15 28.44 240.60 -370.81 1300.0 34.53 31.87 243.34 -395.01 1400.0 34.87 35.34 245.92 -419.48 1500.0 35.18 38.85 248.33 -444.19 1600.0 35.45 42.38 250.61 -469.14 1700.0 35.69 45.94 252.77 -494.31 1800.0 35.91 49.52 254.81 -519.69 1900.0 36.10 53.12 256.76 -545.27 2000.0 36.27 56.74 258.62 -571.04 2100.0 36.42 60.37 260.39 -596.99 2200.0 36.55 64.02 262.09 -623.11 2300.0 36.67 67.68 263.72 -649.40 2400.0 36.77 71.35 265.28 -675.85 2500.0 36.87 75.04 266.78 -702.46 2600.0 36.95 78.73 268.23 -729.21 2700.0 37.02 82.43 269.62 -756.10 2800.0 37.09 86.13 270.97 -783.13 2900.0 37.16 89.84 272.28 -810.30 3000.0 37.21 93.56 273.54 -837.59 3100.0 37.27 97.29 274.76 -865.00 3200.0 37.32 101.02 275.94 -892.54 3300.0 37.37 104.75 277.09 -920.19 3400.0 37.42 108.49 278.21 -947.95 3500.0 37.47 112.24 279.29 -975.83 3600.0 37.52 115.98 280.35 -1003.81 3700.0 37.57 119.74 281.38 -1031.90 3800.0 37.62 123.50 282.38 -1060.09 3900.0 37.67 127.26 283.36 -1088.37 4000.0 37.72 131.03 284.31 -1116.76 56 D IDEAL GAS PROPERTIES UP TO 4000K

Thermodynamic Properties of N2 Molar mass W (g/mol) 28.01 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) .00 Entropy So(298.15) (J/mol-K) 191.51 Gibbs energy Go(298.15) (kJ/mol) -57.10 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 29.07 .00 191.51 -57.10 300.0 29.08 .05 191.69 -57.45 400.0 29.32 2.97 200.09 -77.06 500.0 29.64 5.92 206.66 -97.41 600.0 30.09 8.90 212.10 -118.36 700.0 30.68 11.94 216.78 -139.80 800.0 31.39 15.05 220.93 -161.69 900.0 32.13 18.22 224.67 -183.98 1000.0 32.76 21.47 228.09 -206.62 1100.0 33.26 24.77 231.23 -229.59 1200.0 33.71 28.12 234.15 -252.86 1300.0 34.11 31.51 236.86 -276.41 1400.0 34.48 34.94 239.40 -300.22 1500.0 34.81 38.40 241.79 -324.28 1600.0 35.10 41.90 244.05 -348.58 1700.0 35.36 45.42 246.18 -373.09 1800.0 35.59 48.97 248.21 -397.81 1900.0 35.80 52.54 250.14 -422.73 2000.0 35.99 56.13 251.98 -447.84 2100.0 36.15 59.74 253.74 -473.12 2200.0 36.30 63.36 255.43 -498.58 2300.0 36.43 67.00 257.05 -524.21 2400.0 36.54 70.65 258.60 -549.99 2500.0 36.65 74.30 260.09 -575.92 2600.0 36.74 77.97 261.53 -602.00 2700.0 36.82 81.65 262.92 -628.23 2800.0 36.90 85.34 264.26 -654.59 2900.0 36.96 89.03 265.56 -681.08 3000.0 37.03 92.73 266.81 -707.70 3100.0 37.09 96.44 268.02 -734.44 3200.0 37.14 100.15 269.20 -761.30 3300.0 37.20 103.87 270.35 -788.28 3400.0 37.25 107.59 271.46 -815.37 3500.0 37.30 111.32 272.54 -842.57 3600.0 37.35 115.05 273.59 -869.88 3700.0 37.40 118.79 274.61 -897.29 3800.0 37.45 122.53 275.61 -924.80 3900.0 37.50 126.28 276.59 -952.41 4000.0 37.55 130.03 277.54 -980.11 57

Thermodynamic Properties of N Molar mass W (g/mol) 14.01 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) 472.63 Entropy So(298.15) (J/mol-K) 153.19 Gibbs energy Go(298.15) (kJ/mol) 426.96 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 20.79 .00 153.19 426.96 300.0 20.79 .04 153.32 426.67 400.0 20.79 2.12 159.30 411.03 500.0 20.79 4.20 163.94 394.86 600.0 20.79 6.27 167.72 378.27 700.0 20.79 8.35 170.93 361.33 800.0 20.79 10.43 173.70 344.10 900.0 20.79 12.51 176.15 326.60 1000.0 20.79 14.59 178.34 308.87 1100.0 20.79 16.67 180.32 290.94 1200.0 20.79 18.75 182.13 272.82 1300.0 20.79 20.83 183.80 254.52 1400.0 20.79 22.91 185.34 236.06 1500.0 20.79 24.99 186.77 217.45 1600.0 20.79 27.06 188.12 198.71 1700.0 20.78 29.14 189.38 179.83 1800.0 20.78 31.22 190.56 160.84 1900.0 20.78 33.30 191.69 141.72 2000.0 20.78 35.38 192.75 122.50 2100.0 20.78 37.45 193.77 103.17 2200.0 20.78 39.53 194.73 83.75 2300.0 20.79 41.61 195.66 64.23 2400.0 20.80 43.69 196.54 44.62 2500.0 20.82 45.77 197.39 24.92 2600.0 20.84 47.85 198.21 5.14 2700.0 20.86 49.94 199.00 -14.72 2800.0 20.89 52.03 199.75 -34.66 2900.0 20.93 54.12 200.49 -54.67 3000.0 20.97 56.21 201.20 -74.75 3100.0 21.02 58.31 201.89 -94.91 3200.0 21.08 60.42 202.56 -115.13 3300.0 21.14 62.53 203.20 -135.42 3400.0 21.21 64.65 203.84 -155.77 3500.0 21.29 66.77 204.45 -176.19 3600.0 21.38 68.91 205.05 -196.66 3700.0 21.47 71.05 205.64 -217.20 3800.0 21.57 73.20 206.22 -237.79 3900.0 21.69 75.36 206.78 -258.44 4000.0 21.81 77.54 207.33 -279.14 58 D IDEAL GAS PROPERTIES UP TO 4000K

Thermodynamic Properties of O2 Molar mass W (g/mol) 32.00 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) .00 Entropy So(298.15) (J/mol-K) 205.04 Gibbs energy Go(298.15) (kJ/mol) -61.13 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 29.32 .00 205.04 -61.13 300.0 29.33 .05 205.22 -61.51 400.0 30.21 3.03 213.78 -82.48 500.0 31.11 6.10 220.62 -104.21 600.0 32.03 9.25 226.37 -126.57 700.0 32.93 12.50 231.38 -149.46 800.0 33.76 15.84 235.83 -172.83 900.0 34.45 19.25 239.85 -196.62 1000.0 34.94 22.72 243.51 -220.79 1100.0 35.27 26.23 246.85 -245.31 1200.0 35.59 29.77 249.94 -270.15 1300.0 35.90 33.35 252.80 -295.29 1400.0 36.20 36.96 255.47 -320.70 1500.0 36.49 40.59 257.98 -346.37 1600.0 36.77 44.25 260.34 -372.29 1700.0 37.04 47.94 262.58 -398.44 1800.0 37.30 51.66 264.70 -424.80 1900.0 37.55 55.40 266.72 -451.37 2000.0 37.79 59.17 268.66 -478.14 2100.0 38.02 62.96 270.51 -505.10 2200.0 38.25 66.77 272.28 -532.24 2300.0 38.47 70.61 273.98 -559.56 2400.0 38.68 74.47 275.63 -587.04 2500.0 38.89 78.35 277.21 -614.68 2600.0 39.09 82.24 278.74 -642.48 2700.0 39.29 86.16 280.22 -670.43 2800.0 39.48 90.10 281.65 -698.52 2900.0 39.67 94.06 283.04 -726.76 3000.0 39.85 98.04 284.39 -755.13 3100.0 40.02 102.03 285.70 -783.63 3200.0 40.19 106.04 286.97 -812.27 3300.0 40.36 110.07 288.21 -841.02 3400.0 40.53 114.11 289.42 -869.91 3500.0 40.69 118.17 290.59 -898.91 3600.0 40.84 122.25 291.74 -928.02 3700.0 40.99 126.34 292.86 -957.25 3800.0 41.14 130.45 293.96 -986.60 3900.0 41.29 134.57 295.03 -1016.05 4000.0 41.43 138.71 296.08 -1045.60 59

Thermodynamic Properties of O Molar mass W (g/mol) 16.00 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) 249.20 Entropy So(298.15) (J/mol-K) 160.94 Gibbs energy Go(298.15) (kJ/mol) 201.21 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 21.90 .00 160.94 201.21 300.0 21.89 .04 161.08 200.91 400.0 21.50 2.21 167.32 184.48 500.0 21.26 4.35 172.09 167.50 600.0 21.11 6.46 175.95 150.09 700.0 21.03 8.57 179.20 132.33 800.0 20.99 10.67 182.00 114.26 900.0 20.95 12.77 184.47 95.94 1000.0 20.92 14.86 186.68 77.38 1100.0 20.90 16.95 188.67 58.61 1200.0 20.88 19.04 190.49 39.65 1300.0 20.87 21.13 192.16 20.52 1400.0 20.85 23.21 193.71 1.22 1500.0 20.84 25.30 195.14 -18.22 1600.0 20.83 27.38 196.49 -37.80 1700.0 20.83 29.47 197.75 -57.52 1800.0 20.82 31.55 198.94 -77.35 1900.0 20.82 33.63 200.07 -97.30 2000.0 20.82 35.71 201.14 -117.36 2100.0 20.82 37.79 202.15 -137.53 2200.0 20.82 39.88 203.12 -157.79 2300.0 20.83 41.96 204.05 -178.15 2400.0 20.84 44.04 204.93 -198.60 2500.0 20.85 46.13 205.78 -219.14 2600.0 20.86 48.21 206.60 -239.76 2700.0 20.88 50.30 207.39 -260.46 2800.0 20.90 52.39 208.15 -281.23 2900.0 20.92 54.48 208.88 -302.09 3000.0 20.94 56.57 209.59 -323.01 3100.0 20.97 58.67 210.28 -344.00 3200.0 21.00 60.77 210.95 -365.06 3300.0 21.03 62.87 211.59 -386.19 3400.0 21.06 64.97 212.22 -407.38 3500.0 21.10 67.08 212.83 -428.64 3600.0 21.13 69.19 213.43 -449.95 3700.0 21.17 71.31 214.01 -471.32 3800.0 21.21 73.43 214.57 -492.75 3900.0 21.25 75.55 215.12 -514.23 4000.0 21.30 77.68 215.66 -535.77 60 D IDEAL GAS PROPERTIES UP TO 4000K

Thermodynamic Properties of H2 Molar mass W (g/mol) 2.02 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) .00 Entropy So(298.15) (J/mol-K) 130.59 Gibbs energy Go(298.15) (kJ/mol) -38.93 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 28.87 .00 130.59 -38.93 300.0 28.88 .05 130.77 -39.18 400.0 29.12 2.95 139.12 -52.69 500.0 29.28 5.87 145.63 -66.94 600.0 29.38 8.81 150.98 -81.78 700.0 29.46 11.75 155.51 -97.11 800.0 29.58 14.70 159.45 -112.86 900.0 29.79 17.67 162.95 -128.98 1000.0 30.16 20.66 166.11 -145.44 1100.0 30.62 23.70 169.00 -162.20 1200.0 31.08 26.79 171.69 -179.23 1300.0 31.52 29.92 174.19 -196.53 1400.0 31.94 33.09 176.54 -214.07 1500.0 32.36 36.31 178.76 -231.83 1600.0 32.76 39.56 180.86 -249.81 1700.0 33.15 42.86 182.86 -268.00 1800.0 33.52 46.19 184.76 -286.38 1900.0 33.89 49.56 186.59 -304.95 2000.0 34.24 52.97 188.33 -323.70 2100.0 34.57 56.41 190.01 -342.62 2200.0 34.90 59.88 191.63 -361.70 2300.0 35.22 63.39 193.19 -380.94 2400.0 35.52 66.93 194.69 -400.33 2500.0 35.81 70.49 196.15 -419.88 2600.0 36.09 74.09 197.56 -439.56 2700.0 36.36 77.71 198.93 -459.39 2800.0 36.62 81.36 200.25 -479.35 2900.0 36.87 85.03 201.54 -499.44 3000.0 37.11 88.73 202.80 -519.65 3100.0 37.34 92.46 204.02 -539.99 3200.0 37.57 96.20 205.21 -560.46 3300.0 37.78 99.97 206.37 -581.03 3400.0 37.99 103.76 207.50 -601.73 3500.0 38.19 107.57 208.60 -622.53 3600.0 38.38 111.39 209.68 -643.45 3700.0 38.57 115.24 210.73 -664.47 3800.0 38.76 119.11 211.76 -685.59 3900.0 38.94 122.99 212.77 -706.82 4000.0 39.12 126.90 213.76 -728.15 61

Thermodynamic Properties of H Molar mass W (g/mol) 1.01 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) 217.98 Entropy So(298.15) (J/mol-K) 114.60 Gibbs energy Go(298.15) (kJ/mol) 183.81 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 20.79 .00 114.60 183.81 300.0 20.79 .04 114.73 183.60 400.0 20.79 2.12 120.71 171.81 500.0 20.79 4.20 125.35 159.50 600.0 20.79 6.27 129.14 146.77 700.0 20.79 8.35 132.35 133.69 800.0 20.79 10.43 135.12 120.31 900.0 20.79 12.51 137.57 106.68 1000.0 20.79 14.59 139.76 92.81 1100.0 20.79 16.67 141.74 78.73 1200.0 20.79 18.75 143.55 64.47 1300.0 20.79 20.82 145.21 50.03 1400.0 20.79 22.90 146.75 35.43 1500.0 20.79 24.98 148.19 20.68 1600.0 20.79 27.06 149.53 5.79 1700.0 20.79 29.14 150.79 -9.22 1800.0 20.79 31.22 151.98 -24.36 1900.0 20.79 33.30 153.10 -39.62 2000.0 20.79 35.37 154.17 -54.98 2100.0 20.79 37.45 155.18 -70.45 2200.0 20.79 39.53 156.15 -86.01 2300.0 20.79 41.61 157.07 -101.68 2400.0 20.79 43.69 157.96 -117.43 2500.0 20.79 45.77 158.81 -133.27 2600.0 20.79 47.85 159.62 -149.19 2700.0 20.79 49.92 160.40 -165.19 2800.0 20.79 52.00 161.16 -181.27 2900.0 20.79 54.08 161.89 -197.42 3000.0 20.79 56.16 162.59 -213.65 3100.0 20.79 58.24 163.28 -229.94 3200.0 20.79 60.32 163.94 -246.30 3300.0 20.79 62.40 164.58 -262.73 3400.0 20.79 64.48 165.20 -279.21 3500.0 20.79 66.55 165.80 -295.76 3600.0 20.79 68.63 166.38 -312.37 3700.0 20.79 70.71 166.95 -329.04 3800.0 20.79 72.79 167.51 -345.76 3900.0 20.79 74.87 168.05 -362.54 4000.0 20.79 76.95 168.57 -379.37 62 D IDEAL GAS PROPERTIES UP TO 4000K

Thermodynamic Properties of OH Molar mass W (g/mol) 17.01 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) 38.99 Entropy So(298.15) (J/mol-K) 183.60 Gibbs energy Go(298.15) (kJ/mol) -15.76 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 29.93 .00 183.60 -15.76 300.0 29.93 .06 183.79 -16.10 400.0 29.72 3.04 192.37 -34.92 500.0 29.57 6.00 198.98 -54.50 600.0 29.53 8.95 204.37 -74.68 700.0 29.61 11.91 208.93 -95.35 800.0 29.84 14.88 212.89 -116.45 900.0 30.21 17.88 216.43 -137.92 1000.0 30.68 20.93 219.63 -159.72 1100.0 31.19 24.02 222.58 -181.83 1200.0 31.66 27.16 225.32 -204.23 1300.0 32.11 30.35 227.87 -226.89 1400.0 32.54 33.59 230.26 -249.80 1500.0 32.94 36.86 232.52 -272.94 1600.0 33.32 40.17 234.66 -296.30 1700.0 33.68 43.52 236.69 -319.87 1800.0 34.02 46.91 238.63 -343.63 1900.0 34.34 50.33 240.48 -367.59 2000.0 34.64 53.78 242.24 -391.73 2100.0 34.92 57.25 243.94 -416.04 2200.0 35.18 60.76 245.57 -440.51 2300.0 35.42 64.29 247.14 -465.15 2400.0 35.66 67.84 248.65 -489.94 2500.0 35.87 71.42 250.11 -514.88 2600.0 36.07 75.02 251.52 -539.96 2700.0 36.26 78.63 252.89 -565.18 2800.0 36.44 82.27 254.21 -590.54 2900.0 36.60 85.92 255.49 -616.02 3000.0 36.76 89.59 256.74 -641.63 3100.0 36.90 93.27 257.94 -667.37 3200.0 37.04 96.97 259.12 -693.22 3300.0 37.17 100.68 260.26 -719.19 3400.0 37.29 104.40 261.37 -745.27 3500.0 37.40 108.14 262.45 -771.46 3600.0 37.50 111.88 263.51 -797.76 3700.0 37.60 115.64 264.54 -824.17 3800.0 37.70 119.40 265.54 -850.67 3900.0 37.79 123.18 266.52 -877.27 4000.0 37.88 126.96 267.48 -903.97 63

Thermodynamic Properties of NO Molar mass W (g/mol) 30.01 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) 90.30 Entropy So(298.15) (J/mol-K) 210.65 Gibbs energy Go(298.15) (kJ/mol) 27.49 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 29.73 .00 210.65 27.49 300.0 29.73 .06 210.84 27.10 400.0 30.10 3.05 219.44 5.57 500.0 30.57 6.08 226.20 -16.73 600.0 31.17 9.16 231.83 -39.64 700.0 31.91 12.32 236.69 -63.07 800.0 32.72 15.55 241.00 -86.95 900.0 33.49 18.86 244.90 -111.25 1000.0 34.08 22.24 248.46 -135.92 1100.0 34.48 25.67 251.73 -160.94 1200.0 34.85 29.14 254.75 -186.26 1300.0 35.18 32.64 257.55 -211.88 1400.0 35.47 36.17 260.17 -237.76 1500.0 35.74 39.73 262.62 -263.91 1600.0 35.97 43.32 264.94 -290.28 1700.0 36.18 46.93 267.12 -316.89 1800.0 36.36 50.55 269.20 -343.71 1900.0 36.53 54.20 271.17 -370.72 2000.0 36.67 57.86 273.05 -397.94 2100.0 36.80 61.53 274.84 -425.33 2200.0 36.91 65.22 276.55 -452.90 2300.0 37.01 68.91 278.19 -480.64 2400.0 37.10 72.62 279.77 -508.54 2500.0 37.17 76.33 281.29 -536.59 2600.0 37.24 80.05 282.75 -564.79 2700.0 37.31 83.78 284.15 -593.14 2800.0 37.36 87.51 285.51 -621.62 2900.0 37.41 91.25 286.82 -650.24 3000.0 37.46 95.00 288.09 -678.99 3100.0 37.51 98.74 289.32 -707.86 3200.0 37.56 102.50 290.51 -736.85 3300.0 37.60 106.26 291.67 -765.96 3400.0 37.64 110.02 292.79 -795.18 3500.0 37.69 113.78 293.88 -824.52 3600.0 37.73 117.55 294.95 -853.96 3700.0 37.77 121.33 295.98 -883.50 3800.0 37.81 125.11 296.99 -913.15 3900.0 37.86 128.89 297.97 -942.90 4000.0 37.90 132.68 298.93 -972.75 64 D IDEAL GAS PROPERTIES UP TO 4000K

Thermodynamic Properties of Ar Molar mass W (g/mol) 39.95 Standard State Values o Enthalpy of formation ∆f H (kJ/mol) .00 Entropy So(298.15) (J/mol-K) 154.73 Gibbs energy Go(298.15) (kJ/mol) -46.13 o o o TCP H − H(T ) S G (K) (J/mol-K) (kJ/mol) (J/mol-K) (kJ/mol)

298.15 20.79 .00 154.73 -46.13 300.0 20.79 .04 154.86 -46.42 400.0 20.79 2.12 160.84 -62.22 500.0 20.79 4.20 165.48 -78.54 600.0 20.79 6.27 169.27 -95.29 700.0 20.79 8.35 172.47 -112.38 800.0 20.79 10.43 175.25 -129.77 900.0 20.79 12.51 177.70 -147.42 1000.0 20.79 14.59 179.89 -165.30 1100.0 20.79 16.67 181.87 -183.39 1200.0 20.79 18.75 183.68 -201.66 1300.0 20.79 20.82 185.34 -220.12 1400.0 20.79 22.90 186.88 -238.73 1500.0 20.79 24.98 188.31 -257.49 1600.0 20.79 27.06 189.65 -276.39 1700.0 20.79 29.14 190.91 -295.42 1800.0 20.79 31.22 192.10 -314.57 1900.0 20.79 33.30 193.23 -333.83 2000.0 20.79 35.37 194.29 -353.21 2100.0 20.79 37.45 195.31 -372.69 2200.0 20.79 39.53 196.27 -392.27 2300.0 20.79 41.61 197.20 -411.95 2400.0 20.79 43.69 198.08 -431.71 2500.0 20.79 45.77 198.93 -451.56 2600.0 20.79 47.85 199.75 -471.49 2700.0 20.79 49.92 200.53 -491.51 2800.0 20.79 52.00 201.29 -511.60 2900.0 20.79 54.08 202.02 -531.77 3000.0 20.79 56.16 202.72 -552.00 3100.0 20.79 58.24 203.40 -572.31 3200.0 20.79 60.32 204.06 -592.68 3300.0 20.79 62.40 204.70 -613.12 3400.0 20.79 64.48 205.32 -633.62 3500.0 20.79 66.55 205.93 -654.18 3600.0 20.79 68.63 206.51 -674.81 3700.0 20.79 70.71 207.08 -695.49 3800.0 20.79 72.79 207.63 -716.22 3900.0 20.79 74.87 208.17 -737.01 4000.0 20.79 76.95 208.70 -757.86