Chem 340 - Lecture Notes 4 – Fall 2013 – State Function Manipulations
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Lecture 2 the First Law of Thermodynamics (Ch.1)
Lecture 2 The First Law of Thermodynamics (Ch.1) Outline: 1. Internal Energy, Work, Heating 2. Energy Conservation – the First Law 3. Quasi-static processes 4. Enthalpy 5. Heat Capacity Internal Energy The internal energy of a system of particles, U, is the sum of the kinetic energy in the reference frame in which the center of mass is at rest and the potential energy arising from the forces of the particles on each other. system Difference between the total energy and the internal energy? boundary system U = kinetic + potential “environment” B The internal energy is a state function – it depends only on P the values of macroparameters (the state of a system), not on the method of preparation of this state (the “path” in the V macroparameter space is irrelevant). T A In equilibrium [ f (P,V,T)=0 ] : U = U (V, T) U depends on the kinetic energy of particles in a system and an average inter-particle distance (~ V-1/3) – interactions. For an ideal gas (no interactions) : U = U (T) - “pure” kinetic Internal Energy of an Ideal Gas f The internal energy of an ideal gas U = Nk T with f degrees of freedom: 2 B f ⇒ 3 (monatomic), 5 (diatomic), 6 (polyatomic) (here we consider only trans.+rotat. degrees of freedom, and neglect the vibrational ones that can be excited at very high temperatures) How does the internal energy of air in this (not-air-tight) room change with T if the external P = const? f ⎡ PV ⎤ f U =Nin room k= T Bin⎢ N room = ⎥ = PV 2 ⎣ kB T⎦ 2 - does not change at all, an increase of the kinetic energy of individual molecules with T is compensated by a decrease of their number. -
Chapter 3. Second and Third Law of Thermodynamics
Chapter 3. Second and third law of thermodynamics Important Concepts Review Entropy; Gibbs Free Energy • Entropy (S) – definitions Law of Corresponding States (ch 1 notes) • Entropy changes in reversible and Reduced pressure, temperatures, volumes irreversible processes • Entropy of mixing of ideal gases • 2nd law of thermodynamics • 3rd law of thermodynamics Math • Free energy Numerical integration by computer • Maxwell relations (Trapezoidal integration • Dependence of free energy on P, V, T https://en.wikipedia.org/wiki/Trapezoidal_rule) • Thermodynamic functions of mixtures Properties of partial differential equations • Partial molar quantities and chemical Rules for inequalities potential Major Concept Review • Adiabats vs. isotherms p1V1 p2V2 • Sign convention for work and heat w done on c=C /R vm system, q supplied to system : + p1V1 p2V2 =Cp/CV w done by system, q removed from system : c c V1T1 V2T2 - • Joule-Thomson expansion (DH=0); • State variables depend on final & initial state; not Joule-Thomson coefficient, inversion path. temperature • Reversible change occurs in series of equilibrium V states T TT V P p • Adiabatic q = 0; Isothermal DT = 0 H CP • Equations of state for enthalpy, H and internal • Formation reaction; enthalpies of energy, U reaction, Hess’s Law; other changes D rxn H iD f Hi i T D rxn H Drxn Href DrxnCpdT Tref • Calorimetry Spontaneous and Nonspontaneous Changes First Law: when one form of energy is converted to another, the total energy in universe is conserved. • Does not give any other restriction on a process • But many processes have a natural direction Examples • gas expands into a vacuum; not the reverse • can burn paper; can't unburn paper • heat never flows spontaneously from cold to hot These changes are called nonspontaneous changes. -
Thermodynamics
ME346A Introduction to Statistical Mechanics { Wei Cai { Stanford University { Win 2011 Handout 6. Thermodynamics January 26, 2011 Contents 1 Laws of thermodynamics 2 1.1 The zeroth law . .3 1.2 The first law . .4 1.3 The second law . .5 1.3.1 Efficiency of Carnot engine . .5 1.3.2 Alternative statements of the second law . .7 1.4 The third law . .8 2 Mathematics of thermodynamics 9 2.1 Equation of state . .9 2.2 Gibbs-Duhem relation . 11 2.2.1 Homogeneous function . 11 2.2.2 Virial theorem / Euler theorem . 12 2.3 Maxwell relations . 13 2.4 Legendre transform . 15 2.5 Thermodynamic potentials . 16 3 Worked examples 21 3.1 Thermodynamic potentials and Maxwell's relation . 21 3.2 Properties of ideal gas . 24 3.3 Gas expansion . 28 4 Irreversible processes 32 4.1 Entropy and irreversibility . 32 4.2 Variational statement of second law . 32 1 In the 1st lecture, we will discuss the concepts of thermodynamics, namely its 4 laws. The most important concepts are the second law and the notion of Entropy. (reading assignment: Reif x 3.10, 3.11) In the 2nd lecture, We will discuss the mathematics of thermodynamics, i.e. the machinery to make quantitative predictions. We will deal with partial derivatives and Legendre transforms. (reading assignment: Reif x 4.1-4.7, 5.1-5.12) 1 Laws of thermodynamics Thermodynamics is a branch of science connected with the nature of heat and its conver- sion to mechanical, electrical and chemical energy. (The Webster pocket dictionary defines, Thermodynamics: physics of heat.) Historically, it grew out of efforts to construct more efficient heat engines | devices for ex- tracting useful work from expanding hot gases (http://www.answers.com/thermodynamics). -
Thermodynamic Stability: Free Energy and Chemical Equilibrium ©David Ronis Mcgill University
Chemistry 223: Thermodynamic Stability: Free Energy and Chemical Equilibrium ©David Ronis McGill University 1. Spontaneity and Stability Under Various Conditions All the criteria for thermodynamic stability stem from the Clausius inequality,cf. Eq. (8.7.3). In particular,weshowed that for anypossible infinitesimal spontaneous change in nature, d− Q dS ≥ .(1) T Conversely,if d− Q dS < (2) T for every allowed change in state, then the system cannot spontaneously leave the current state NO MATTER WHAT;hence the system is in what is called stable equilibrium. The stability criterion becomes particularly simple if the system is adiabatically insulated from the surroundings. In this case, if all allowed variations lead to a decrease in entropy, then nothing will happen. The system will remain where it is. Said another way,the entropyofan adiabatically insulated stable equilibrium system is a maximum. Notice that the term allowed plays an important role. Forexample, if the system is in a constant volume container,changes in state or variations which lead to a change in the volume need not be considered eveniftheylead to an increase in the entropy. What if the system is not adiabatically insulated from the surroundings? Is there a more convenient test than Eq. (2)? The answer is yes. To see howitcomes about, note we can rewrite the criterion for stable equilibrium by using the first lawas − d Q = dE + Pop dV − µop dN > TdS,(3) which implies that dE + Pop dV − µop dN − TdS >0 (4) for all allowed variations if the system is in equilibrium. Equation (4) is the key stability result. -
Estimation of Internal Pressure of Liquids and Liquid Mixtures
Available online at www.ilcpa.pl International Letters of Chemistry, Physics and Astronomy 5 (2012) 1-7 ISSN 2299-3843 Estimation of internal pressure of liquids and liquid mixtures N. Santhi 1, P. Sabarathinam 2, G. Alamelumangai 1, J. Madhumitha 1, M. Emayavaramban 1 1Department of Chemistry, Government Arts College, C.Mutlur, Chidambaram-608102, India 2Retd. Registrar & Head of the Department of Technology, Annamalai University, Annamalainagar-608002, India E-mail address: [email protected] ABSTRACT By combining the van der Waals’ equation of state and the Free Length Theory of Jacobson, a new theoretical model is developed for the prediction of internal pressure of pure liquids and liquid mixtures. It requires only the molar volume data in addition to the ratio of heat capacities and critical temperature. The proposed model is simple, reliably accurate and capable of predicting internal pressure of pure liquids with an average absolute deviation of 4.24% in the predicted internal pressure values compared to those given in literature. The average absolute deviation in the predicted internal pressure values through the proposed model for the five binary liquid mixtures tested varies from 0.29% to 1.9% when compared to those of literature values. Keywords: van der Waals, pressure of liquids, theory of Jacobson, capacity of heat liquids, equation of Srivastava and Berkowitz 1. INTRODUTION The importance of internal pressure in understanding the properties of liquids and the full potential of internal pressure as a structural probe did become apparent with the pioneering work of Hildebrand 1, 2 and the first review of the subject by Richards 3 appeared in 1925. -
Thermodynamic Temperature
Thermodynamic temperature Thermodynamic temperature is the absolute measure 1 Overview of temperature and is one of the principal parameters of thermodynamics. Temperature is a measure of the random submicroscopic Thermodynamic temperature is defined by the third law motions and vibrations of the particle constituents of of thermodynamics in which the theoretically lowest tem- matter. These motions comprise the internal energy of perature is the null or zero point. At this point, absolute a substance. More specifically, the thermodynamic tem- zero, the particle constituents of matter have minimal perature of any bulk quantity of matter is the measure motion and can become no colder.[1][2] In the quantum- of the average kinetic energy per classical (i.e., non- mechanical description, matter at absolute zero is in its quantum) degree of freedom of its constituent particles. ground state, which is its state of lowest energy. Thermo- “Translational motions” are almost always in the classical dynamic temperature is often also called absolute tem- regime. Translational motions are ordinary, whole-body perature, for two reasons: one, proposed by Kelvin, that movements in three-dimensional space in which particles it does not depend on the properties of a particular mate- move about and exchange energy in collisions. Figure 1 rial; two that it refers to an absolute zero according to the below shows translational motion in gases; Figure 4 be- properties of the ideal gas. low shows translational motion in solids. Thermodynamic temperature’s null point, absolute zero, is the temperature The International System of Units specifies a particular at which the particle constituents of matter are as close as scale for thermodynamic temperature. -
The Carnot Cycle, Reversibility and Entropy
entropy Article The Carnot Cycle, Reversibility and Entropy David Sands Department of Physics and Mathematics, University of Hull, Hull HU6 7RX, UK; [email protected] Abstract: The Carnot cycle and the attendant notions of reversibility and entropy are examined. It is shown how the modern view of these concepts still corresponds to the ideas Clausius laid down in the nineteenth century. As such, they reflect the outmoded idea, current at the time, that heat is motion. It is shown how this view of heat led Clausius to develop the entropy of a body based on the work that could be performed in a reversible process rather than the work that is actually performed in an irreversible process. In consequence, Clausius built into entropy a conflict with energy conservation, which is concerned with actual changes in energy. In this paper, reversibility and irreversibility are investigated by means of a macroscopic formulation of internal mechanisms of damping based on rate equations for the distribution of energy within a gas. It is shown that work processes involving a step change in external pressure, however small, are intrinsically irreversible. However, under idealised conditions of zero damping the gas inside a piston expands and traces out a trajectory through the space of equilibrium states. Therefore, the entropy change due to heat flow from the reservoir matches the entropy change of the equilibrium states. This trajectory can be traced out in reverse as the piston reverses direction, but if the external conditions are adjusted appropriately, the gas can be made to trace out a Carnot cycle in P-V space. -
Thermodynamics: First Law, Calorimetry, Enthalpy
Thermodynamics: First Law, Calorimetry, Enthalpy Monday, January 23 CHEM 102H T. Hughbanks Calorimetry Reactions are usually done at either constant V (in a closed container) or constant P (open to the atmosphere). In either case, we can measure q by measuring a change in T (assuming we know heat capacities). Calorimetry: constant volume ∆U = q + w and w = -PexΔV If V is constant, then ΔV = 0, and w = 0. This leaves ∆U = qv, again, subscript v indicates const. volume. Measure ΔU by measuring heat released or taken up at constant volume. Calorimetry Problem When 0.2000 g of quinone (C6H4O2) is burned with excess oxygen in a bomb (constant volume) calorimeter, the temperature of the calorimeter increases by 3.26°C. The heat capacity of the calorimeter is known to be 1.56 kJ/°C. Find ΔU for the combustion of 1 mole of quinone. Calorimetry: Constant Pressure Reactions run in an open container will occur at constant P. Calorimetry done at constant pressure will measure heat: qp. How does this relate to ∆U, etc.? qP & ΔU ∆U = q + w = q - PΔV At constant V, PΔV term was zero, giving ∆U = qv. If P is constant and ΔT ≠ 0, then ΔV ≠ 0. – In particular, if gases are involved, PV = nRT wp ≠ 0, so ∆U ≠ qp Enthalpy: A New State Function It would be nice to have a state function which is directly related to qp: ∆(?) = qp We could then measure this function by calorimetry at constant pressure. Call this function enthalpy, H, defined by: H = U + PV Enthalpy Enthalpy: H = U + PV U, P, V all state functions, so H must also be a state function. -
Pressure Vessels
PRESSURE VESSELS David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 August 23, 2001 Introduction A good deal of the Mechanics of Materials can be introduced entirely within the confines of uniaxially stressed structural elements, and this was the goal of the previous modules. But of course the real world is three-dimensional, and we need to extend these concepts accordingly. We now take the next step, and consider those structures in which the loading is still simple, but where the stresses and strains now require a second dimension for their description. Both for their value in demonstrating two-dimensional effects and also for their practical use in mechanical design, we turn to a slightly more complicated structural type: the thin-walled pressure vessel. Structures such as pipes or bottles capable of holding internal pressure have been very important in the history of science and technology. Although the ancient Romans had developed municipal engineering to a high order in many ways, the very need for their impressive system of large aqueducts for carrying water was due to their not yet having pipes that could maintain internal pressure. Water can flow uphill when driven by the hydraulic pressure of the reservoir at a higher elevation, but without a pressure-containing pipe an aqueduct must be constructed so the water can run downhill all the way from the reservoir to the destination. Airplane cabins are another familiar example of pressure-containing structures. They illus- trate very dramatically the importance of proper design, since the atmosphere in the cabin has enough energy associated with its relative pressurization compared to the thin air outside that catastrophic crack growth is a real possibility. -
CHAPTER 17 Internal Pressure and Internal Energy of Saturated and Compressed Phases Ainstitute of Physics of the Dagestan Scient
CHAPTER 17 Internal Pressure and Internal Energy of Saturated and Compressed Phases ILMUTDIN M. ABDULAGATOV,a,b JOSEPH W. MAGEE,c NIKOLAI G. POLIKHRONIDI,a RABIYAT G. BATYROVAa aInstitute of Physics of the Dagestan Scientific Center of the Russian Academy of Sciences, Makhachkala, Dagestan, Russia. E-mail: [email protected] bDagestan State University, Makhachkala, Dagestan, Russia cNational Institute of Standards and Technology, Boulder, Colorado 80305 USA. E- mail: [email protected] Abstract Following a critical review of the field, a comprehensive analysis is provided of the internal pressure of fluids and fluid mixtures and its determination in a wide range of temperatures and pressures. Further, the physical meaning is discussed of the internal pressure along with its microscopic interpretation by means of calorimetric experiments. A new relation is explored between the internal pressure and the isochoric heat capacity jump along the coexistence curve near the critical point. Various methods (direct and indirect) of internal pressure determination are discussed. Relationships are studied between the internal pressure and key thermodynamic properties, namely expansion coefficient, isothermal compressibility, speed of sound, enthalpy increments, and viscosity. Loci of isothermal, isobaric, and isochoric internal pressure maxima and minima were examined in addition to the locus of zero internal pressure. Details were discussed of the new method of direct internal pressure determination by a calorimetric experiment that involves simultaneous measurement of the thermal pressure coefficient (∂P / ∂T )V , i.e. internal pressure Pint = (∂U / ∂V )T and heat capacity cV = (∂U / ∂T )V . The dependence of internal pressure on external pressure, temperature and density for pure fluids, and on concentration for binary mixtures is considered on the basis of reference (NIST REFPROP) and crossover EOS. -
Ideal Gas Temperature.)
FUNDAMENTALS OF CLASSICAL AND STATISTICAL THERMODYNAMICS SPRING 2005 1 1. Basic Concepts of Thermodynamics The basic concepts of thermodynamics such as system, energy, property, state, process, cycle, pressure, and temperature are explained. Thermodynamics can be defined as the science of energy. Energy can be viewed as the ability to cause changes. Thermodynamics is concerned with the transfer of heat and the appearance or disappearance of work attending various chemical and physical processes. • The natural phenomena that occur about us; • Controlled chemical reactions; • The performance of engines • Hypothetical processes such as chemical reactions that do not occur but can be imagined. HISTORY: Thermodynamics emerged as a science with the construction of the first successful atmospheric steam engines in England by Thomas Savery in 1697 and Thomas Newcomen in 1712. These engines were very slow and inefficient, but they opened the way for the development of a new science. The first and second laws of thermodynamics emerged 2 simultaneously in the 1850s primarily out of the works of William Rankine, Rudolph Clausius, and Lord Kelvin (formerly William Thomson). The term thermodynamics was first used in a publication by Lord Kelvin in 1849. The first thermodynamic textbook was written in 1859 by William Rankine, a professor at the University of Glasgow. 3 Classical vs. Statistical Thermodynamics Substances consist of large number of particles called molecules. The properties of the substance naturally depend on the behavior of these particles. For example, the pressure of a gas in a container is the result of momentum transfer between the molecules and the walls of the container. -
Lecture452 2 12.Pdf
University of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2012 Lecture 2. 6/20/12 A. Differentials and State Functions • Thermodynamics allows us to calculate energy transformations occurring in systems composed of large numbers of particles, systems so complex as to make rigorous mechanical calculations impractical. • The limitation is that thermodynamics does not allow us to calculate trajectories. We can only calculate energy changes that occur when a system passes from one equilibrium state to another. • In thermodynamics, we do not calculate the internal energy U. We calculate the change in internal energy ∆U, taken as the difference between the internal energy of the initial equilibrium state and the internal energy of the final equilibrium state: ∆=UUfinal − U initial • The internal energy U belongs to a class of functions called state functions. State function changes ∆F are only dependent on the initial state of the system, described by Pinitial, Vinitial, Tinitial, ninitial, and the final state of the system, described by Pfinal, Vfinal, Tfinal, and nfinal. State function changes do not depend upon the details of the specific path followed (i.e. the trajectory). • Quantities that are path-dependent are not state functions. • Thermodynamic state functions are functions of several state variables including P, V, T, and n. changes in the state variable n is are not considered at present. We will concentrate at first on single component/single phase systems with no loss or gain of material. Of the three remaining state variables P, V, T only two are independent because of the constraint imposed by the equation of state.