Ideal Gas State Functions
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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. -
Lecture 4: 09.16.05 Temperature, Heat, and Entropy
3.012 Fundamentals of Materials Science Fall 2005 Lecture 4: 09.16.05 Temperature, heat, and entropy Today: LAST TIME .........................................................................................................................................................................................2� State functions ..............................................................................................................................................................................2� Path dependent variables: heat and work..................................................................................................................................2� DEFINING TEMPERATURE ...................................................................................................................................................................4� The zeroth law of thermodynamics .............................................................................................................................................4� The absolute temperature scale ..................................................................................................................................................5� CONSEQUENCES OF THE RELATION BETWEEN TEMPERATURE, HEAT, AND ENTROPY: HEAT CAPACITY .......................................6� The difference between heat and temperature ...........................................................................................................................6� Defining heat capacity.................................................................................................................................................................6� -
Entropy: Ideal Gas Processes
Chapter 19: The Kinec Theory of Gases Thermodynamics = macroscopic picture Gases micro -> macro picture One mole is the number of atoms in 12 g sample Avogadro’s Number of carbon-12 23 -1 C(12)—6 protrons, 6 neutrons and 6 electrons NA=6.02 x 10 mol 12 atomic units of mass assuming mP=mn Another way to do this is to know the mass of one molecule: then So the number of moles n is given by M n=N/N sample A N = N A mmole−mass € Ideal Gas Law Ideal Gases, Ideal Gas Law It was found experimentally that if 1 mole of any gas is placed in containers that have the same volume V and are kept at the same temperature T, approximately all have the same pressure p. The small differences in pressure disappear if lower gas densities are used. Further experiments showed that all low-density gases obey the equation pV = nRT. Here R = 8.31 K/mol ⋅ K and is known as the "gas constant." The equation itself is known as the "ideal gas law." The constant R can be expressed -23 as R = kNA . Here k is called the Boltzmann constant and is equal to 1.38 × 10 J/K. N If we substitute R as well as n = in the ideal gas law we get the equivalent form: NA pV = NkT. Here N is the number of molecules in the gas. The behavior of all real gases approaches that of an ideal gas at low enough densities. Low densitiens m= enumberans tha oft t hemoles gas molecul es are fa Nr e=nough number apa ofr tparticles that the y do not interact with one another, but only with the walls of the gas container. -
HEAT and TEMPERATURE Heat Is a Type of ENERGY. When Absorbed
HEAT AND TEMPERATURE Heat is a type of ENERGY. When absorbed by a substance, heat causes inter-particle bonds to weaken and break which leads to a change of state (solid to liquid for example). Heat causing a phase change is NOT sufficient to cause an increase in temperature. Heat also causes an increase of kinetic energy (motion, friction) of the particles in a substance. This WILL cause an increase in TEMPERATURE. Temperature is NOT energy, only a measure of KINETIC ENERGY The reason why there is no change in temperature at a phase change is because the substance is using the heat only to change the way the particles interact (“stick together”). There is no increase in the particle motion and hence no rise in temperature. THERMAL ENERGY is one type of INTERNAL ENERGY possessed by an object. It is the KINETIC ENERGY component of the object’s internal energy. When thermal energy is transferred from a hot to a cold body, the term HEAT is used to describe the transferred energy. The hot body will decrease in temperature and hence in thermal energy. The cold body will increase in temperature and hence in thermal energy. Temperature Scales: The K scale is the absolute temperature scale. The lowest K temperature, 0 K, is absolute zero, the temperature at which an object possesses no thermal energy. The Celsius scale is based upon the melting point and boiling point of water at 1 atm pressure (0, 100o C) K = oC + 273.13 UNITS OF HEAT ENERGY The unit of heat energy we will use in this lesson is called the JOULE (J). -
Heat Energy a Science A–Z Physical Series Word Count: 1,324 Heat Energy
Heat Energy A Science A–Z Physical Series Word Count: 1,324 Heat Energy Written by Felicia Brown Visit www.sciencea-z.com www.sciencea-z.com KEY ELEMENTS USED IN THIS BOOK The Big Idea: One of the most important types of energy on Earth is heat energy. A great deal of heat energy comes from the Sun’s light Heat Energy hitting Earth. Other sources include geothermal energy, friction, and even living things. Heat energy is the driving force behind everything we do. This energy gives us the ability to run, dance, sing, and play. We also use heat energy to warm our homes, cook our food, power our vehicles, and create electricity. Key words: cold, conduction, conductor, convection, energy, evaporate, fire, friction, fuel, gas, geothermal heat, geyser, heat energy, hot, insulation, insulator, lightning, liquid, matter, particles, radiate, radiant energy, solid, Sun, temperature, thermometer, transfer, volcano Key comprehension skill: Cause and effect Other suitable comprehension skills: Compare and contrast; classify information; main idea and details; identify facts; elements of a genre; interpret graphs, charts, and diagram Key reading strategy: Connect to prior knowledge Other suitable reading strategies: Ask and answer questions; summarize; visualize; using a table of contents and headings; using a glossary and bold terms Photo Credits: Front cover: © iStockphoto.com/Julien Grondin; back cover, page 5: © iStockphoto.com/ Arpad Benedek; title page, page 20 (top): © iStockphoto.com/Anna Ziska; pages 3, 9, 20 (bottom): © Jupiterimages Corporation; -
IB Questionbank
Topic 3 Past Paper [94 marks] This question is about thermal energy transfer. A hot piece of iron is placed into a container of cold water. After a time the iron and water reach thermal equilibrium. The heat capacity of the container is negligible. specific heat capacity. [2 marks] 1a. Define Markscheme the energy required to change the temperature (of a substance) by 1K/°C/unit degree; of mass 1 kg / per unit mass; [5 marks] 1b. The following data are available. Mass of water = 0.35 kg Mass of iron = 0.58 kg Specific heat capacity of water = 4200 J kg–1K–1 Initial temperature of water = 20°C Final temperature of water = 44°C Initial temperature of iron = 180°C (i) Determine the specific heat capacity of iron. (ii) Explain why the value calculated in (b)(i) is likely to be different from the accepted value. Markscheme (i) use of mcΔT; 0.58×c×[180-44]=0.35×4200×[44-20]; c=447Jkg-1K-1≈450Jkg-1K-1; (ii) energy would be given off to surroundings/environment / energy would be absorbed by container / energy would be given off through vaporization of water; hence final temperature would be less; hence measured value of (specific) heat capacity (of iron) would be higher; This question is in two parts. Part 1 is about ideal gases and specific heat capacity. Part 2 is about simple harmonic motion and waves. Part 1 Ideal gases and specific heat capacity State assumptions of the kinetic model of an ideal gas. [2 marks] 2a. two Markscheme point molecules / negligible volume; no forces between molecules except during contact; motion/distribution is random; elastic collisions / no energy lost; obey Newton’s laws of motion; collision in zero time; gravity is ignored; [4 marks] 2b. -
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). -
Thermodynamics of Ideal Gases
D Thermodynamics of ideal gases An ideal gas is a nice “laboratory” for understanding the thermodynamics of a fluid with a non-trivial equation of state. In this section we shall recapitulate the conventional thermodynamics of an ideal gas with constant heat capacity. For more extensive treatments, see for example [67, 66]. D.1 Internal energy In section 4.1 we analyzed Bernoulli’s model of a gas consisting of essentially 1 2 non-interacting point-like molecules, and found the pressure p = 3 ½ v where v is the root-mean-square average molecular speed. Using the ideal gas law (4-26) the total molecular kinetic energy contained in an amount M = ½V of the gas becomes, 1 3 3 Mv2 = pV = nRT ; (D-1) 2 2 2 where n = M=Mmol is the number of moles in the gas. The derivation in section 4.1 shows that the factor 3 stems from the three independent translational degrees of freedom available to point-like molecules. The above formula thus expresses 1 that in a mole of a gas there is an internal kinetic energy 2 RT associated with each translational degree of freedom of the point-like molecules. Whereas monatomic gases like argon have spherical molecules and thus only the three translational degrees of freedom, diatomic gases like nitrogen and oxy- Copyright °c 1998{2004, Benny Lautrup Revision 7.7, January 22, 2004 662 D. THERMODYNAMICS OF IDEAL GASES gen have stick-like molecules with two extra rotational degrees of freedom or- thogonally to the bridge connecting the atoms, and multiatomic gases like carbon dioxide and methane have the three extra rotational degrees of freedom. -
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. -
TEMPERATURE, HEAT, and SPECIFIC HEAT INTRODUCTION Temperature and Heat Are Two Topics That Are Often Confused
TEMPERATURE, HEAT, AND SPECIFIC HEAT INTRODUCTION Temperature and heat are two topics that are often confused. Temperature measures how hot or cold an object is. Commonly this is measured with the aid of a thermometer even though other devices such as thermocouples and pyrometers are also used. Temperature is an intensive property; it does not depend on the amount of material present. In scientific work, temperature is most commonly expressed in units of degrees Celsius. On this scale the freezing point of water is 0oC and its boiling point is 100oC. Heat is a form of energy and is a phenomenon that has its origin in the motion of particles that make up a substance. Heat is an extensive property. The unit of heat in the metric system is called the calorie (cal). One calorie is defined as the amount of heat necessary to raise 1 gram of water by 1oC. This means that if you wish to raise 7 g of water by 4oC, (4)(7) = 28 cal would be required. A somewhat larger unit than the calorie is the kilocalorie (kcal) which equals 1000 cal. The definition of the calorie was made in reference to a particular substance, namely water. It takes 1 cal to raise the temperature of 1 g of water by 1oC. Does this imply perhaps that the amount of heat energy necessary to raise 1 g of other substances by 1oC is not equal to 1 cal? Experimentally we indeed find this to be true. In the modern system of international units heat is expressed in joules and kilojoules. -
5.5 ENTROPY CHANGES of an IDEAL GAS for One Mole Or a Unit Mass of Fluid Undergoing a Mechanically Reversible Process in a Closed System, the First Law, Eq
Thermodynamic Third class Dr. Arkan J. Hadi 5.5 ENTROPY CHANGES OF AN IDEAL GAS For one mole or a unit mass of fluid undergoing a mechanically reversible process in a closed system, the first law, Eq. (2.8), becomes: Differentiation of the defining equation for enthalpy, H = U + PV, yields: Eliminating dU gives: For an ideal gas, dH = and V = RT/ P. With these substitutions and then division by T, As a result of Eq. (5.11), this becomes: where S is the molar entropy of an ideal gas. Integration from an initial state at conditions To and Po to a final state at conditions T and P gives: Although derived for a mechanically reversible process, this equation relates properties only, and is independent of the process causing the change of state. It is therefore a general equation for the calculation of entropy changes of an ideal gas. 1 Thermodynamic Third class Dr. Arkan J. Hadi 5.6 MATHEMATICAL STATEMENT OF THE SECOND LAW Consider two heat reservoirs, one at temperature TH and a second at the lower temperature Tc. Let a quantity of heat | | be transferred from the hotter to the cooler reservoir. The entropy changes of the reservoirs at TH and at Tc are: These two entropy changes are added to give: Since TH > Tc, the total entropy change as a result of this irreversible process is positive. Also, ΔStotal becomes smaller as the difference TH - TC gets smaller. When TH is only infinitesimally higher than Tc, the heat transfer is reversible, and ΔStotal approaches zero. Thus for the process of irreversible heat transfer, ΔStotal is always positive, approaching zero as the process becomes reversible. -
Chapter 3 3.4-2 the Compressibility Factor Equation of State
Chapter 3 3.4-2 The Compressibility Factor Equation of State The dimensionless compressibility factor, Z, for a gaseous species is defined as the ratio pv Z = (3.4-1) RT If the gas behaves ideally Z = 1. The extent to which Z differs from 1 is a measure of the extent to which the gas is behaving nonideally. The compressibility can be determined from experimental data where Z is plotted versus a dimensionless reduced pressure pR and reduced temperature TR, defined as pR = p/pc and TR = T/Tc In these expressions, pc and Tc denote the critical pressure and temperature, respectively. A generalized compressibility chart of the form Z = f(pR, TR) is shown in Figure 3.4-1 for 10 different gases. The solid lines represent the best curves fitted to the data. Figure 3.4-1 Generalized compressibility chart for various gases10. It can be seen from Figure 3.4-1 that the value of Z tends to unity for all temperatures as pressure approach zero and Z also approaches unity for all pressure at very high temperature. If the p, v, and T data are available in table format or computer software then you should not use the generalized compressibility chart to evaluate p, v, and T since using Z is just another approximation to the real data. 10 Moran, M. J. and Shapiro H. N., Fundamentals of Engineering Thermodynamics, Wiley, 2008, pg. 112 3-19 Example 3.4-2 ---------------------------------------------------------------------------------- A closed, rigid tank filled with water vapor, initially at 20 MPa, 520oC, is cooled until its temperature reaches 400oC.