REVIEW PROBLEMS
4.10 SUMMARY
The Helmholtz energy A and the Gibbs energy G are state functions defined by A⬅ U TS and G ⬅ H TS. The condition that the total entropy of system plus sur-roundings be maximized at equilibrium leads to the condition that A or G of a closed system with only P-V work be minimized if equilibrium is reached in a system held at fixed T and V or fixed T and P, respectively.
The first law dU dq dw combined with the second-law expression dqrev T dS gives dU T dS P dV (the Gibbs equation for dU) for a reversible change in a closed system with P-V work only. This equation, the definitions H⬅ U PV, A ⬅ U TS, G⬅ H TS, and the heat-capacity equations CP P Pand CV ( V Vare the basic equations for a closed system in equilibrium.
Section 4.10 Summary 135 lev38627_ch04.qxd 2/29/08 3:13 PM Page 135
Chapter 4 Material Equilibrium
136
From the Gibbs equations for dU, dH, and dG, expressions for the variations in U, H, and G with respect to T, P, and V were found in terms of the readily measured prop-erties CP, a, and k. Application of the Euler reciprocity relation to dG S dT V dP gives ( T P; ( Tis found similarly from the Gibbs equa-tion for dA. These relaequa-tions allow calculaequa-tion of U, H, and S for arbitrary changes of state.
For a system (open or closed) in mechanical and thermal equilibrium with P-V work only, one has dG S dT V dP a imiadnai, where the chemical poten-tial of substance i in phase a is defined as mia⬅ This expression for dG applies during an irreversible chemical reaction or transport of matter between phases.
The condition for equilibrium between phases is that, for each substance i, the chemical potential mimust be the same in every phase in which i is present: mai mib. The condition for reaction equilibrium is that inimi 0, where the ni’s are the reac-tion’s stoichiometric numbers, negative for reactants and positive for products. The chemical potentials are the key properties in chemical thermodynamics, since they determine phase and reaction equilibrium.
Important kinds of calculations dealt with in this chapter include:
• Calculation of U, H, and S for changes in system temperature and pressure and calculation of G and A for isothermal processes (Sec. 4.5).
• Calculation of CP CV, ( T, ( T, ( P, ( T, etc., from read-ily measured properties (CP, a, k) (Sec. 4.4).
Although this has been a long, mathematical chapter, it has presented concepts and results that lie at the heart of chemical thermodynamics and that will serve as a foun-dation for the remaining thermodynamics chapters.
FURTHER READING
Zemansky and Dittman, chaps. 9, 14; Denbigh, chap. 2; Andrews (1971), chaps. 13, 15, 20, 21; Van Wylen and Sonntag, chap. 10; Lewis and Randall, App. 6; McGlashan, chaps. 6, 8.
10Ga>0nia2T,P,naji.
Section 4.3
4.1 True or false? (a) The quantities U, H, A, and G all have the same dimensions. (b) The relation G H T S is valid for all processes. (c) G A PV. (d) For every closed system in thermal and mechanical equilibrium and capable of only P-V work, the state function G is minimized when material equilibrium is reached. (e) The Gibbs energy of 12 g of ice at 0°C and 1 atm is less than the Gibbs energy of 12 g of liquid water at 0°C and 1 atm. ( f ) The quantities S dT, T dS, V dP, and 兰12V dP all have dimensions of energy.
4.2 Calculate G, A, and Sunivfor each of the following processes and state any approximations made: (a) reversible melting of 36.0 g of ice at 0°C and 1 atm (use data from Prob.
2.49); (b) reversible vaporization of 39 g of C6H6at its normal
boiling point of 80.1°C and 1 atm; (c) adiabatic expansion of 0.100 mol of a perfect gas into vacuum (Joule experiment) with initial temperature of 300 K, initial volume of 2.00 L, and final volume of 6.00 L.
Section 4.4
4.3 Express each of the following rates of change in terms of state functions. (a) The rate of change of U with respect to tem-perature in a system held at constant volume. (b) The rate of change of H with respect to temperature in a system held at con-stant pressure. (c) The rate of change of S with respect to tem-perature in a system held at constant pressure.
4.4 The relation ( V T [Eq. (4.37)] is notable because is relates the three fundamental thermodynamic state functions
PROBLEMS
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137 U, S, and T. The reciprocal of this relation, ( V 1/T,
shows that entropy always increases when internal energy in-creases at constant volume. Use the Gibbs equation for dU to show that ( U P/T.
4.5 Verify the Maxwell relations (4.44) and (4.45).
4.6 For water at 30°C and 1 atm, use data preceding Eq. (4.54) to find (a) ( T; (b) mJT.
4.7 Given that, for CHCl3at 25°C and 1 atm, r 1.49 g/cm3, CP,m 116 J/(mol K), a 1.33 103K1, and k 9.8 105atm1, find CV,mfor CHCl3at 25°C and 1 atm.
4.8 For a liquid with the typical values a 103K1, k 104atm1, Vm 50 cm3/mol, CP,m 150 J/mol-K, calculate at 25°C and 1 atm (a) (Hm/T )P; (b) (Hm/P)T; (c) ( T; (d ) (Sm/T )P; (e) (Sm/P)T; ( f ) CV,m; (g) ( T.
4.9 Show that ( T TVa PVk (a) by starting from the Gibbs equation for dU; (b) by starting from (4.47) for
( T.
4.10 Show that ( P CP PVa (a) by starting from dU T dS P dV; (b) by substituting (4.26) into (4.30).
4.11 Starting from dH T
aT/k 1/k.
4.12 Consider solids, liquids, and gases not at high pressure.
For which of these is CP,m CV,musually largest? Smallest?
4.13 Verify that [ P H/T2. This is the Gibbs–Helmholtz equation.
4.14 Derive the equations in (4.31) for ( Pand ( V from the Gibbs equations (4.33) and (4.34) for dU and dH.
4.15 Show that mJ (P aTk1)/CV, where mJis the Joule coefficient.
4.16 A certain gas obeys the equation of state PVm RT(1 bP), where b is a constant. Prove that for this gas (a) ( T bP2; (b) CP,m CV,m R(1 bP)2; (c) mJT 0.
4.17 Use Eqs. (4.30), (4.42), and (4.48) to show that (CP/P)T T (2V/T2)P. The volumes of substances in-crease approximately linearly with T, so 2V/T2 is usually quite small. Consequently, the pressure dependence of CPcan usually be neglected unless one is dealing with high pressures.
4.18 The volume of Hg in the temperature range 0°C to 100°C at 1 atm is given by V V0(1 at bt2), where a 0.18182 103°C1, b 0.78 108°C2, and where V0is the vol-ume at 0°C and t is the Celsius temperature. The density of mercury at 1 atm and 0°C is 13.595 g/cm3. (a) Use the result of Prob. 4.17 to calculate (CP,m/P)Tfor Hg at 25°C and 1 atm.
(b) Given that CP,m 6.66 cal mol1K1for Hg at 1 atm and 25°C, estimate CP,mof Hg at 25°C and 104atm.
4.19 For a liquid obeying the equation of state Vm c1 c2T
c3T2 c4P c5PT [Eq. (1.40)], find expressions for each of the following properties in terms of the c’s, CP, P, T, and V:
(a) CP CV; (b) ( T; (c) ( T; (d ) mJT; (e) ( P; ( f ) ( T.
4.20 A reversible adiabatic process is an isentropic (constant-entropy) process. (a) Let aS ⬅ V1( S. Use the first Maxwell equation in (4.44) and Eqs. (1.32), (1.35), and (4.31) to show that aS CVk/T Va. (b) Evaluate aS for a perfect gas. Integrate the result, assuming that CVis constant, and ver-ify that you obtain Eq. (2.76) for a reversible adiabatic process in a perfect gas. (c) The adiabatic compressibility is kS ⬅
V1( S. Starting from ( S S( S,
prove that kS CVk/CP.
4.21 Since all ideal gases are perfect (Sec. 4.4) and since for a
perfect gas ( T T
0 for an ideal gas. Verify this directly from (4.48).
4.22 This problem finds an approximate expression for Uintermol, the contribution of intermolecular interactions to U. As the volume V changes at constant T, the average distance be-tween molecules changes and so the intermolecular interaction energy changes. The translational, rotational, vibrational, and electronic contributions to U depend on T but not on V (Sec. 2.11). Infinite volume corresponds to infinite average distance between molecules and hence to Uintermol 0.
Therefore U(T, V) U(T, ) Uintermol(T, V). (a) Verify that Uintermol(T, V) 兰V ( TdV, where the integration is at constant T, and V is some particular volume. (b) Use (4.57) to show that for a van der Waals gas Uintermol,m a/Vm. (This is only a rough approximation since it omits the effects of inter-molecular repulsions, which become important at high densi-ties.) (c) For small to medium-size molecules, the van der Waals a values are typically 106to 107 cm6 atm mol2(Sec. 8.4).
Calculate the typical range of Uintermol, min a gas at 25°C and 1 atm. Repeat for 25°C and 40 atm.
4.23 (a) For liquids at 1 atm, the attractive intermolecular forces make the main contribution to Uintermol. Use the van der Waals expression in Prob. 4.22b and the van der Waals a value of 1.34 106cm6atm mol2for Ar to show that for liquid or gaseous Ar,
(b) Calculate the translational and intermolecular energies in liq-uid and in gaseous Ar at 1 atm and 87.3 K (the normal boiling point). The liquid’s density is 1.38 g/cm3at 87 K. (c) Estimate
Umfor the vaporization of Ar at its normal boiling point and compare the result with the experimental value 5.8 kJ/mol.
Section 4.5
4.24 True or false? (a) G is undefined for a process in which T changes. (b) G 0 for a reversible phase change at constant T and P.
4.25 Calculate G and A when 2.50 mol of a perfect gas with CV,m 1.5R goes from 28.5 L and 400 K to 42.0 L and 400 K.
4.26 For the processes of Probs. 2.45a, b, d, e, and f, state whether each of A and G is positive, zero, or negative.
4.27 Calculate A and G when a mole of water vapor ini-tially at 200°C and 1 bar undergoes a cyclic process for which w 145 J.
112.5 J>mol-K2T const.
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138
4.28 (a) Find G for the fusion of 50.0 g of ice at 0°C and 1 atm. (b) Find G for the supercooled-water freezing process of Prob. 3.14.
4.29 Find A and G when 0.200 mol of He(g) is mixed at constant T and P with 0.300 mol of O2(g) at 27°C. Assume ideal gases.
4.30 Suppose 1.00 mol of water initially at 27°C and 1 atm undergoes a process whose final state is 100°C and 50 atm. Use data given preceding Eq. (4.54) and the approximation that the temperature and pressure variations of a, k, and CP can be neglected to calculate: (a) H; (b) U; (c) S.
4.31 Calculate G for the isothermal compression of 30.0 g of water from 1.0 atm to 100.0 atm at 25°C; neglect the varia-tion of V with P.
4.32 A certain gas obeys the equation of state PVm RT(1 bP cP2), where b and c are constants. Find expres-sions for Hmand Smfor a change of state of this gas from (P1, T1) to (P2, T2); neglect the temperature and pressure depen-dence of CP,m.
4.33 If 1.00 mol of water at 30.00°C is reversibly and adiabat-ically compressed from 1.00 to 10.00 atm, calculate the final volume by using expressions from Prob. 4.20 and neglecting the temperature and pressure variation in kS. Next calculate the final temperature. Then use the first law and the ( S ex-pression in Prob. 4.20 to calculate U; compare the result with the approximate answer of Prob. 2.47. See Eq. (4.54) and data preceding it.
4.34 Use a result of the example after Eq. (4.55) to derive an expression for U for a gas obeying the van der Waals equation and undergoing a change of state.
Section 4.6
4.35 True or false? (a) The chemical potential miis a state function. (b) miis an intensive property. (c) miin a phase must remain constant if T, P, and xiremain constant in the phase.
(d ) The SI units of miare J/mol. (e) The definition of mifor a single-phase system is mi ( f ) The chemical potential of pure liquid acetone at 300 K and 1 bar equals Gmof liquid acetone at 300 K and 1 bar. (g) The chemical potential of benzene in a solution of benzene and toluene at 300 K and 1 bar must be equal to Gmof pure benzene at 300 K and 1 bar.
4.36 Show that
4.37 Use Eq. (4.75) to show that dq T dS imidnifor a one-phase closed system with P-V work only in mechanical and thermal equilibrium. This expression gives dq during a chemi-cal reaction. Since the reaction is irreversible, dq T dS.
Section 4.7
4.38 True or false? (a) The chemical potential of benzene in a solution of benzene and toluene must equal the chemical poten-tial of toluene in that solution. (b) The chemical potenpoten-tial of su-crose in a solution of susu-crose in water at 300 K and 1 bar must 10A>0ni2T, V, nji.
mi10U>0ni2S,V,nji10H>0ni2S,P, nji 10Gi>0ni2T,P,nji.
equal the molar Gibbs energy of solid sucrose at 300 K and 1 bar. (c) The chemical potential of sucrose in a saturated solu-tion of sucrose in water at 300 K and 1 bar must equal the molar Gibbs energy of solid sucrose at 300 K and 1 bar. (d ) If phases a and b are in equilibrium with each other, the chemical poten-tial of phase a must equal the chemical potenpoten-tial of phase b.
4.39 For each of the following closed systems, write the con-dition(s) for material equilibrium between phases: (a) ice in equilibrium with liquid water; (b) solid sucrose in equilibrium with a saturated aqueous solution of sucrose; (c) a two-phase system consisting of a saturated solution of ether in water and a saturated solution of water in ether; (d ) ice in equilibrium with an aqueous solution of sucrose. (e) Solid sucrose and solid glu-cose in equilibrium with an aqueous solution of these two solids.
4.40 For each of the following pairs of substances, state which substance, if either, has the higher chemical potential:
(a) H2O(l) at 25°C and 1 atm vs. H2O(g) at 25°C and 1 atm;
(b) H2O(s) at 0°C and 1 atm vs. H2O(l) at 0°C and 1 atm;
(c) H2O(s) at 5°C and 1 atm vs. supercooled H2O(l) at 5°C and 1 atm; (d ) C6H12O6(s) at 25°C and 1 atm vs. C6H12O6(aq) in an unsaturated aqueous solution at 25°C and 1 atm;
(e) C6H12O6(s) at 25°C and 1 atm vs. C6H12O6(aq) in a saturated solution at 25°C and 1 atm; ( f ) C6H12O6(s) at 25°C and 1 atm vs. C6H12O6(aq) in a supersaturated solution at 25°C and 1 atm. (g) Which substance in (a) has the higher Gm?
4.41 Show that for ice in equilibrium with liquid water at 0°C and 1 atm the condition of equality of chemical potentials is equivalent to G 0 for H2O(s) → H2O(l).
Section 4.8
4.42 Give the value of the stoichiometric number n for each species in the reaction C3H8(g) 5O2(g)→ 3CO2(g) 4H2O(l).
4.43 Write the reaction equilibrium condition for N2 3H2∆
2NH3in a closed system.
4.44 Suppose that in the reaction 2O3→ 3O2, a closed system initially contains 5.80 mol O2and 6.20 mol O3. At some later time, 7.10 mol of O3is present. What is j at this time?
General
4.45 For H2O(s) at 0°C and 1 atm and H2O(l) at 0°C and 1 atm, which of the following quantities must be equal for the two phases? (a) Sm; (b) Um; (c) Hm; (d ) Gm; (e) m; ( f ) Vm. 4.46 Consider a two-phase system that consists of liquid water in equilibrium with water vapor; the system is kept in a constant-temperature bath. (a) Suppose we reversibly increase the system’s volume, holding T and P constant, causing some of the liquid to vaporize. State whether each of H, S, Suniv, and
G is positive, zero, or negative. (b) Suppose we suddenly re-move some of the water vapor, holding T and V constant. This reduces the pressure below the equilibrium vapor pressure of water, and liquid water will evaporate at constant T and V until the equilibrium vapor pressure is restored. For this evaporation lev38627_ch04.qxd 2/29/08 3:13 PM Page 138
139 process state whether each of U, S, Suniv, and A is
posi-tive, zero, or negative.
4.47 For each of the following processes, state which of U,
H, S, Suniv, A, and G must be zero. (a) A nonideal gas undergoes a Carnot cycle. (b) Hydrogen is burned in an adia-batic calorimeter of fixed volume. (c) A nonideal gas undergoes a Joule–Thomson expansion. (d ) Ice is melted at 0°C and 1 atm.
4.48 Give an example of a liquid with a negative ( T. 4.49 Give the name of each of these Greek letters and state the thermodynamic quantity that each stands for: (a) n; (b) m; (c) j;
(d ) a; (e) k; ( f ) r.
4.50 Give the conditions of applicability of each of these equations: (a) dU dq dw; (b) dU T dS P dV; (c) dU T dS P dV i amai dnia.
4.51 Give the SI units of (a) G; (b) Sm; (c) CP; (d ) mi. 4.52 For a closed system with P-V work only, (a) write the equation that gives the condition of phase equilibrium; (b) write the equation that gives the condition of reaction equilibrium.
(c) Explain why dG 0 is not the answer to (a) and (b).
4.53 For a closed system with P-V work only and held at con-stant T and P, show that dS dq/T dG/T for an irreversible material change. (Hint: Start with G⬅ H TS.)
4.54 An equation for Gm of a pure substance as a function of T and P (or of Am as a function of T and V ) is called a fundamental equation of state. From a fundamental equation of state, one can calculate all thermodynamic properties of a substance. Express each of the following properties in terms of Gm, T, P, (Gm/T )P, (Gm/P)T, (2Gm/T2)P, (2Gm/P2)T, and 2Gm/ m; (b) Vm; (c) Hm; (d ) Um; (e) CP,m; ( f ) CV ,m; (g) a; (h) k. [Using Eqs. like (4.60) and (4.63) for H and
S and experimental CP, a, and k data, one can construct a fun-damental equation of state of the form Gm f(T, P), where U and S have each been arbitrarily assigned a value of zero in some reference state, which is usually taken as the liquid at the triple point. Accurate fundamental equations of state have been constructed for several fluids. For fluid H2O, fundamental equa-tions of state contain about 50 parameters whose values are ad-justed to give good fits of experimental data; see A. Saul and W.
Wagner, J. Phys. Chem. Ref. Data, 18, 1537 (1989); P. G. Hill, ibid., 19, 1233 (1990).]
4.55 When 3.00 mol of a certain gas is heated reversibly from 275 K and 1 bar to 375 K and 1 bar, S is 20.0 J/K. If 3.00 mol of this gas is heated irreversibly from 275 K and 1 bar to 375 K and 1 bar, will S be less than, the same as, or greater than 20.0 J/K?
4.56 For each of the following sets of quantities, all the quan-tities except one have something in common. State what they have in common and state which quantity does not belong with the others. (In some cases, more than one answer for the prop-erty in common might be possible.) (a) CV, CP, U, T, S, G, A, V;
(b) H, U, G, S, A.
4.57 For each of the following statements, tell which state function(s) is (are) being described. (a) It enables one to find the rates of change of enthalpy and of entropy with respect to tem-perature at constant pressure. (b) They determine whether sub-stance i in phase a is in phase equilibrium with i in phase b.
(c) It enables one to find the rates of change of U and of S with respect to T at constant V. (d ) It is maximized when an isolated system reaches equilibrium. (e) It is maximized when a system reaches equilibrium. ( f ) It is minimized when a closed system capable of P-V work only and held at constant T and P reaches equilibrium.
4.58 True or false? (a) CP ,m CV,m R for all gases. (b) CP
CV TVa2/k for every substance. (c) G is always zero for a reversible process in a closed system capable of P-V work only. (d ) The Gibbs energy of a closed system with P-V work only is always minimized at equilibrium. (e) The work done by a closed system can exceed the decrease in the system’s internal energy. ( f ) For an irreversible, isothermal, isobaric process in a closed system with P-V work only, G must be negative. (g) Gsyst
Gsurris constant for any process. (h) S is positive for every irreversible process. (i) Ssyst Ssurris positive for every ir-reversible process. ( j) (TS) S T T S. (k) (U TS)
P V/T for a constant-pressure process. (m) If a system remains in thermal and mechanical equilibrium during a process, then its T and P are constant dur-ing the process. (n) The entropy S of a closed system with P-V work only is always maximized at equilibrium. (o) If a b, then we must have ka kb, where k is a nonzero constant.
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Standard
Thermodynamic Functions of
Reaction
For the chemical reaction aA bB ∆cC dD, we found the condition for reaction equilibrium to be amA bmB cmC dmD[Eq. (4.98)]. To effectively apply this con-dition to reactions, we will need tables of thermodynamic properties (such as G, H, and S) for individual substances. The main topic of this chapter is how one uses ex-perimental data to construct such tables. In these tables, the properties are for sub-stances in a certain state called the standard state, so this chapter begins by defining the standard state (Sec. 5.1). From tables of standard-state thermodynamic properties, one can calculate the changes in standard-state enthalpy, entropy, and Gibbs energy for chemical reactions. Chapters 6 and 11 show how equilibrium constants for reactions can be calculated from such standard-state property changes.