Hints for solving the problems

*2. Use eq. (3.2) in the differential form. Divide the work done into two steps, one at constant x, the other at constant y. Expressions for X and Y are thus obtained. When these are differentiated and then subtracted, eq. (P3) is obtained.

*5. Use the equation

obtained from electrodynamics, for the radiation pressure. Then use the expression (§8)

w - ( w ) vi T + \ Q T+ p] i y ,H2> to derive an explicit equation for dQ. Find the integrating factor N(T) for dQ and calculate the efficiency of the Carnot cycle.

6. Start from eq. (4.17) and represent S as a function of T and V. Since dS is a complete differential, the mixed second derivatives must be the same. Equation (P5) is then obtained directly.

10. If T is not a reversible one-one, i.e. strictly monotonic, function of t, there must be at least one pair of values tv t2with tx > t2such that T(tx) = T(t2). Now use the Carnot process to show that eq. (4.14) is not obeyed under these circumstances and that the entropy is there­ fore not a function of state, which contradicts the Second Law.

*11. Let the given process correspond to n Carnot processes. Formulate the statement to be proved and show that it is equivalent to the statement: the sum of the areas of the curved triangles by which the two processes differ vanishes for n-> oo. For the proof, use the argument that the curvatures can be neglected for sufficiently large n and that neighbouring adiabatics may, therefore, be regarded as being parallel.

where ds is the line element and d/ the vectorial area element. The integration on the left-hand side must be carried out over a closed curve, the integral on the right over the area bounded by this curve. Use (3.6) to transform the integral §(dQ/T) stretching over an arbitrary reversible cyclic process in the P-F-plane into the form of the left-hand side of eq. (H3). Now apply Stokes’ theorem and use the continuity properties of thermodynamic functions (homogeneous system!) and eq. (4.14) to show that the integrand vanishes every­ where in the P-F-plane. Equation (4.15) follows immediately.

13. Define efficiency as

Calculate W from the part processes and remember that the heat Q22 is introduced at constant pressure. Eliminate the temperatures with the aid of the equation of state (Pi).

14. Calculate the entropies of the substances in the initial and final state by means of eq. (4.17). Calculate the total entropy change using (4.20) and (4.21). In order to show that AS > 0, first prove the auxiliary

*21. Divide the path into three parts, such that у = const, along the first part, z = const, along the second, and z = const, also along the third.

*23. Consider the change from empirical variables of state s,t measured on an arbitrary scale to the values s*,t* measured on a

Write eqs. (11.5) and (11.8) for the asterisked variables and express the Jacobi determinant of the right-hand side of (11.8) in terms of the Jacobi determinant of the original variables.

*24. Refer to theorem 4 of § 13 and show that the half volumes generated by an adiabatic may be distinguished physically by means of the empirical entropy for the special case of an ideal gas even though they are generally indistinguishable. The sign of Ф, and therefore that of r, is thus clearly fixed for the present case as can be seen from (10.23), (11.10), and (11.20).

(a)Use eq. (10.3) and the definition preceding eq. (13.1),

(b)Use eqs. (10.3)-(10.5) together with the theorem of experience formulated at the end of § 13.

28.

(a) Use the fact that the entropy is a complete differential.

30. First derive the equations of state (20.22) and (20.23) in the entropy representation. Use these to eliminate the intensive para­ meters of the entropy representation

(a)from eq. (20.42).

(b)from eq. (20.46).

32.

(a)Use (P5).

(b)Use (P18).

(c)Use (Pi) and (P6). *37.

(b)Eliminate the volume by means of the approximate expression

(H7)

(H8)

and integrate eq. (25.36) between the ice point T0, or t0, and the unknown thermodynamic temperature T. Since the difference between the ice point and the steam point is 100°, T0 can be elimi­ nated by means of the integral to this steam point.

*51.

(a)In order to obtain comparable forms, introduce the pressure change dP corresponding to the volume change dv in the JouleThomson experiment.

(b)Derive the expression for the Joule-Thomson coefficient from eqs. (21.15) and (20.20).

(c)Use (25.9), (24.24), and (25.36).

Remember for purposes of comparison that, in general, [d(Pv)jdP]h > 0. *57. Use (25.3) and (26.16).

*61.

(a)Derive the analogues to the Clausius-Clapeyron equation for the pairs of variables /z/Р, 1 \T and /х, T. Remember that the finite difference quotients on the right-hand sides become differential quotients for neighbouring phases.

(b) Apply the condition pjT = 0 and the result of (a) to the infinite number of Gibbs-Duhem equations.

*70. Put

in the form of eq. (P3) valid for quasi-static processes. Remember that the saturation pressure is independent of n. Use (A 117) and the Clausius-Clapeyron equation.

*71. Introduce the standard pressure P+ and eliminate P0/P0 by means of (31.5), (31.7), and (A50).

*72. Start with the differential form of the equilibrium conditions (27.8) and represent the chemical potentials as functions of T, P, and the xt. The derivatives of /x2 with respect to the mole fractions are eliminated by using (26.16).

*74. Derive first a differential equation for the change in composi­ tion of the azeotropic mixture with temperature. Start with the equations mentioned in the hints for problem 72 and use the condition

for the azeotropic point. Find the second derivative of the isobaric boiling point curve at the point (H10) and thus deduce a criterion which shows whether the boiling point curve has a maximum or a minimum at the azeotropic point. Simplify the differential equation derived first by introducing the partial heats of vaporization

Z ,i = Ai-7*I>0, L2 = h'2-h l> 0 (H ll)

and by using the assumptions mentioned in the problem.

*81. Consider on the one hand a reaction at the temperature T followed by heating of the reaction products to the temperature T + dP and, on the other, heating of the reactants from T to T + dT followed by reaction at P + dP. Take the work done into considera­ tion for the cases (b) and (c). In case (c), replace the pressure change by a constant mean value. Use the progress variable f defined by eq. (14.3) as the variable for the chemical reaction.

*82. In the form of eq. (P3) valid for quasi-static processes, put

У = Т, X = Q , x = £

(H12)

where the summation is to be taken over the participants in the reaction with the sign convention of eq. (33.15), and the denote the molar heat capacities for the conditions given in problem 81.

*85.

(a)Start with eq. (A31) with x = P. Remember that (dujdv)T Ф 0 for the dissociating ideal gas. Calculate (docldv)T from (A143) and (dvldT)p from the thermal equation of state of the dissociating gas, (A142), and (34.29). Take (34.30) and (34.31) into consideration.

(b)Use (A146) and (A150).

98.First carry out the calculation for V = const, and use the general theory to show that the result is also valid for P = const.

*99. Represent 82u as a sum of squares by the method of

completion of squares.

*100. Apply the method of problem 99 to the general case and

variables.

*110. A tetragonal crystal has a four-fold axis of symmetry. If this is chosen as the z-axis, the transformation of coordinates correspond­ ing to this symmetry property is

—y->x, z->z.

The derivation is analogous to that of § 56 for a monoclinic crystal. *111. A cubic crystal has three mutually perpendicular four-fold

axes of symmetry.

the matrix A the cofactor of Ait from the determinant |A |and then transposing the matrix so obtained. Remember that \c\ is a step determinant.

113.Express the elastic stiffness coefficients in terms of the elastic compliance coefficients (i.e. reverse problem 112).

114.Use (52.25), (55.12), and (56.26).

115.Use eq. (55.2).

116. Useeq. (55.5).

Put the variables in a sequence such that the temperature comes last.

117. Use eq. (55.2).

*118.

(b)The energy of the electric field in the dielectric must be included in the fundamental equation in order to obtain any statement about the dielectric constant. The second term of (57.19) must therefore be introduced into the fundamental equation and the stability conditions must then be applied.

Use a thermodynamic potential analogous to (59.4).

121.

(a)Express the chemical potential of component 1 as a function of T , P, E, and xv Use the differential form of the equilibrium condition (27.8).

(b)Find the limiting value of

dzj ldx[ dE/dE

for

x[ —x\-> 0.

126. Use eqs. (67.14) and (67.15).