2.9 Problems for This Chapter
127
K P (T ), is given in terms of the Gibbs energy of formation, f G
◦ , for gaseous
water from elemental hydrogen and oxygen as K P (T ) = exp{−2 f G
◦ /(RT )}.
39. Use the expression for the affinity as a function of the extent of reaction, ξ , for
the formation of water from hydrogen and oxygen (cf. Problem 37) to determine
the equilibrium value, ξ e , for the extent of reaction for temperatures T = 2000
and 5000 K for reaction mixtures for which the total pressure is maintained at
1 bar. Employ these equilibrium values of ξ to calculate the partial pressures of
the three components, namely, P H 2 O (ξ e ), P H 2 (ξ e ), P O 2 (ξ e ), in the equilibrium
mixtures, and confirm that the values thereby obtained give K P (T ) values that
are, as they should be, consistent with the values of K P (T ) obtained using the
Gibbs energy of formation expression. The Gibbs energy of formation for H 2 O
has values f G
◦ = −135.528 kJ mol
−1 at T = 2000 K and 40.949 kJ mol
−1 at
5000 K.
40. At temperature T = 500 K both Br 2 and I 2 are gases, and react to form the
interhalogen IBr(g) according to the chemical reaction
Br 2 (g) + I 2 (g) 2IBr(g) .
Obtain an expression for the Gibbs energy and its first derivative as functions
of the extent of reaction ξ for a starting mixture of one mole each of Br 2 (g)
and I 2 (g) and no IBr(g). Given that G
◦
IBr = −8.748 kJ mol
−1 at 500 K, plot
G(ξ ) as a function of ξ , obtain ξ e for this reaction, and evaluate the equilibrium
constant K P . Determine the composition of the equilibrium mixture of Br 2 (g),
I 2 (g), and IBr(g) at 500 K.
41. According to Eq. (2.2.12), which is the expression obtained upon combining
the First and Second Laws of thermodynamics, the internal energy U is a
function of entropy, S, and volume, V . For an ideal monatomic gas, for which
only translational energy states are thermally accessible, use Eqs. (2.3.13b)
and (2.3.14) to obtain an expression giving the translational entropy as an
explicit function of T , V , and N (a result that was first obtained by purely
thermodynamic reasoning by Sackur and Tetrode). Show that S trans (T , V , N)
may be rewritten as a function of U trans , V , and N and that, by inverting it,
the internal energy U trans can be expressed as an explicit function of S trans , V ,
and N .
42. Employ the expression obtained in Problem 41 for U trans (S, V , N) for an
ideal monatomic gas and the extended First and Second Laws expression
dU(S, V , N) = T dS −P dV +μ dN for a pure substance to obtain expressions
for P (S, V , N), T (S, V , N), and the chemical potential μ trans (S, V , N). Show
that the expressions thereby obtained for P , T , and μ are consistent, respectively, with the equation of state, the expression U trans =
3
2 Nk B T for an ideal
gas, and the chemical potential, μ, as the Gibbs energy per particle for a pure
substance.
127
K P (T ), is given in terms of the Gibbs energy of formation, f G
◦ , for gaseous
water from elemental hydrogen and oxygen as K P (T ) = exp{−2 f G
◦ /(RT )}.
39. Use the expression for the affinity as a function of the extent of reaction, ξ , for
the formation of water from hydrogen and oxygen (cf. Problem 37) to determine
the equilibrium value, ξ e , for the extent of reaction for temperatures T = 2000
and 5000 K for reaction mixtures for which the total pressure is maintained at
1 bar. Employ these equilibrium values of ξ to calculate the partial pressures of
the three components, namely, P H 2 O (ξ e ), P H 2 (ξ e ), P O 2 (ξ e ), in the equilibrium
mixtures, and confirm that the values thereby obtained give K P (T ) values that
are, as they should be, consistent with the values of K P (T ) obtained using the
Gibbs energy of formation expression. The Gibbs energy of formation for H 2 O
has values f G
◦ = −135.528 kJ mol
−1 at T = 2000 K and 40.949 kJ mol
−1 at
5000 K.
40. At temperature T = 500 K both Br 2 and I 2 are gases, and react to form the
interhalogen IBr(g) according to the chemical reaction
Br 2 (g) + I 2 (g) 2IBr(g) .
Obtain an expression for the Gibbs energy and its first derivative as functions
of the extent of reaction ξ for a starting mixture of one mole each of Br 2 (g)
and I 2 (g) and no IBr(g). Given that G
◦
IBr = −8.748 kJ mol
−1 at 500 K, plot
G(ξ ) as a function of ξ , obtain ξ e for this reaction, and evaluate the equilibrium
constant K P . Determine the composition of the equilibrium mixture of Br 2 (g),
I 2 (g), and IBr(g) at 500 K.
41. According to Eq. (2.2.12), which is the expression obtained upon combining
the First and Second Laws of thermodynamics, the internal energy U is a
function of entropy, S, and volume, V . For an ideal monatomic gas, for which
only translational energy states are thermally accessible, use Eqs. (2.3.13b)
and (2.3.14) to obtain an expression giving the translational entropy as an
explicit function of T , V , and N (a result that was first obtained by purely
thermodynamic reasoning by Sackur and Tetrode). Show that S trans (T , V , N)
may be rewritten as a function of U trans , V , and N and that, by inverting it,
the internal energy U trans can be expressed as an explicit function of S trans , V ,
and N .
42. Employ the expression obtained in Problem 41 for U trans (S, V , N) for an
ideal monatomic gas and the extended First and Second Laws expression
dU(S, V , N) = T dS −P dV +μ dN for a pure substance to obtain expressions
for P (S, V , N), T (S, V , N), and the chemical potential μ trans (S, V , N). Show
that the expressions thereby obtained for P , T , and μ are consistent, respectively, with the equation of state, the expression U trans =
3
2 Nk B T for an ideal
gas, and the chemical potential, μ, as the Gibbs energy per particle for a pure
substance.