(a) If C^{E} is a constant, independent of T, find expressions for G^E, S^E, and H^E as functions of T.(b) From the equations developed in part (a), find values for G^E, S^E, and H^E for an equimolar solution of benzene(1)/n-hexane(2) at 323.15 K, given the following excess-property values for

Question

(a) If [latex] C_{P}^{E}[/latex] is a constant, independent of T, find expressions for [latex]G^E, S^E, and H^E[/latex] as functions of T.

(b) From the equations developed in part (a), find values for [latex] G^E, S^E, and H^E[/latex] for an equimolar solution of benzene(1)/n-hexane(2) at 323.15 K, given the following excess-property values for an equimolar solution at 298.15 K:

[latex] C_P^E=-2.86 \mathrm{~J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1} \quad H^E=897.9 \mathrm{~J} \cdot \mathrm{mol}^{-1} \quad G^E=384.5 \mathrm{~J} \cdot \mathrm{mol}^{-1} [/latex]

Accepted Answer

(a) Let [latex]C_{P}^{E} = a,[/latex] where a is a constant. From the last column of Table 10.1:

10.1: Summary of Equations for the Gibbs Energy and Related Properties

M in Relation to G	[latex]M^R in Relation to G^R[/latex]	[latex] M^E ext { in Relation to } G^E [/latex]
V = (∂ G / ∂ P )_{T, x} (10.4)	[latex] V^R=\left(\partial G^R / \partial P ight)_{T, x} [/latex]	[latex] V^E=\left(\partial G^E / \partial P ight)_{T, x} [/latex]
[latex]S = − (∂ G / ∂ T )_{P, x}[/latex] (10.5)	[latex] S^R=-\left(\partial G^R / \partial T ight)_{P, x} [/latex]	[latex] S^E=-\left(\partial G^E / \partial T ight)_{P, x} [/latex]
[latex]H = G + TS =G-T(\partial G / \partial T)_{P, x} =-R T^2\left[\frac{\partial(G / R T)}{\partial T} ight]_{P, x} [/latex]	[latex] H^R=G^R+T S^R =G^R-T\left(\partial G^R / \partial T ight)_{P, x} =-R T^2\left[\frac{\partial\left(G^R / R T ight)}{\partial T} ight]_{P, x} [/latex]	[latex] H^E=G^E+T S^E =G^E-T\left(\partial G^E / \partial T ight)_{P, x} =-R T^2\left[\frac{\partial\left(G^E / R T ight)}{\partial T} ight]_{P, x} [/latex]
[latex] C_P=(\partial H / \partial T)_{P, x}=-T\left(\partial^2 G / \partial T^2 ight) P, x [/latex]	[latex] C_P^R=\left(\partial H^R / \partial T ight)_{P, x} =-T\left(\partial^2 G^R / \partial T^2 ight)_{P, x} [/latex]	[latex] C_P^E=\left(\partial H^E / \partial T ight)_{P, x}=-T\left(\partial^2 G^E / \partial T^2 ight)_{P, x} [/latex]

[latex] C_P^E=-T\left(\frac{\partial^2 G^E}{\partial T^2} ight)_{P, x} \quad ext { whence }\left(\frac{\partial^2 G^E}{\partial T^2} ight)_{P, x}=-\frac{a}{T} [/latex]

Integration yields:

[latex] \left(\frac{\partial G^E}{\partial T} ight)_{P, x}=-a \ln T+b [/latex]

where b is a constant of integration. A second integration gives:

[latex] G^E=-a(T \ln T-T)+b T+c [/latex] (A)

where c is another integration constant. With [latex]S^E = − (∂ G^E / ∂ T )_{ P, x} [/latex] (Table 10.1),

[latex]S^E = a ln T − b [/latex] (B)

Because [latex] H^E=G^E+T S^E [/latex] combination of Eqs. (A) and (B) yields:

[latex]H^E = aT + c [/latex] (C)

(b) Let [latex] C_{P_0}^E, H_0^E, ext { and } G_0^E [/latex] represent the given values at [latex]T_0 = 298.15 K. But C_{P}^{E}[/latex] is constant, and therefore [latex] a=C_{P_0}^E=-2.86 \mathrm{~J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1} . [/latex]
By Eq. (A),

[latex] c=H_0^E-a T_0=1750.6 [/latex]

By Eq. (C),

[latex] b=\frac{G_0^E+a\left(T_0 \ln T_0-T_0 ight)-c}{T_0}=-18.0171 [/latex]

Substitution of known values into Eqs. (A), (B), and (C) for T = 323.15 yields:

[latex] G^E=344.4 \mathrm{~J} \cdot \mathrm{mol}^{-1} \quad S^E=1.492 \mathrm{~J} \cdot \mathrm{mol}^{-1} \cdot \mathrm{K}^{-1} \quad H^E=826.4 \mathrm{~J} \cdot \mathrm{mol}^{-1} [/latex]

Question 10.11: (a) If C^{E} is a constant, independent of T, find expressio...

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