# 8.9: Putting the Second Law to Work (Exercises)

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- Contributed by Patrick Fleming
- Assistant Professor (Chemistry and Biochemistry) at California State University East Bay

## Q6.1

Using Table T1, calculate the standard reaction Gibbs functions (\(\Delta G^o\)) for the following reactions at 298 K.

- \(CH_3CH_2OH(l) + 3 O_2(g) \rightarrow 2 CO_2(g) + 3 H_2O(l)\)
- \(C_6H_{12}O_6(s) + 6 O_2 \rightarrow 6 CO_2(g) + 6 H_2O(l)\)
- \(2 POCl_3(l) \rightarrow 2 PCl_3(l) + O_2(g)\)
- \(2 KBr(s) + Cl_2(g) \rightarrow 2 KCl(s) + Br_2(l)\)
- \(SiH_4(g) + 2 Cl(g) \rightarrow SiCl_4(l) + 2 H_2(g)\)

## Q6.2

Estimate \(\Delta G\) at 1000 K from its value at 298 K for the reaction

\[C(s) + 2 H_2(g) \rightarrow CH_4(g)\]

with \(\Delta G = -50.75\, kJ \,at\, 298\, K\)/

## Q6.3

The standard Gibbs function for formation (\(\Delta G_f^o\)) of \(PbO_2(s)\) is -217.4 kJ/mol at 298 K. Assuming \(O_2\) is an ideal gas, find the standard Helmholtz function for formation (\(\Delta A_f^o\) for \(PbO_2\) at 298K.

## Q6.4

Calculate the entropy change for 1.00 mol of an ideal monatomic gas (C_{V} = 3/2 R) undergoing an expansion and simultaneous temperature increase from 10.0 L at 298 K to 205.0 L at 455 K.

## Q6.5

Consider a gas that obeys the equation of state

\[ p =\dfrac{nRT}{V-nb}\]

- Find expressions for \(\alpha\) and \(\kappa_T\) for this gas.
- Evaluate the difference between \(C_p\) and \(C_V\) for the gas.

## Q6.6

Show that

\[\left( \dfrac{\partial C_p}{\partial p} \right)_T=0\]

for an ideal gas.

## Q6.7

Derive the thermodynamic equation of state

\[\left( \dfrac{\partial H}{\partial p} \right)_T = V( 1- T \alpha)\]

## Q6.8

Derive the thermodynamic equation of state

\[\left( \dfrac{\partial U}{\partial V} \right)_T = T \dfrac{ \alpha}{\kappa_T} -p\]

## Q6.9

The “Joule Coefficient” is defined by

\[ \mu_J = \left( \dfrac{\partial T}{\partial V} \right)_U \]

Show that

\[ \mu_J = \dfrac{1}{C_V} \left( p - \dfrac{T \alpha}{\kappa_T }\right)\]

and evaluate the expression for an ideal gas.

## Q6.10

Derive expressions for the pressure derivatives

\[ \left( \dfrac{\partial X}{\partial p} \right)_T\]

where\(X\) is \(U\), \(H\), \(A\), \(G\), and \(S\) at constant temperature in terms of measurable properties. (The derivation of \( \left( \dfrac{\partial H}{\partial p} \right)_T\) was done in problem Q6.7).

Evaluate the expressions for

- \( \left( \dfrac{\partial S}{\partial p} \right)_T\)
- \( \left( \dfrac{\partial H}{\partial p} \right)_T\)
- \( \left( \dfrac{\partial U}{\partial p} \right)_T\)

for a van der Waals gas.

## Q6.11

Derive expressions for the volume derivatives

\[ \left( \dfrac{\partial X}{\partial V} \right)_T\]

where \(X\) is \(U\), \(H\), \(A\), \(G\), and \(S\) at constant temperature in terms of measurable properties. (The derivation of \( \left( \dfrac{\partial U}{\partial V} \right)_T\) was done in problem Q8.8.)

Evaluate the expressions for

- \( \left( \dfrac{\partial X}{\partial V} \right)_T\)
- \( \left( \dfrac{\partial X}{\partial V} \right)_T\)

for a van der Waals gas.

## Q6.12

Evaluate the difference between \(C_p\) and \(C_V\) for a gas that obeys the equation of state

\[ p =\dfrac{nRT}{V-nb}\]

## Q6.13

The adiabatic compressibility (\(k_S\)) is defined by

\[ \kappa_S = \dfrac{1}{V} \left( \dfrac{\partial V}{\partial p} \right)_S\]

Show that for an ideal gas,

\[ \kappa_S = \dfrac{1}{p \gamma}\]