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

## Q6.1

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

1. $$CH_3CH_2OH(l) + 3 O_2(g) \rightarrow 2 CO_2(g) + 3 H_2O(l)$$
2. $$C_6H_{12}O_6(s) + 6 O_2 \rightarrow 6 CO_2(g) + 6 H_2O(l)$$
3. $$2 POCl_3(l) \rightarrow 2 PCl_3(l) + O_2(g)$$
4. $$2 KBr(s) + Cl_2(g) \rightarrow 2 KCl(s) + Br_2(l)$$
5. $$SiH_4(g) + 2 Cl(g) \rightarrow SiCl_4(l) + 2 H_2(g)$$

## 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 (CV = 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

$p =\dfrac{nRT}{V-nb}$

1. Find expressions for $$\alpha$$ and $$\kappa_T$$ for this gas.
2. 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,

$\kappa_S = \dfrac{1}{p \gamma}$