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 (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
\[ p =\dfrac{nRT}{V-nb}\]
- 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,
\[ \kappa_S = \dfrac{1}{p \gamma}\]
