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8.9: Putting the Second Law to Work (Exercises)

  • Page ID
    199226
  • 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,

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

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