One of the most useful applications of thermodynamics is predicting how the Gibbs function changes with temperature.
Standard Gibbs functions are typically tabulated at \(298\ \mathrm{K}\), but many reactions occur at temperatures far from room temperature. It is therefore useful to derive an expression that allows Gibbs functions to be estimated at other temperatures.
This expression is known as the Gibbs–Helmholtz equation.
Begin with the definition of the Gibbs function:
\[ G = H - TS \]
Dividing by temperature gives
\[ \frac{G}{T} = \frac{H}{T} - S \]
Differentiate both sides with respect to temperature at constant pressure:
\[ \left( \frac{\partial (G/T)} {\partial T} \right)_p = \left( \frac{\partial (H/T)} {\partial T} \right)_p - \left( \frac{\partial S} {\partial T} \right)_p \]
Expanding the derivative of \(H/T\):
\[ \left( \frac{\partial (H/T)} {\partial T} \right)_p = \frac{1}{T} \left( \frac{\partial H} {\partial T} \right)_p - \frac{H}{T^2} \]
Since
\[ \left( \frac{\partial H} {\partial T} \right)_p = C_p \]
and
\[ \left( \frac{\partial S} {\partial T} \right)_p = \frac{C_p}{T} \]
the \(C_p/T\) terms cancel, leaving
\[ \boxed{ \left( \frac{\partial (G/T)} {\partial T} \right)_p = -\frac{H}{T^2} } \]
The equation becomes easier to integrate if we change variables from \(T\) to \(1/T\).
Using the chain rule,
\[ \boxed{ \left( \frac{\partial (G/T)} {\partial (1/T)} \right)_p = H } \]
This is the differential form of the Gibbs–Helmholtz equation.
It relates the temperature dependence of the Gibbs function directly to the enthalpy of the system.
Suppose a reaction has a Gibbs function \(\Delta G_1\) at temperature \(T_1\) and \(\Delta G_2\) at temperature \(T_2\).
Assuming the enthalpy change remains approximately constant over the temperature interval, integration gives
\[ \int_{\Delta G_1/T_1}^{\Delta G_2/T_2} d\!\left(\frac{\Delta G}{T}\right) = \Delta H \int_{1/T_1}^{1/T_2} d\!\left(\frac{1}{T}\right) \]
which evaluates to
\[ \boxed{ \frac{\Delta G_2}{T_2} - \frac{\Delta G_1}{T_1} = \Delta H \left( \frac{1}{T_2} - \frac{1}{T_1} \right) } \]
This form of the Gibbs–Helmholtz equation allows Gibbs functions at one temperature to be estimated from values known at another temperature.
The Gibbs–Helmholtz equation provides a direct connection between free energy and enthalpy.
For an exothermic process (\(\Delta H < 0\)), increasing the temperature generally makes \(\Delta G\) less negative, reducing the tendency of the reaction to proceed toward products.
For an endothermic process (\(\Delta H > 0\)), increasing the temperature generally makes \(\Delta G\) more negative, increasing the tendency of the reaction to proceed toward products.
Big picture: The Gibbs–Helmholtz equation provides a powerful method for predicting how reaction free energies change with temperature. This is particularly useful because thermodynamic data are often available at only a single temperature, while real systems frequently operate under different conditions.
Estimate \(\Delta G_f^\circ\) for \(CuI_2(s)\) at \(500\ \mathrm{K}\) using the Gibbs–Helmholtz equation. Assume \(\Delta H_f^\circ\) is constant over this temperature range.
| Quantity | Value at \(298.15\ \mathrm{K}\) |
|---|---|
| \(\Delta G_f^\circ\) | \(-64.0\ \mathrm{kJ\,mol^{-1}}\) |
| \(\Delta H_f^\circ\) | \(-80.0\ \mathrm{kJ\,mol^{-1}}\) |
The integrated Gibbs–Helmholtz equation is
\[ \frac{\Delta G_2}{T_2} - \frac{\Delta G_1}{T_1} = \Delta H \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]
Solving for \(\Delta G_2\):
\[ \Delta G_2 = T_2 \left[ \frac{\Delta G_1}{T_1} + \Delta H \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \right] \]
Substitute \(T_1=298.15\ \mathrm{K}\) and \(T_2=500\ \mathrm{K}\):
\[ \Delta G_2 = \left(500 K \right) \left[ \frac{-64.0 kJ/mol}{298.15 K} + (-80.0 kJ/mol) \left( \frac{1}{500 K} - \frac{1}{298.15 K} \right) \right] \]
\[ \Delta G_2 = -53.2\ \mathrm{kJ\,mol^{-1}} \]
\[ \boxed{ \Delta G_f^\circ(500\ \mathrm{K}) = -53.2\ \mathrm{kJ\,mol^{-1}} } \]
Interpretation: The value of \(\Delta G_f^\circ\) becomes less negative at the higher temperature. This is consistent with an exothermic formation process, where increasing temperature tends to make formation less favorable.
\[ \frac{\Delta G_{T_2}}{T_2} - \frac{\Delta G_{T_1}}{T_1} = \Delta H \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]
A thermodynamic system will be generated at random. Use the Gibbs–Helmholtz equation to calculate \(\Delta G\) at the new temperature.
Big picture: The Gibbs–Helmholtz equation allows free energies at one temperature to be estimated from values known at another temperature. It provides one of the most important links between enthalpy and Gibbs free energy, making it possible to predict how temperature influences the spontaneity of physical and chemical processes.