推导
吉布斯自由能随温度和压力变化的方程为:
\[
\begin{align}
\text{d} G = V \text{d} p - S \text{d} T
\end{align}
\]
因此,有:
\[
\begin{align}
\left ( \frac{\partial G}{\partial T} \right)_p = -S
\end{align}
\]
又根据吉布斯自由能的定义\(G = H - TS\),得:
\[
\begin{align}
S = \frac{H-G}{T}
\end{align}
\]
所以,(2)和(3)结合得:
\[
\begin{align}
\left ( \frac{\partial G}{\partial T} \right)_p = \frac{G-H}{T}
\end{align}
\]
更进一步,为了表达的简洁,或因为在使用中,可能更关注\(G/T\)随温度的变化,所以,整体对温度求导:
\[
\begin{align}
\left( \frac{\partial}{\partial T} \frac{G}{T} \right)_p &= \frac{1}{T} \frac{\partial G}{\partial T} + G \frac{\partial (1/T)}{\partial T} \\
&= \frac{1}{T} \frac{\partial G}{\partial T} - \frac{G}{T^2} \quad \text{将(4)代入这里} \\
&= -\frac{H}{T^2}
\end{align}
\]
即得吉布斯-亥姆霍兹方程:
\[
\begin{align}
\left( \frac{\partial}{\partial T} \frac{G}{T} \right)_p = -\frac{H}{T^2}
\end{align}
\]
用途
吉布斯-亥姆霍兹对恒压条件下的物理状态的变化或者化学反应非常有用。如果这是一个等温过程,那么还可以写成:
\[
\begin{align}
\left( \frac{\partial}{\partial T} \frac{\Delta G}{T} \right)_p = -\frac{\Delta H}{T^2}
\end{align}
\]
即一个系统从初始状态到最终状态直接的吉布斯自由能变为:
\[
\begin{align}
\Delta G = G_f - G_i
\end{align}
\]
- \(f\): final
- \(i\): initial
\[
\begin{align}
\left( \frac{\partial \Delta G}{\partial T} \right)_p= \left( \frac{\partial G_f}{\partial T}\right)_p - \left( \frac{\partial G_i}{\partial T}\right)_p
\end{align}
\]
可以得到吉布斯自由能变与熵变、焓变的两种关系:
\[
\begin{align}
\left( \frac{\partial \Delta G}{\partial T} \right)_p &= \left( \frac{\partial G_f}{\partial T}\right)_p - \left( \frac{\partial G_i}{\partial T}\right)_p \\
&= - S_f - (-S_i) \\
&= - \Delta S
\end{align}
\]
\[
\begin{align}
\left( \frac{\partial}{\partial T} \frac{\Delta G}{T} \right)_p &= \left( \frac{\partial }{\partial T} \frac{G_f}{T}\right)_p - \left( \frac{\partial}{\partial T} \frac{G_i}{T} \right)_p \\
&= - \frac{H_f}{T^2} - (- \frac{H_i}{T^2}) \\
&= - \frac{\Delta H}{T^2}
\end{align}
\]
因此,在已知一个反应/变化在\(T_1\)时的吉布斯自由能变,我们就可以对上述关系(14或17)进行积分以求在\(T_2\)时该反应的吉布斯自由能变:
\[
\int_{T_1}^{T_2} \left( \frac{\partial \Delta G}{\partial T} \right)_p dT = \Delta G(T_2) - \Delta G(T_1) = \int_{T_1}^{T_2} \Delta S dT
\]
\[
\int_{T_1}^{T_2}\left( \frac{\partial}{\partial T} \frac{\Delta G}{T} \right)_p d T = \frac{\partial \Delta G(T_2)}{\partial T_2} - \frac{\partial \Delta G(T_1)}{\partial T_1} = \Delta H \left( \frac{1}{T_2} - \frac{1}{T_1} \right)
\]