Kirchhoff’s law (thermodynamics)

Kirchhoff’s law describes the change in enthalpy of a reaction with respect to the change in temperature.

From eq37, the heat capacity at constant pressure $C_p$ is defined as the change of enthalpy with respect to the change in temperature at constant pressure, i.e. $C_p=\left(\frac{\partial H}{\partial T}\right)_p$ . In other words, $C_p$ is the gradient of the curve of enthalpy versus temperature at constant pressure.

$C_p$ for a perfect gas is independent of temperature and we get a straight line with a constant gradient when the enthalpy of a perfect gas at constant pressure is plotted against temperature,. However, for an ideal or real gas, $C_p$ varies with temperature and eq37 becomes:

$C_p(T)=\left(\frac{\partial H}{\partial T}\right)_p\;\;\;\;\;\;\;\;(64)$

Integrating both sides of eq64 with respect to temperature,

$\int_{T_1}^{T_2}C_p(T)dT=\int_{T_1}^{T_2}\left(\frac{\partial H}{\partial T}\right)_pdT\;\;\;\;\;\;\;\;(65)$

Since $H$ is a state function, the line integral of eq65 gives

$H(T_2)- H(T_1)=\int_{T_1}^{T_2}C_p(T)dT\;\;\;\;\;\;\;\;(66)$

or, for the change in enthalpy of a reaction,

$\Delta H_r(T_2)-\Delta H_r(T_1)=\int_{T_1}^{T_2}C_p(T)dT\;\;\;\;\;\;\;\;(66a)$

Eq66 (or eq66a) is known as Kirchhoff’s law, which is used to calculate the change in enthalpy of a substance (or a reaction) from $T_1$ to $T_2$ .

The computation of eq65 to eq66 (or eq66a) is only valid if the function is continuous and differentiable within the limits of $T_1$ to $T_2$ . If eq66a has a temperature range that includes phase transitions, which result in points of discontinuity in $H$ (see diagram above), it has to be modified as follows:

$\Delta H_r(T_2)- \Delta H_r(T_1)=\int_{T_1}^{T_{fus}}C_p(T)dT+\Delta H_{fus}+\int_{T_{fus}}^{T_{vap}}C_p(T)dT+\Delta H_{vap}+\int_{T_{vap}}^{T_2}C_p(T)dT\;\;\;\;\;\;\;\;(67)$

To find an expression for $C_p(T)$ , we use a power series to generate curves that fit experimental values of $C_p$ for different substances on $C_p$ versus temperature plots:

$C_p(T)=a+bT+cT^2+dT^3+\cdots$

The set of constants $a$ , $b$ , $c$ , $d$ , etc. are specific to each chemical species and are the outcome of the polynomial regression. As the contribution of higher powers of $T$ to $C_p$ is small, the expression is fairly well represented by: