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 
is defined as the change of enthalpy with respect to the change in temperature at constant pressure, i.e. _p)
. In other words, is the gradient of the curve of enthalpy versus temperature at constant pressure.
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, varies with temperature and eq37 becomes:
=\left(\frac{\partial H}{\partial T}\right)_p\;\;\;\;\;\;\;\;(64))
Integrating both sides of eq64 with respect to temperature,
dT=\int_{T_1}^{T_2}\left(\frac{\partial H}{\partial T}\right)_pdT\;\;\;\;\;\;\;\;(65))
Since 
is a state function, the line integral of eq65 gives
- 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_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 
to 
.
The computation of eq65 to eq66 (or eq66a) is only valid if the function is continuous and differentiable within the limits of to . If eq66a has a temperature range that includes phase transitions, which result in points of discontinuity in (see diagram above), it has to be modified as follows:
- \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 )
, we use a power series to generate curves that fit experimental values of for different substances on versus temperature plots:
=a+bT+cT^2+dT^3+\cdots)
The set of constants 
, 
, 
, 
, etc. are specific to each chemical species and are the outcome of the polynomial regression. As the contribution of higher powers of 
to is small, the expression is fairly well represented by:
=a+bT+cT^2+dT^3\;\;\;\;\;\;\;\;(68))
