# Kirchhoff's Law

Kirchhoff's Law describes the enthalpy of a reaction's variation with temperature changes. In general, enthalpy of any substance increases with temperature, which means both the products and the reactants' enthalpies increase. The overall enthalpy of the reaction will change if the increase in the enthalpy of products and reactants is different.

### Introduction

At constant pressure, the heat capacity is equal to change in enthalpy divided by the change in temperature.

\[ c_p = \dfrac{\Delta H}{\Delta T} \label{1}\]

Therefore, if the heat capacities do not vary with temperature then the change in enthalpy is a function of the difference in temperature and heat capacities. The amount that the enthalpy changes by is proportional to the product of temperature change and change in heat capacities of products and reactants. A weighted sum is used to calculate the change in heat capacity to incorporate the ratio of the molecules involved since all molecules have different heat capacities at different states.

\[ H_{T_f}=H_{T_i}+\int_{T_i}^{T_f} c_{p} dT \label{2}\]

If the heat capacity is temperature independent over the temperature range, then Equation \ref{1} can be approximated as

\[ H_{T_f}=H_{T_i}+ c_{p} (T_{f}-T_{i}) \label{3}\]

with

- \( c_{p} \) is the (assumed constant) heat capacity and
- \( H_{T_{i}}\) and \(H_{T_{f}} \) are the enthalpy at the respective temperatures.

Equation \ref{3} can only be applied to small temperature changes, (<100 K) because over a larger temperature change, the heat capacity is not constant. There are many biochemical applications because it allows us to predict enthalpy changes at other temperatures by using standard enthalpy data.

### Contributors

- Janki Patel (UCD), Kostia Malley (UCD)