**Effect of Temperature on Heat of Reaction: The Kirchhoff Equation**

The heat change accompanying chemical or physical processes generally depends on the temperature at which the process takes place. This dependence is mathematically expressed in the form of what is known as Kirchhoff equation after G. R. Kirchhoff (1858) who first developed this equation. The equation may easily be derived with the help of the first law of thermodynamics.

Consider the process in which the reactants in state A at temperature T_{1} are converted into products in state B a temperature T_{2}. Assume that all operations are carried out at constant pressure. The conversion may be carried out in two ways but according to Hess’s law the total heat change must be the same in both cases.

(1) The reactants in state A at temperature T_{1} are heated to a temperature T_{2}. The heat absorbed is (∆T) (C_{P})_{A}. where ∆T = T_{2} – T_{1}, and (C_{P})_{A} is the heat capacity of the reactants in the state A. The reaction is now allowed to take place at this temperature and the heat change for the process is (H_{B} – H_{A})_{2} = ∆H_{2}

The total heat change for the process = **(∆T) (C _{P})_{A} + ∆H_{2}**.

(2) The reactants in state A at temperature T_{1} are considered to products in state B at the same temperature. The heat cement change = (H_{B} – H_{A})_{1} = ∆H_{1}. The temperature of the products is then raised from T_{1} to T_{2} and the heat absorbed is (∆T) (C_{P})_{B}, where (C_{P})_{B} is the heat capacity of the products.

The total heat change for the process = **(∆T) (C _{P})_{B} + ∆H_{1}.**

From Hess’s law,

**(∆T) (C _{P})_{A} + ∆H_{2} = (∆T) (C_{P})_{B} + ∆H_{1}**

or, ∆H_{2} – ∆H_{1} = [(C_{P})_{B} – (C_{P})_{A}] x (∆T)

= ∆C_{P} (∆T)

where, ∆C_{P} = (C_{P})_{B} – (C_{P})_{A}

or, **[∆H _{2} – ∆H_{1}] / ∆T = ∆C_{P}**

For an infinitesimally small change in temperature one can write,

**[d∆H / dT] = ∆C _{P}** … … … … (1)

Similarly, it may be shown that if the process is carried out at constant volume the relationship is;

**[d∆U / dT] = ∆C** … … … … (2)

The relationships (1) and (2) are different forms of the Kirchhoff equation. The equations are useful for calculating the heat 41 reaction at a given temperature when the value is known at another temperature provided the heat capacities of the reactants and products are also known.