Kirchoff's Equation: Variation of Heat of Reaction with Temperature

Kirchhoff's Equation

The variation of heat of reaction with temperature is defined by Kirchhoff's equation. This mathematical relationship was developed by G.R. Kirchhoff in 1858.

Let us consider a generic chemical reaction: A → B

If this reaction is carried out at constant pressure, the heat of reaction ($\Delta H$) at any given temperature is expressed as:

ΔH = H_B − H_A

Where H_A and H_B are the enthalpies of the reactants and products respectively.

Differentiating the above equation with respect to temperature at constant pressure, we obtain:

Where (C_p)_A and (C_p)_B are the heat capacities of the reactants and products respectively at constant pressure.

Thus, the change in heat of reaction per degree change in temperature equals the difference in heat capacities of products and reactants (ΔC_P), provided the reaction happens under constant pressure conditions.

Integrating the differential equation between temperature thresholds T₁ and T₂ yields:

Where ΔH₁ and ΔH₂ are the enthalpies of the reaction at temperatures T₁ and T₂ respectively.

Reaction at Constant Volume

If the reaction takes place at constant volume, the heat of reaction is equivalent to the internal energy change:

ΔE = E_B − E_A

Where E_A and E_B represent the internal energies of reactants and products.

Differentiating this expression with respect to temperature at constant volume, we secure:

Where (C_V)_A and (C_V)_B are the heat capacities at constant volume.

Integrating this equation between T₁ and T₂ gives:

From these derivations, it is obvious that if the molar heat capacities of reactants and products are equal (i.e., ΔC_V = 0 or ΔC_P = 0), the heat of reaction will remain completely independent of temperature alterations.