Kirchhoff's Equation
Understanding the variation of heat of reaction with temperature.
In thermodynamics, the heat of reaction (enthalpy change) is not a constant value; it varies with temperature. Kirchhoff's Equation provides the mathematical relationship to calculate the enthalpy change ($\Delta H$) or internal energy change ($\Delta U$) of a reaction at one temperature, provided the value at another temperature and the heat capacities are known.
1. Derivation at Constant Pressure
Consider a general chemical reaction: $$ \text{Reactants} \rightarrow \text{Products} $$
Let $H_{reactants}$ and $H_{products}$ be the total enthalpies of the reactants and products, respectively. The heat of reaction is: $$ \Delta H = H_{products} - H_{reactants} $$
Differentiating with respect to temperature ($T$) at constant pressure ($P$): $$ \left( \frac{\partial \Delta H}{\partial T} \right)_P = \left( \frac{\partial H_{products}}{\partial T} \right)_P - \left( \frac{\partial H_{reactants}}{\partial T} \right)_P $$
We know that heat capacity at constant pressure is $C_p = \left( \frac{\partial H}{\partial T} \right)_P$. Substituting this: $$ \left( \frac{\partial \Delta H}{\partial T} \right)_P = (C_p)_{products} - (C_p)_{reactants} $$
Differential Form (Constant Pressure):
Integrated Form
Integrating between temperatures $T_1$ and $T_2$:
$$ \int_{T_1}^{T_2} d(\Delta H) = \int_{T_1}^{T_2} \Delta C_p \, dT $$ $$ \Delta H_2 - \Delta H_1 = \int_{T_1}^{T_2} \Delta C_p \, dT $$Case 1: If $\Delta C_p$ is constant (independent of T over small ranges):
2. Derivation at Constant Volume
Similarly, for internal energy change ($\Delta U$), we use the heat capacity at constant volume ($C_v$). $$ \Delta U = U_{products} - U_{reactants} $$ Differentiating with respect to T at constant Volume (V):
Differential Form (Constant Volume):
Integrated Form (assuming constant $C_v$):
$$ \Delta U_2 = \Delta U_1 + \Delta C_v (T_2 - T_1) $$3. Importance of $\Delta C_p$
The term $\Delta C_p$ represents the difference between the sum of molar heat capacities of products and reactants, multiplied by their stoichiometric coefficients.
Note: Be careful with units! Heat capacity is usually given in $J \, K^{-1} \, mol^{-1}$, while enthalpy is often in $kJ \, mol^{-1}$. Convert units before adding.
Knowledge Check
Test your understanding of Kirchhoff's Equation
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