Question

At a temperature of 298 K, the molar enthalpy change for the conversion of red phosphorus to white phosphorus is 17.6 kJ mol-1. The molar heat capacities of red and white phosphorus are, respectively, 21.19 and 23.82 J mol-1 K-1. Calculate the molar enthalpy change at 198 K.

Answer 1

According to the Kirchhoff's law, the dependence of enthalpy change of the reaction $∆H$ on temperature $T$ is related to the change in heat capacity $c_p$ as follows:

$∆H_{T_2}= ∆H_{T_1} + \int_{T_1}^{T_2}{∆c_p}dT$ .

The change in heat capacity for the conversion of red phosphorus to white phosphorus is:

$∆c_p = c_{p, white P} - c_{p, red P} = 23.82 - 21.19 =2.63$ J mol^-1 K^-1.

Therefore, the change in enthalpy at 198 K is:

$∆H_{198 K} = ∆H_{298 K} + \int_{298}^{198}{2.63}dT$

$∆H_{198 K} = 17.6·10^{3} \text{ (J/mol)} + \int_{298}^{198}{2.63}dT$

$∆H_{198 K} = 17.6·10^{3} \text{ (J/mol)} + 2.63·(198-298)\text{ (J/mol)}$

$∆H_{198 K} = 17600 -263 = 17337 \text{ (J/mol)}$ , or 17.3 kJ/mol.

Answer: the molar enthalpy change at 198 K for the conversion of red phosphorus to white phosphorus is 17.3 kJ/mol.