Question

what is the enthalpy of a gas at 20 K given that its heat capacity as a function of temperature is given by C=2T^2, over the range T= 0K to 20K ?

Answer 1

At constant pressure, the heat capacity is

$$C_P=\dfrac{dH}{dT}$$

Then

$$dH=C_PdT$$

Integrate both sides

$$\Delta H=\displaystyle\int_{T_i}^{T_f}C_PdT$$

Given $C_P=2T^2, T_i=0K, T_f=20K$

$$\Delta H=\displaystyle\int_{0}^{20}2T^2dT=\bigg[\dfrac{2T^3}{3}\bigg]\begin{matrix} 20 \\ 0 \end{matrix}$$

$$=\dfrac{16}{3}\times10^3 \ J$$