# Enthalpy contribution and heat capacity

Imagine having at the input of a burner 105 mol/h of $$\ce{O_2}$$ at 50 °C. The enthalpy contribution of this input mass is calculated using the Kirchhoff equation

$$\Delta H = \Delta H^° + \int_{25 °C}^{50 °C} C_p \,dT$$

where

$$\Delta H^° = 0 \dfrac{\text{cal}}{\text{mol}}$$

$$C_p = (7.129 + 0.1407 \cdot 10^{-2} \hspace{3pt} T - 0.6438 \cdot 10^{-5} \hspace{3pt} T^2) \dfrac{\text{cal}}{\text{mol} \cdot °\text{C}}$$

Compute the definite integral

$$\int_{25 °C}^{50 °C} C_p \,dT = \Big( 7.129 \hspace{3pt} \Big[T\Big]^{50 °C}_{25 °C} + 0.1407 \cdot 10^{-2} \hspace{3pt} \Big[\dfrac{T^2}{2}\Big]^{50 °C}_{25 °C} - 0.6438 \cdot 10^{-5} \hspace{3pt} \Big[\dfrac{T^3}{3}\Big]^{50 °C}_{25 °C} \Big)$$

The result will be a number whose unit of measure will be $$\dfrac{\text{cal}}{\text{mol} \cdot °\text{C}}$$. While enthalpy is measured in $$\dfrac{\text{cal}}{\text{mol}}$$, there is a discrepancy between the units of measurement.

Can you help me?

• You have omitted the multiplication by T within the integration. E.g. $\int{\vec F \vec {\mathrm{d}l}}$ does not have dimension N, but Nm=J. Similarly C_p.dT has dimension in your case cal/mol. Commented Feb 4, 2022 at 15:43