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?
