Definition of Specific Heat: Molar Specific or Mass Specific Quantity?

Question

Five kilograms of air, initially at 0 °C, are heated until the temperature reaches 50 °C. The internal energy increases by 180 kJ during this process. Find the specific heat of air.

I know this is a basic question, but I am confused on the dimensional analysis. The answer provided is $0.72 \ kJ/kg$. From my textbook we defined specific heat of an ideal gas as:

$$u_1 - u_2 = c (T_2 - T_1)$$

Where $u$ is the specific internal energy (ie: kJ/kmol). They call it a molar specific internal energy. Thus, the units for specific heat should be $\frac{[kJ]}{[kmol \cdot K]}$. Are these two valid definitions for specific heat? How can I tell which one is of interest ?

Answer 1

Specific values are those normalized to the amount of material (mass in kg or quantity in moles) and are thus useful as material properties. "Specific heat" typically means the heat capacity normalized to the mass, but if "molar" is specified, it means the heat capacity normalized to the number of moles ("molar heat capacity" is also used for the latter). One can easily convert from one to the other for a certain material (e.g., the molar mass of air is 0.02897 kg/mol).