The heat capacity of a closed system in an infinitesimal process is defined as the ratio between the heat exchanged and the temperature change produced.
For a process at P=cte, the heat capacity is given by the equation:

The heat capacity at constant volume is defined in the same way:

As , the above equations can be written as:

Therefore, C p and C v give us the variation of U and H with respect to temperature and must always take positive values.

We define molar heat capacities for pure substances as the ratio between C p and the number of moles of the substance.

What is the relationship between C p and C v ?

To simplify this equation we consider the internal energy as a function of temperature and volume , and find the differential

Dividing all terms by dT to P = cte

Substituting in the relation C p - C v we get:

Taking the common factor of the derivative of V with respect to T gives us: