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:

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