Perhaps the simplest process to consider is one which
occurs at constant volume. Such a process is called isochoric and corresponds
to a vertical line in the p-V diagram. Because there is no area under such a line or in other words
no work is done by or on the thermodynamic system in an isochoric process.
The first law of thermodynamics thus implies that the change in internal energy
exactly matches the heat flow:
Because the internal energy changes there must be a change in the
temperature of the system. How much the temperature changes for
a given heat flow is a materials property which we call the heat capacity
of the system. For an isochoric process we define:
Where the subscript V means that the expression holds for a constant
volume heat transfer. We can now continue Eq. 6
In general depends on temperature and so we cannot go further without
knowing materials properties. In the special case of an ideal gas, however,
is temperature independent and we obtain
This allows us to write a very important formula for the temperature
dependence of the
internal energy of an ideal gas:
Where for simplicity and to be in accordance with the later predictions
of the kinetic theory of gases we have chosen U(T=0)=0.
It is important to note that although a subscript V appears in this
expression it is a general expression which is true for any
thermodynamic equilibrium state of an ideal gas.