The change in the internal energy is given by the sum of the work and
heat contributions. At constant volume, work contribution is zero and so
the change in internal energy is given by heat, which is the product of the
specific heat and the temperature change. Thus, the change in internal
energy divided by the change in temperature is the specific heat:
(2.23)
A more precise statement is to write the specific heat at constant volume
in terms of incremental changes in internal energy, or equivalently as a
partial derivative, with the subscript denoting that the changes occur with
the volume fixed:
(2.24)
For an ideal gas at constant volume, we can relate the change in enthalpy
to the temperature change and specific heat, just as was done for the change
in internal energy. The change in enthalpy is directly proportional to the
change in temperature:
ΔH=ΔU+Δ(PV) =CVΔT+Δ(nRT) =CVΔT+nRΔT=(CV+nR)ΔT (2.25)
Notice that when there is no change in the temperature then both the
internal energy and the enthalpy remain constant for an ideal gas. This
is not necessarily true for other systems.
The specific heat at constant pressure has a different dependence. Heat
results in not only a temperature increase of the system but also in work
that can be performed. As the system heats up it will expand in order to
keep the pressure constant, resulting in a −PΔVcontribution of work. The
change in the internal energy is then given by:
ΔU=q−PΔV (2.26)
The change in enthalpy at constant pressure (eqn 2.27) is given by the
contributions of the change in internal energy and change in volume since
the change in pressure is zero:
ΔH=ΔU+Δ(PV) =ΔU+PΔV (2.27)
Substituting the expression for the change in internal energy (eqn 2.26)
gives the simple result that the change in enthalpy is equal to the heat:
ΔH=ΔU+PΔV=(q−PΔV) +PΔV=q (2.28)