## Specific Heat Capacities of an Ideal Gas

For a simple system, internal energy (u) is a
function of two independant variables, thus we assume it to be a
function of temperature T and specific volume v, hence:

Substituting equation (2) in the energy equation (1)
and simplifying, we obtain:

Now for a constant volume process (**d**v = 0):

That is, the specific constant volume heat capacity
of a system is a function only of its internal energy and
temperature. Now in his classic experiment of 1843 Joule showed that
the internal energy of an ideal gas is a function of temperature
only, and not of pressure or specific volume. Thus for an ideal gas
the partial derivatives can be replaced by ordinary derivatives, and
the change in internal energy can be expressed as:

Consider now the enthalpy. By definition h = u + P v,
thus differentiating we obtain:

Again for a simple system, enthalpy (h) is a function
of two independant variables, thus we assume it to be a function of
temperature T and pressure P, hence:

Substituting equation (6) in the energy equation (5),
and simplifying:

Hence for a constant pressure process, since **d**P
= 0:

That is, the specific constant pressure heat capacity
of a system is a function only of its enthalpy and temperature. Now
by definition

Now since for an ideal gas Joule showed that internal
energy is a function of temperature only, it follows from the above
equation that enthalpy is a function of temperature only. Thus for an
ideal gas the partial derivatives can be replaced by ordinary
derivatives, and the differential changes in enthalpy can be
expressed as

Finally, from the definition of enthalpy for an ideal
gas we have:

Values of R, C_{P}, C_{v}
and k for ideal gases are presented (at 300K) in the
table on **Properties
of Various Ideal Gases**. Note that the
values of C_{P}, C_{v} and
k are constant with temperature only for mon-atomic gases such as
helium and argon. For all other gases their temperature dependence
can be considerable and needs to be considered. We find it convenient
to express this dependence in tabular form and have provided a table
of **Specific
Heat Capacities of Air**.

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Engineering Thermodynamics by
Israel Urieli is licensed under a
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