## 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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