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The relative amount of compression/expansion energy that goes into
temperature
versus pressure
can be characterized by the heat capacity ratio
where
is the specific heat (also called heat
capacity) at constant pressure, while
is the specific heat at
constant volume. The specific heat, in turn, is the amount of
heat required to raise the temperature of the gas by one degree. It
is derived in statistical thermodynamics [139]
that, for an ideal gas, we have
, where
is the ideal
gas constant (introduced in Eq.(B.45)). Thus,
for any
ideal gas. The extra heat absorption that occurs when heating a gas
at constant pressure is associated with the work (§B.2)
performed on the volume boundary (force times distance = pressure times
area times distance) as it expands to keep pressure constant. Heating
a gas at constant volume involves increasing the kinetic energy of the
molecules, while heating a gas at constant pressure involves both that
and pushing the boundary of the volume out. The reason not all
gases have the same
is that they have different
internal degrees of freedom, such as those associated with
spinning and vibrating internally. Each degree of freedom can store
energy.
In terms of
, we have
![$\displaystyle P_1V_1^\gamma \eqsp P_2V_2^\gamma, \protect$](img3123.png) |
(B.46) |
where
for dry air at normal temperatures. Thus,
if a volume of ideal gas is changed from
to
, the pressure
change is given by
![$\displaystyle P_2 \eqsp P_1 \left(\frac{V_1}{V_2}\right)^{\gamma}$](img3127.png) |
(B.47) |
and, converting pressure to temperature via the
ideal gas law
, the temperature change is
![$\displaystyle T_2 \eqsp T_1 \left(\frac{V_1}{V_2}\right)^{\gamma-1}.$](img3129.png) |
(B.48) |
These equations follow from Eq.(B.46) and the ideal gas law
Eq.(B.45).
The value
is typical for any diatomic
gas.B.31 Monatomic inert gases, on the other hand,
such as Helium, Neon, and Argon, have
. Carbon
dioxide, which is triatomic, has a heat capacity ratio
. We see that more complex molecules have lower
values because they can store heat in more degrees of freedom.
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