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Speed of Sound in Real Gas
The ideal gas model can be improved by introducing the
compressibility factor.
The compressibility factor represents the deviation from the ideal gas.
Figure:
The Compressibility Chart

Thus, a real gas equation can be expressed in many cases as
The speed of sound of any gas is provided by
equation (
3.7).
To obtain the expression for a gas that obeys the law expressed by
(
3.19) some mathematical expressions are
needed.
Recalling from thermodynamics, the Gibbs function
(
3.20) is used to obtain
The definition of pressure specific heat for a pure substance
is
The definition of volumetric specific heat for a pure substance
is
From thermodynamics, it can be shown
^{3.4}
The specific volumetric is the inverse of the density as
and thus
Substituting the equation (
3.24) into
equation (
3.23) results
Simplifying equation (
3.25) to became

(3.26) 
Utilizing Gibbs equation (
3.20)
Letting
for isentropic process results in
Equation (
3.28) can be integrated by parts.
However, it is more convenient to express
in terms
of
and
as follows
Equating the right hand side of equations
(
3.28) and (
3.29) results
in
Rearranging equation (
3.30) yields
If the terms in the braces are constant in the range
under interest in this study, equation
(
3.31) can be integrated.
For short hand writing convenience,
is defined as
Note that
approaches
when
and when
is constant.
The integration of equation (
3.31) yields
Equation (
3.33) is similar to equation
(
3.11).
What is different in these derivations is
that a relationship between coefficient
n
and
was established.
This relationship (
3.33) isn't new, and
infact any thermodynamics book shows this relationship.
But the definition of
n
in equation (
3.32)
provides a tool to estimate
n
.
Now, the speed of sound for a real gas can be
obtained in the same manner as for an ideal gas.
SOLUTION
According to the ideal gas model the speed of sound should be
For the real gas first coefficient
has
Solution
According to the ideal gas model the speed of sound should be
For the real gas first coefficient
has

The correction factor for air under normal conditions
(atmospheric conditions or even increased pressure) is minimal
on the speed of sound.
However, a change in temperature can have a dramatical change in
the speed of sound.
For example, at relative moderate pressure but low temperature
common in atmosphere, the compressibility factor,
and
which means that speed of sound is only
about factor of (0.5) to calculated by ideal gas model.
Next: Speed of Sound in
Up: Speed of Sound
Previous: Speed of sound in
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Created by:Genick BarMeir, Ph.D.
On:
20071121