Adiabatic Compressibility

The speed of sound in a gas depends on the "springiness" of the gas. That is, it depends on how the volume of the gas responds to changes in pressure. We have already seen one measure of this response, called the isothermal compressibility,

                (1)
Equation one gives a parameter that determines how the gas responds to changes in pressure if the temperature remains constant.

Sir Isaac Newton assumed that the speed of sound was an isothermal process and used the parameter defined by Equation 1 to calculate the speed of sound in a gas. His answer did not agree with experiment.

It turns out that sound transmission in a gas is an adiabatic process rather than an isothermal process. The sound wave causes oscillations in pressure but the oscillations are fast enough that heat can not move from compressed regions to rarified regions in order to keep the temperature constant. Before the heat can be conducted away from the compressed regions the compression has moved on so that sound propagation is adiabatic.

We define the adiabatic compressibility as,

                (2)
We can calculate the adiabatic compressibility in terms of quantities that we already know (using the Euler cyclic rule twice, once the normal way and once in reverse).
                (3a, b, c)
where we have used Euler's cyclic relation to go from Equation 3a to 3b and the chain rule in the denominator and numerator of 3b to go to 3c.

We already know that

                (4a)
and
                (4b)
so
                (5a, b)
In Equation 5b we can use Euler's cyclic rule in reverse to write
                (6a,b)
Since Cp > CV The isothermal compressibility is always greater than the adiabatic compressibility.

For a monatomic ideal gas, where Cp = 5nR/2 and CV = 3nR/2 we see that

                (7)


Adiabatic Gas Expansion Revisited

Early in the course we derived the equation for the adiabatic expansion of an ideal gas,

          (8a, b)
where,
                (9)
You may recall that the derivation was sort of "round-about." Here we would like to use some of our new thermodynamics tools to provide a much more direct derivation.

The goal is to discover how the pressure changes with volume if entropy is held constant. That is, we would like to find the derivative,

Then we can integrate this derivative to find an expression for p as a function of V. First, let's find the derivative. Notice that this derivative is just the reciprocal of the derivative in Equation 2 so most of the work has already been done. Using the same procedures we used above we fin that,
                (10a, b, c, d, e, f)
Equation 10f is a general thermodynamic relationship. It contains no approximations. To proceed further we must decide what material we want to consider. The equation we were trying to derive was based on the ideal gas for which
                (11a, b)
So,
                (12)
Set up Equation 12 for integration, (don't forget to put all the p stuff on one side and all the V stuff on the other)
                (13)
and integrate between p1, V1 and p2, V2,
                (14)
where we have made the approximations that γ is independent of volume and factored it out of the integral. Equation 14 integrates to
                (15a, b, c)
from which we conclude that
                (16)
or
                (8b)


Just for the record, we add that the correct formula for the speed of sound is

               (17)
where ρ is the density of the gas (in kg/m3).

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Copyright 2004, W. R. Salzman
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Last updated 5 Nov 04
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