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

For a monatomic ideal gas, where *C _{p}* = 5

(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(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

(13)and integrate between

(14)where we have made the approximations that

(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

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