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Next: Speed of Sound in Up: Speed of Sound Previous: Speed of sound in   Index

# 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.
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 in-fact 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   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21