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The speed of sound in air derived from ideal and real gas behaviorSpenser Joyce
11-5-2014
Physics Comprehensive
Ideal vs. Non-Ideal GasesIdeal – hard little spheres
Non-ideal – real molecules with different masses and shapes
We use virial coefficients, which are a representation of the sum total of energy, to help understand corrections for a non-ideal system
(a) Derive from first principles the coefficients tabulated in the Cramer paper of the polynomial expansion for the speed of sound.
= 1 + …Truncate at B because corrections made afterward are negligible in regard to our analysis of the speed of sound
Sub in V = for simplification
This equation is our starting point in the derivation:
We Taylor expand about B so we can see how the total energy and temperature contribute to correction factors:[1+}]
Explanation of quantities
– ratio of specific heat capacities of gas
B – second virial coefficient (for dry air in this case)
and - first and second derivatives of B with respect to temperature
The virial coefficient represents deviations between forces/energies between molecules
B is temperature dependent
From Cramer paper and references
= 1.4029
R = 8.31448
M = 28.9643704
p = 101325 Pa
T = 273.15 K
B = -13.5 = 66 = -155
= 331.6648
-11.51
= 10.25
-9.382
Results from source paper
Calculated results
* The results for , and are likely due to a scaling factor I did not take into account, however, the signs are correct as well as the unit quantities.
Departures examplesMany terms in the previous equations are squared, or quadratic indicating a non-ideal (linear) behavior of the gas
Specifically temperature terms, evident in both equations
In the first equation, we see terms that are both pressure and temperature dependent
(c) To see the relative importance of the various terms and the temperature dependence thereof, plot
(i) the speed of sound for an ideal gas (ii) the speed of sound for the leading order
non ideal behaviors in Cramer (iii) the speed of sound using the next-higher
order terms in your derivation
Ideal gas behavior
Non-ideal behavior including second virial coefficient B
Temperature (K)
Speed o
f Sound (
m/s
)
For constant pressure at 101325 Pa and variable T in K temps:
The speed of sound for an ideal gas and for a non-ideal gas
including B
References• Cramer, Owen. "The variation of the specific-heat ratio and the speed
of sound in air with temperature, pressure, humidity, and CO2 concentrations." Journal of the Acoustical Society of America 93.5 (1993): 2510-2516. Print.
• Bird, R. Byron, Charles F. Curtiss, and Joseph O. Hirschfelder. Molecular Theory of Gases and Liquids. New York: n.p., 1954. Print.
• Wong, George S. K. “Speed of sound in standard air.” Journal of the Acoustical Society of America 79 (1986): 1359-1366. Print.
• Greenspan, Martin. “Comments on “Speed of sound in standard air” [J. Acoust. Soc. Am. 79, 1359-1366 (1986)].” Journal of the Acoustical Society of America 82 (1987): 370-372. Print.