The speed of sound in air derived from ideal and real gas behaviorSpenser Joyce

11-5-2014

Physics Comprehensive

OverviewIdeal vs. Non-ideal gases

Pipe organ project review

Problem solutions

Questions

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.

(b) Note and explain all departures from ideal gas behavior in the derivation.

Departures from ideal gas behavior:

[1+}]

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

*Greenspan paper

Physical representations of virial coefficents B and C:

–2.0739

Why aren’t we using C in our derivation?The corrections with this term are

negligible

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.

Questions?