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Peter Macinnis posted, on 30/8/2002:

Quick check:

Can anybody tell me what the relationship is between the root mean square velocity in a gas and the speed of sound in the same gas?  Are there any fudge factors for monatomic and diatomic, or elasticity?

Specifically, how would I calculate the speed of sound in hot helium? (if anybody can go that far)
 

Zero Sum replied:

The speed of sound depends on the state of the gas--more specifically, the square root of the temperature of the gas. The speed of sound (a) is equal to the square root of the ratio of specific heats (gamma) times the gas constant (R) times the absolute temperature (T).
 

a = sqrt [gamma * R * T]

No mon/diatomic fudge factors, but the gas constant would vary for mon/diatomic varieties of the same gas.

see http://www.grc.nasa.gov/WWW/K-12/airplane/sound.html

Glen Moore responded

(Speed of sound)/(rms speed of molecules) = sqrt(1.66)/sqrt(3) -  for helium.

The 1.66 is gamma  which does depend on the geometry of the molecule. 1.66 for monatomic, 1.4 for diatomic, 1.3 for polyatomic.

Regards
Glen
 
 

Donald Lang added:

Slow trawl of memory and bookshelf.

speed of sound = square root of ( gamma *  k  *  T / mu)

root mean square of molecular speed = square root of  (3  *  k * T /mu)

Note that the 3 in the RMS speed comes from living in 3 dimensions; gamma is a fudge factor MISSED by Isaac Newton {That makes it by definition a fudge factor!} It takes account of the rest of the ways a molecule can
divide up the energy it is carrying. For a monatomic gas such as Helium gamma is 5/3 or 1.67, and hence the RMS molecular speed/ speed of sound is sqrt (3 / (5/3)) = 3 / root (5)  ~ 1.338.
 

Toby Fiander added

I have done some background reading already, I notice that the speed of sounds (at sea level and everything) is 1116ft/s whereas Newton calculated 979ft/s, a factor of about 1.14 [noting that Jann's calculator has pretty coloured rocks for buttons, that three digits are green, two are blue, two: orange, two: red and one is purple(ish)].

Using the method that Newton is supposed to have used more or less, I got 943ft/s. which gives to the measured value of about 1.18.  If I follow the ideal gas discussion correctly, the difference is equal to the square root of the ratio of specific heats, ie.  (1.4)^0.5, which is 1.18.  The site shown below has an interesting discussion of it all:

http://www.mathpages.com/home/kmath109/kmath109.htm

I follow most of this (I think).  The partial differential equations of the last half are tantalisingly close to making sense practically, so that eventually (quoting from the text):

c  =  sqrt [dp/d(rho)]

Where (if I have indeed followed the discussion) c is the celerity of a (sound) wave and the rest of it appears to be the square root of the rate of change of pressure WRT density.... most interesting, but I now have a question of a more practical nature.

I deal (sorry) with lots of gases - sewers, oxygen, nitrogen, carbon dioxide, carbon monoxide, methane, hydrogen sulfide (sic) and sulfur dioxide?  ... and speed of sound is sometimes important (well, I can think of two such occasions, one of which involved an explosion front as well).  But perhaps worst of all, Murphy is a work companion, so it seems very likely to me that at (probably crucial) times, the energy transfer will not be quasi-static or adiabatic.

So, my question is (preferably in practice, but in a general sense anyway) when do gases behave badly (ie.  non-ideal) and, hence, what other values can gamma take?

Aha!  and helium having a specific heat of 1.24, compared with oxygen at 0.219 and nitrogen at 0.249, explains why hot helium is used in the T4 shock tunnel at University of Queensland.  Hydrogen has a specific heat of 3.41, which would be even better, but I suppose hot hydrogen has some drawbacks to it :-)

These people whack a piston down on hot helium and burst through a 3mm steel "diaphragm" to produce a shock wave when they are doing scramjet stuff, and I was wondering about the speeds possible -- Attwood machines were never that much fun, not even Wimshurst machines.  I want one of these, Gerald may have one with the business end inside a parliamentary chamber of his choice.

(All s.h. figures from column 1 of my old CRC handbook that I keep at home -- values of 1.242 and  3.42 are also quoted for helium and hydrogen respectively, from an older reference.)

>see http://www.grc.nasa.gov/WWW/K-12/airplane/sound.html

Ah good, that gives me a clear account with numbers (I don't plan to admit that I was about to slip in the Rydberg constant :-)

 
On 17/9/2002, Toby Fiander posted:

Recently, the list looked at some of the energy behaviour of gases in connection with Scramjets...

However, I raised the issue of when a gas is not "ideal".  Barb herself wrote a quite helpful email to me privately, which I am hoping she will post to the list and then put on the site.  Essentially, it says gases are only ideal when the assumptions behind idealness are not violated ...

My question is, at Barb's urging, when is a gas likely to behave in such a way that it is not ideal?  In other words, what real world situations exist (or even laboratory ones) where a gas behaves badly?

Real gases behave badly any time the assumptions that underlie the gas equations break down - i.e. that all gases consist of point sources, with no inter-particle (atomic, molecular, whatever...) forces.

So if the volume of the actual particles in the gas is no longer negligible compared to the volume of the gas - this will mean for all gases at high densities or close to the phase transition to liquid (or solid, in a few cases) - the gas behaves less and less like an ideal one.  Also if there are appreciable forces between the particles in the gas then again there will be a deviation from ideal behaviour - the differences between hydrogen sulfide and water vapour are a good case there - hydrogen bonding is a particularly strong force (in intermolecular terms), but at low temperatures and/or high densities dipole-dipole interactions and even the Van der Waals dispersion forces become important.

Sorry, it's the old Chem teacher talking - and I took my last Y12 chem class in '91 - so the terminology may be a bit archaic.

I'm sure there are others on the list who could explain it much better - perhaps you should repost your original query on this?