On the speed of sound

The next thing you're going to try to convince me of is that there is a speed of light. 

nicely done.

bitflipper


I think I get what you're saying, drew. Assuming that the statement that "air density is proportional to temperature" is true, then wouldn't it still be temperature that's the critical variable? That statement wouldn't be true if air density could vary independently of temperature.

They're all interrelated by pV=nRT

p=air pressure
V=volume
n=amount of stuff
R=gas constant
T=temperature

and note that
n/V=air density

So, if you change any one thing in that formula, at least one other thing has got to change.

And if you say, "The speed of sound is slower at high altitudes because of the lower temperature, not the lower pressure", well I would ask:

Is the air pressure lower because the temperature is lower?
Or is the temperature lower because the air pressure is lower?

I would say that the answer to both of those questions is "yes". 


But because the speed of sound is based (in part) on the ratio of pressure to density, and "pV=nRT", you're left with just the R and the T.  IOW, the ratio of pressure to density is equal to R * T.

He kept repeating about the "ratio being constant" at a constant temperature, but he didn't make clear (at least to me) that because the two are related to each other by R * T, they cancel out and you're left with just the constant R and the variable T. The only other argument in the formula is a constant (for a given medium), so temperature is indeed the only variable. 

  I'm just the type where if I read someone say, "Everyone else is wrong! Pressure doesn't matter!" and then they show me the formula - and pressure is one of the arguments, well I need to take a closer look at "where did it go?" and how it can't matter.   

You might find this amusing too. When I wiki'd the Gas constant, it says:

As of 2006, the most precise measurement of R is obtained by measuring the speed of sound...

You might find this amusing too. When I wiki'd the Gas constant, it says: As of 2006, the most precise measurement of R is obtained by measuring the speed of sound...

LOL!

Fortunately, the speed of sound can be accurately measured, whether we know how to calculate it or not.

Jimbo21


One of the (many) things that I don't get is pressurized gas like R-22 or canned air cleaner is basically (I assume) at ambient temperature until released to a lower pressure. When that happens it's temp obviously drops and thus we get refrigeration and air conditioning. How does the density/temperature thing work here or is this apples and oranges in that speed of sound is totally unrelated in the example above?

Jim 

Look up Boyle's Law.

It is all interrelated.

Not that anyone cares, but for the sake of correctness:

I said earlier that in pV=nRT, n/V would equal the density.

While this is more or less true, you end up with a "units problem" if you try to replace the ratio of pressure to volume with "R * T" in the speed of sound equation as given:

The speed of sound,
c=( γ *p/ρ)^1/2
where  
p = air pressure
ρ = air density
γ = the adiabatic constant, characteristic of the specific gas. It's about 1.4 for air at reasonable temperatures.

To get the density in the proper units, you need to multiply it by the molecular weight (M) of the substance. You can get the molecular weight by adding together the atomic weights that make up the molecule and dividing it by 1000 (to convert from g to kg).

So then:
density ρ = (n/V)*M

and:
pV=nRT

where:
p=air pressure  (in Pascals)
V=volume  (in cubic meters)
n=amount of stuff  (in moles)
R=gas constant
T=temperature (in °K)  

so:
p=(n/V)*R*T
p=(ρ/M)*R*T  
p/(ρ/M)=R*T
(p/ρ)*M=R*T
p/ρ=(R*T)/M  


So solving the speed of sound equation for temperature:
c=(γ*p/ρ)^1/2
c=(γ*R*T/M)^1/2
where  
γ = the adiabatic constant, characteristic of the specific gas (about 1.4 for air).  
T = temperature in °K (0°C=273.15°K)
R = the Ideal Gas Constant (8.3144621)
M = the Molecular weight (about .02896 for air)

Alternatively, you can use the Specific Gas Constant for the gas in question, which (not surprisingly) equals the Ideal Gas Constant divided by the molecular weight of that gas.

Many years ago as an undergraduate, we did an elegant pchem lab that I have not seen done since.  Hopefully this description does it justice:

We had a large glass jar with an inlet valve on the bottom.  The top of the jar had a long glass cylinder (that was polished on the inside analogous to a cylinder in an engine) mounted vertically.  We also had a polished steel ball that fit into the glass cylinder (analogous to a piston).

When the bottom inlet was sealed and the steel ball was dropped into the cylinder the ball would fall but before it reached the bottom of the cylinder (and fell into the jar) it would bounce back up on a “cushion” of air … and then it would continue to bounce back up and down.

What we did was purge the jar with different gases and observe the frequency of oscillation of the steel ball which ultimately was functionally dependent upon the molar mass of the gas.

This is probably a better illustration of the inhaling helium from a balloon phenomenon but ….

I think my problem is my thermometer doesn't read out in Kelvin.

OK, but how do I get inside the airtight container with my guitar, amp, mic and DAW? Plus, whenever I get around helium I get kinda goofy.

quantumeffect

One of the (many) things that I don't get is pressurized gas like R-22 or canned air cleaner is basically (I assume) at ambient temperature until released to a lower pressure. When that happens it's temp obviously drops and thus we get refrigeration and air conditioning. How does the density/temperature thing work here or is this apples and oranges in that speed of sound is totally unrelated in the example above?

That drop in temperature is due to the absorption of heat when a substance expands. It's rapid evaporation that makes alcohol feel cold on your skin, and why sweating cools you down. Lizards don't sweat, which is why they must seek refuge in a cool bar and sip iced beverages. Hence the phrase "lounge lizard".

Just to elaborate a bit on Bit's comments ... when a liquid evaporates (a phase change) at a constant temperature, molecules at the surface have enough kinetic energy in excess of the bulk liquid to overcome intermolecular forces ... and so they enter the gas phase.  As the liquid vaporizes the average kinetic energy of the bulk liquid decreases and consequently the temperature decreases.  Thus, vaporization (liquid -> gas) is endothermic and the reverse process (condensation) is exothermic.  Refrigeration systems function by cycling through vaporization and condensation in a closed system.

Yes! that's it! I actually did take physical chemistry (OK, it was the baby Pchem that was trig based, not calculus based) 22years ago. Time flies along with the brain cells.

So if the speed of sound is 4 times faster in hydrogen than in oxygen, does that mean sound moves slower in an oxygen-rich atmosphere such as a rain forest?

I know I certainly move slower in the tropics.

quantumeffect

I would like to comment on a property of materials (stiffness) and one of the problems I have is with a statement from the article … which is either incorrect or incomplete:  

Since sound is more easily transmitted between close molecules,
it travels faster in the denser substance. Thus the speed of sound increases with the
stiffness of the material.

A change in pressure (a mechanical stress) in the gas is going to be related to the stiffness of the gas (I guess you would use the adiabatic elastic modulus and yes, I know the modulus is an intensive property and the stiffness is an extensive property) and it is true that the speed of sound increases with the stiffness of the material.  If you compare materials in different phases (i.e., air to water to steel) you may draw the conclusion the speed of sound also increases with density but, the speed of sound in a given phase is actually inversely proportional to density (or decreases with increasing density).  For example:

Hydrogen gas, density = 0.08988 g/L and the S.O.S. is 1290 m/s

Oxygen gas, density = 1.429 g/L and the S.O.S. is 316 m/s

I think I’ve babbled on too long and am loosing focus ….

I think that passage was one of the ones that I questioned in the context of the article. But I would say it's incomplete (and thus misleading).

But the lower density is due to the much lower molecular weight of H₂, which does determine the speed of sound, along with temperature and the adiabatic constant, not the density or pressure (at least for gases).


And I enjoy your babbling.

[EDIT: Remove to cloudy brain errors]

So the point I was trying to make in the pressure / density relation comments in my previous post is that the speed of sound is often given by an equation that relates it to the stiffness of the material whether it be a solid, liquid or gas.  I’ve intentionally not used any equations but this relationship is straightforward.

stiffness has a specific meaning … it is how much a material deforms when a pressure is applied to it so:

speed of sound = the square root of a stiffness coefficient divided by the density

c = (C/d)^1/2

If you go to the wiki page on speed of sound this is the 1^st equation shown and is given for solids, liquids and gases.

The derivation of this relationship is actually pretty complicated and the pchem manual I taught it out of uses all different letters for the variables but I will try to give it an English translation. The goal is to show that pressure and density are directly proportional to each other … so as one goes up the other goes up … but, to do it in terms of a model that describes an acoustical wave moving through a material and not starting with the ideal gas law.  That said (again, Drew showed the pressure / density relationship from the ideal gas equation) … I’m just discussing it from a different perspective, the end result is the same.

Another way of visualizing the acoustical wave is as a spring.  If you held a Slinky at one end and a friend held it stretched out horizontally at the other end and you tapped one of the ends … you would see a repeating pattern of compressed regions of spring moving down the length of the spring completely analogous to the acoustical wave.  And if you think about it this way, you can see how the stiffness of the spring will affect how fast the compressed region moves down the length of the spring. 

A pchem book would probably do the following:

Start with a wave equation and if you will allow me a circular definition … it is an equation that describes the behavior of a wave in a material of a specific density.  This will have in it, particle displacement and the speed of sound.

If you consider an imaginary cross section of the material, you can get equations that relate the particle displacement to changes in density and changes in volume in the material (for some reason physicists and physical chemists like to evaluate cross-sectional areas).

Then deal with the pressure (which is force per unit area) by using Hooke’s Law (that’s the spring one), this gives you pressure changes in the material in terms of not only particle displacement but a term that characterizes the stiffness of the material (the adiabatic elastic modulus which I am also calling the stiffness coefficient) which is analogous to the spring constant or I guess it is the spring constant.

If we consider the net acoustic force acting on our imaginary cross-section and use Newton’s second law of motion we get an equation that relates pressure to the density and from the use of Hooke’s Law, the stiffness coefficient to density 

So, from the wave equation, Hooke’s Law and Newton’s Law of Motion we end up with:

speed of sound  = (stiffness coefficient/density)^1/2

… this one is often the first equation found in physics texts to characterize the SOS.

Through Hooke’s Law we show the relationship between force (or in this case pressure) and the stiffness coefficient giving you:

speed of sound  = (pressure/density)^1/2       {SEE EDIT AT BOTTOM}

and we also know the relationship between the root mean square speed, the speed of sound, temperature, the heat capacity and the molar mass,:

speed of sound = rms-speed = (heat capacity x temperature / molar mass)^1/2

I kinda’ wanted to show it this way as an alternative to the PV=nRT interpretation because of the experiment that I described in post # 16 where a steel ball was bouncing on a cushion of air.  The frequency depended upon the gas that was acting as the “cushion”.  Imagine the ball bouncing up and down in the glass cylinder, the glass and the gas were clear and colorless making it look as though there was nothing supporting it either from below or above … except maybe with a little imagination … the ball was actually suspended from an invisible spring.
 
[EDIT]
I really should write for a reversible adiabatic process ...
speed of sound = (heat capacity x pressure/density)^1/2
the word adiabatic means that the process is not exchanging heat with the surrounding like a hot liquid in a Thermos bottle.

This is a great thread... I really appreciate the minimal use of equations.

It's been great to learn why so many articles about this subject often times peter out and don't go further.

I was always left thinking I didn't get the whole story and there were often times subtle implications suggesting that the full explanation takes too long to write in a short article.

Quantum has a gift for teaching and I just want to voice my appreciation.

I've moved a little further forward today.

A big thanks to quantum and all you guys that inspire him with great comments and questions.


best regards,
mike