# Tbr sound

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#### Meredith92

##### Full Member
7+ Year Member

Tbr gives the equation v= sqrt ( (Cp/CV) Pressure/density) for gases... But when answering questions about liquids or solids, it seems as though density increases speed. Why is there this difference? TBR gives the general equation v is proportional to sqrt (restoring F or KE/ molecular inertia) but where does density fit in here? Also I can't figure out if this too is just for a gas?

#### NuttyEngDude

##### Red-Flagville
10+ Year Member
density increases speed of sound for liquids and solids because the closer the molecules are to each other (As opposed to nothingness), then the faster the sound can propagate by one molecule vibrating its neighbor molecule. TBR writes that the speed is based on how fast the molecules can return to their original position which is why Vsolid > Vliquid > Vgas

I think of gases as a special case, that the molecules are so far spread out in a gas as compared to solid or liquid that the density matters much less than other factors such as pressure or temperature, or a factor of energy over inertia (which is what example 6.1a is trying to show).

#### Meredith92

##### Full Member
7+ Year Member
The density not only matters less but it also has the opposite effect.. Do you know why that is? And why would high pressure increase v when density decreases v? I feel like the two concepts are related

Thanks again!

#### NuttyEngDude

##### Red-Flagville
10+ Year Member
Hey there, when it is gas, the only way sound can be transmitted is if there is a collision of gas particles, thus it would depend on how fast the particles can travel and hence the hotter the gas the faster the particles travel, but if the particles are heavier (more dense since the volume can be controlled), then it will travel slower and thereby the sound will travel slower.

When it is a solid (or liquid), then there is contact between the molecules at all times, so it does not matter about collision frequency of the particles.

#### Meredith92

##### Full Member
7+ Year Member
Gotcha! Thank you! I wasnt really thinking about the mass of the molecules I was thinking more about the number of molecules for unit of volume. But what you said makes sense

Just to clarify- An increase in pressure increases velocity because there is more movement/ force per unit volume? Why does that help velocity? More collisions?

#### NuttyEngDude

##### Red-Flagville
10+ Year Member
I dont know if it will increase the velocity of the particles themselves, but yes, it means there will be more collisions, because there is less space for the gas molecules to wander, (time between collisions) so the sound will travel faster (velocity of the sound). In a gas, you need a collision for the longitudinal nature of the wave to propagate.

#### Meredith92

##### Full Member
7+ Year Member
Thanks for your help! Seems much more simple now

#### Meredith92

##### Full Member
7+ Year Member
In ex6.1b it says "lighter particles move faster than heavier particles so speed of sound is faster in Mediums with lighter particles assuming all other factors are equal." I thought we just said that a greater density for Liquids and solids increades velocity... This seems to say the opposite...

Edit- it seems like there is a big diff between molecule density and the density of the substance... I don't understand this concept
See

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#### NuttyEngDude

##### Red-Flagville
10+ Year Member
Hey Meredith,

Two subtle uses of density here.

First use of density:
When you are comparing solids, liquids and gases, just think of density in that case, in the sense of the medium, so a solid is more dense than say a liquid than say a gas. So because the molecules are closer together they will transmit the sound faster than if they are spread out.

Second use of density:
If you are in the same medium, the equation for speed of sound is v = sqrt (restoring force/molecular inertia). What is changing here is what you are using to plug in for "Restoring force" and "molecular inertia". For gases, you are using kinetic energy which pretty much means heat. For solids and liquids, you are using the bulk modulus or young's modulus, it's basically the "stiffness" of the material. For all mediums you are then plugging in density for the denominator. Because gases are more compressible than liquid or solid, it can take into account heat and pressure which is where there starts to be some funny stuff.

You can compare across mediums but you will need the bulk, young, or whichever value to plug into the numerator but there is no point because the general rule of velocity of solid > liquid > gas will apply, especially for something like the mcat.

Technically we are using density correctly, mass over volume, we are just in different units in both uses

Example 6.1b is just testing your understanding of that equation and knowing what to plug into the numerator and denominator.

edit: I suppose they could be tricky and give you a solid with some bulk modulus value and density that comes out to have a speed of sound slower than a very hot light gas or something, shrug. But if you know the equation then you will be fine.

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#### Meredith92

##### Full Member
7+ Year Member
Thanks again for all your help! (youre awesome)

This may be a stupid question but if something has a higher molecular weight is it necessarily more dense? Does molecular inertia relate directly to density? Are they the same thing?

#### NuttyEngDude

##### Red-Flagville
10+ Year Member
Thanks again for all your help! (youre awesome)

This may be a stupid question but if something has a higher molecular weight is it necessarily more dense? Does molecular inertia relate directly to density? Are they the same thing?

You are welcome!

Technically you can not assume mass correlates to density but I think they get around it in the problem when they write "assume all solids have the same lattice structure and all gases have the same molarity". which means they will have the same volumes which means you can directly compare masses in this case

I think it is good to assume that molecular inertia = density, they kind of say this in Example 6.1a.