speed of sound wrt temp, density,and medium change

[cartman1980]

In going through TBR Physics Chapter 6, i wanted to ensure my understanding is correct, along w/ clarify a few questions

Speed of sound is given by

1. v(air) < v(liquid) < v(solid)
This is because the bulk modulus is progressively larger from air to solid, AND the magnitude is far greater than the increase in density. Intuitively, this also makes sense since e.g. in solids the restorative molecular force resisting the compression/rarefaction is larger than in gases.

2. For ideal gases, speed of sound depends only on temp changes

3. Constant temp. Change gas composition
v(lighter gas) > v(heavier gas)
Granted that lighter gas will have lesser density. But, wouldn't it also have lesser restorative force i.e. lesser bulk modulus?

4. Constant gas composition. Change temp
v(higher temp) > v(lesser temp)

High temp --> higher molecular speed --> lesser restorative force? ==> shouldn't speed of sound decrease with increase in temperature?

But, e.g. Q40 asks: If you increase the temp of air by 44%, the speed of sound wave increases by
a. 6.6%
b. 20%
c. 44%
d. 107%

Answer explanation states v proportional to sqrt (temp).
How is this formula arrived at?

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[attixx]

Regarding the issue of temperature.. restorative force causes solids to transmit faster because the molecules are restored to their original condition more rapidly. This means they can participate in another compressional wave more often.

Higher temperature gases have more kinetic energy and vibrate quicker, propagating the sound wave along faster for the same reason - quicker return to original state.

Just know the equation v ~ sqrt(restoring force or molecular KE/molecular inertia). Temperature increases molecular KE without affecting molecular inertia. So increase temp/KE by 44% and increase v in the gas by 6.6%.

 

[BHaus9]

To derive the answer for #4, just consider the simple equations we know for kinetic energy as it relates to a) a body in motion, and b) a gas.
We have: a) KE = 1/2 m v^2 and b) KE = 3/2 R T
Since KE = 1/2 m v^2 = 3/2 R T, a little simplification yields v^2 = 3RT/M.
Thus, v = sqrt(3RT/M), v ~ sqrt(temp/mass). As attixx indicated, holding mass constant gives us v ~ sqrt(temp).

However, I disagree with attixx in that I think the correct answer is B, 20%. You have to realize that "increase by 44%" means "multiply by 1.44." To find the multiplier for v, take the square root of 1.44, which gives 1.2. Multiplying by 1.2 is the same as increasing by 20%. This answer is also appealing because, of all the options provided, it is the easiest to deduce using mental math (144 is the square of 12; all the other numbers make for obscure roots).

 

[attixx]

^^ definitely right sir.

 

[BHaus9]

^^ definitely right sir.

i needed revenge over the losing of the meiosis thing

 

[cartman1980]

restorative force causes solids to transmit faster because the molecules are restored to their original condition more rapidly. This means they can participate in another compressional wave more often.

In solids, the molecular restorative forces are strong(er) due to molecular bonding. In gases, aren't such forces supposed to be weaker such that increased temp would provide more KE to gaseous particles thereby causing them to be more entropic thereby increasing collisions, and hence, reducing effective sound propagation?

We have: a) KE = 1/2 m v^2 and b) KE = 3/2 R T

a) i get.
Where does b) come from? Have yet to encounter it in either TPR (completed first pass @ review) or TBR

 

[attixx]

The "b)" formula that Haus wrote is definitely in TBR. It might be in the Gases section on chemistry when talking about kinetic energy of a gas.

As far as your other question goes, restorative forces is just one means of passing the sound wave along.
Remember the fundamental reason that stronger restorative forces pass song along faster. The molecules spend less time being displaced by the pressure/sound wave and can return to their original position quicker, passing on the sound wave and participating in another pressure wave quicker.
Think of a sound wave as energy, and heated gases as more energetic. They can more effectively pass that sound energy along because they vibrate quicker and can transmit the energy quicker.

 

[ModusProbandi]

In going through TBR Physics Chapter 6, i wanted to ensure my understanding is correct, along w/ clarify a few questions

Speed of sound is given by

1. v(air) < v(liquid) < v(solid)
This is because the bulk modulus is progressively larger from air to solid, AND the magnitude is far greater than the increase in density. Intuitively, this also makes sense since e.g. in solids the restorative molecular force resisting the compression/rarefaction is larger than in gases.

2. For ideal gases, speed of sound depends only on temp changes

3. Constant temp. Change gas composition
v(lighter gas) > v(heavier gas)
Granted that lighter gas will have lesser density. But, wouldn't it also have lesser restorative force i.e. lesser bulk modulus?

4. Constant gas composition. Change temp
v(higher temp) > v(lesser temp)

High temp --> higher molecular speed --> lesser restorative force? ==> shouldn't speed of sound decrease with increase in temperature?

But, e.g. Q40 asks: If you increase the temp of air by 44%, the speed of sound wave increases by
a. 6.6%
b. 20%
c. 44%
d. 107%

Answer explanation states v proportional to sqrt (temp).
How is this formula arrived at?

PV= nRT bro!

increase the temperature by 1.44, increase V by 1.44. This means air is less dense, assuming the same number of moles. Ie divide density by 1.44 in that equation you gave, which is the same as multiplying c by 1.2. 20% increase, of course, this tells you nothing of the bulk factor, since it probably changes with temp too [but it may not if it's intrinsic]

 

[cartman1980]

The "b)" formula that Haus wrote is definitely in TBR. It might be in the Gases section on chemistry when talking about kinetic energy of a gas.

Perhaps it is have to start w/ TBR Chem-2

As far as your other question goes, restorative forces is just one means of passing the sound wave along.
Remember the fundamental reason that stronger restorative forces pass song along faster. The molecules spend less time being displaced by the pressure/sound wave and can return to their original position quicker, passing on the sound wave and participating in another pressure wave quicker.
Think of a sound wave as energy, and heated gases as more energetic. They can more effectively pass that sound energy along because they vibrate quicker and can transmit the energy quicker.

For some reason, I still keep thinking that gas molecules at high temp are like a canon ball and not boomerang i.e. no incentive for it to 'return' to its original position