Before going into why the real answer is what it is, there is a flaw in your reasoning for your original answer. Your reasoning about density assumes the gas is free to expand and contract, which is not an assumption given in the problem. An ideal gas in a box of fixed size does not gain or lose mass or volume when heat or cooled, only T and P change. And the speed of sound in air does not change if you measure it inside with the doors and windows closed and sealed. So what variables can we rule out?
If I make the room bigger or smaller while holding T and P constant, v(sound) does not change. If it did, it would be very obvious in daily life.
If I increase pressure while holding V and T constant, I do it by adding more gas molecules. This increases density, but does not change the velocity of the gas molecules. In this scenario, your answer is correct. Speed of sound varies with P. However, this is only true because real gases are not ideal gases, and there are more complex interactions between molecules. In an ideal gas where all the interactions are perfectly elastic and instantaneous collisions, the density has no effect because mean free path is irrelevant to instantaneous or average speed of movement across the group of molecules as a whole, it just changes which molecules are moving. Side note: Remember sound is a pressure wave. So, if this were not true, then the speed of sound would depend on how loud it is. Also, it would be different at the peak and trough of each cycle of the pressure wave. Again, in real gases there are effects of this kind, and there is a maximum volume sound can be (in air at sea level, around 194 decibels the pressure at the trough hits zero; this is a shockwave and it can move faster than sound). But ideal gases assume all that away.
If we increase or decrease the amount of gas at fixed T, then either we’re just adding volume again, or we’re just increasing pressure again, and we’ve seen that those shouldn’t have an effect on an ideal gas’ speed of sound.
So now let’s increase T. It doesn’t matter what effect this has on P and V and n, as seen in the above. So what’s left? Increasing T linearly increases the average kinetic energy of the gas molecules (PV and NkT both have units of energy, this is why), and velocity increases as the sqrt of kinetic energy. So if gas molecule velocity is what determines v(sound), then it has to be that v(sound) increases as sqrt(T).
I’m not sure what level physics class you’re in, but I have two very general pieces of advice.
The first is that you can often get very far by figuring out what doesn’t or can’t matter, and what has to be held constant. This is why conservation laws are so important! It’s also why boundary conditions of a problem are important. They let you write down equations that constrain what the solution can possibly depend on. As an example, a string attached to fixed points on both ends can only vibrate at frequencies where those ends don’t move. Air vibrating in a tube that’s closed at one end can’t have net motion at the closed end, but has to be moving at the other end for any sound to come out at all. These kinds of considerations let you figure out what resonance frequencies are possible in principle for idealized versions of different types of musical instruments. (If you get to more advanced physics, this is why there’s so much discussion of symmetry. Noether’s theorem tells us that every symmetry of the laws of physics implies a conservation law, and vice versa. “Energy is conserved” is equivalent to saying “The laws of physics don’t change over time.” “Linear and angular momentum are conserved” is equivalent to saying “the laws of physics don’t depend on where you are or what direction you’re facing.” There’s a reason Einstein originally wanted Relativity to be called “The Theory of Invariants”: he arrived at it in large part by thinking about what couldn’t matter and what couldn’t change, then letting everything else vary however it needed to in order to accommodate that.)
The second is something a professor used to like to say: that if you really want an intuition for physics you need to feel it in your bones. Try things out. Look for places in your regular life where different concepts could apply, and try to reason out how they apply. It’s a slow process that doesn’t always match up to a semester-long class schedule, but it’s the kind of thing where a year or two later you’ll find yourself saying, “Oh yeah, that’s what they meant!” Sometimes this takes a long time and a lot more physics than you’d expect! I was in grad school for materials science when one professor wrote down an equation at the end of a handful of math-heavy lectures and said, “And that’s why metals are shiny.” Other times it turns out you can explain seemingly complicated things with high school physics and very careful reasoning.
If I make the room bigger or smaller while holding T and P constant, v(sound) does not change. If it did, it would be very obvious in daily life.
This feels a bit too handwavy to me, I could say the same thing about temperature: if the speed of sound were affected by making a room hotter or colder, it would be very obvious in daily life, therefore the speed of sound doesn’t depend on temperature. But it isn’t obvious in daily life that the speed of sound changes based on temperature either.
So now let’s increase T. It doesn’t matter what effect this has on P and V and n, as seen in the above. So what’s left? Increasing T linearly increases the average kinetic energy of the gas molecules (PV and NkT both have units of energy, this is why), and velocity increases as the sqrt of kinetic energy. So if gas molecule velocity is what determines v(sound), then it has to be that v(sound) increases as sqrt(T).
I think this also falls short of justifying that v(sound) increases as T increases. Why does it have to be that v(sound) increases with gas molecule velocity and not decreases instead? Why is it the case that gas molecule velocity determines v(sound) at all?
To your first point: there is no scenario in daily life where we experience a change in absolute temperature spanning orders of magnitude (at most about 30%, from ~250K to ~325K, but we do experience room sizes that span multiple orders of magnitude (a closet vs. a concert hall vs outdoors in an open grassland). Similarly, our experiences of pressure variation almost never span more than about +/-30% from 1 atm. So I maintain that a dependence on volume would be much more obvious than a dependence on temperature or pressure unless it were something like log(V) or V^.1 (which would be hard to reconcile with ending up with units of m/s), and even then there would be scenarios that should be very odd, like a long, narrow cave with an open mouth where the speed of sound was several times faster along the short axis than the long axis, or vice versa. Or think about it this way. If you stand across a football field in a sealed, airtight indoor stadium bang cymbals together, I hear it about 1⁄4 of a second later. If we do the same thing without the stadium in a place where there’s nothing blocking sound propagation for many miles, I hear it about 1⁄4 of a second later. This happens even though the volume of gas is at least 6 orders of magnitude larger (say, if we call the distance to the horizon the new relevant room size) or much more than that (if we count the whole atmosphere as the room size).
To your second point—I agree this is not obvious. So, we have to dig deeper. What is sound? A pressure wave. What does that mean? Well it means you create a pattern of high and low pressure regions that propagates, for example by vibrating a membrane and imparting force to (initially randomly/thermally moving) molecules. Well why does it propagate? Because all the individual molecules move randomly, so the molecules in the high pressure regions tend to flow into the nearby low pressure regions more than the reverse (diffusion along concentration gradients). Now, could the initial forces creating the high and low pressure regions at the source impart some net kinetic energy in a particular direction or set of directions? After all, sound moves net away from the source (note: diffusion in 2D and 3D is also automatically net away from any point, but not in 1D, that’s another fun derivation). It takes a bit more investigation to figure out why that doesn’t let you make a sound wave that’s faster or slower, but basically that has to do with boundary conditions again. You can totally get sound to a distant point sooner if your source has net velocity in that direction (aka I can theoretically hear a bullet that passes a foot away from me before the sound from the gun itself reaches me), or if your sound is loud enough to make a shockwave, but the former isn’t propagation of the sound itself and the latter violates the ideal gas assumption. If you have a stationary source, though, then that boundary condition applies the constraint that each time you move right to provide rightward force must be balanced by moving left to apply leftward force (or less rightward force, if the source is the wall of your box). So you aren’t applying net force to the gas (that would violate F=ma=0 for reaction forces on the source in the frame of reference where the source is stationary). Therefore the already-existing thermal motion of the gas is the only velocity we have to work with. Now it doesn’t automatically have to be a simple relationship with that velocity. After all, there’s a wide distribution of velocities of individual molecules, and depending on how the source is behaving you can even get into subtle distinctions like group vs phase velocity (it’s the group velocity that matters for what we’re discussing here). But that’s the velocity distribution we’ve got. And somehow we have to end up with something measured in m/s. Those are pretty significant constraints on the form of the resulting equation.
You can also do all sorts of fancy things that make sound have different speeds based on direction and frequency, if you carefully combine materials with different densities and bulk moduli (see acoustic metamaterials). In this case you’re interrupting the movement of molecules on distance scales shorter than the wavelength of sound, which isn’t explicitly listed as an assumption of something forbidden in the problem, but it is one implicitly forbidden in the conventional understanding of what “sound” means at the level where teachers are asking questions about the behavior of ideal gases.
After writing all this I’m reminded of a story/joke where a professor says “It’s easy to show that [something.]” A student asks, “Um… is that easy?” The professor starts writing and checking his notes. He leaves, comes back an hour later, says “Yes, it’s easy,” and then continues on with the lecture.
note: diffusion in 2D and 3D is also automatically net away from any point, but not in 1D, that’s another fun derivation
What? The Green’s function for the 1D heat equation has the same dependance on distance as the 2D and 3D case.
“It’s easy to show that [something.]” A student asks, “Um… is that easy?” The professor starts writing and checking his notes. He leaves, comes back an hour later, says “Yes, it’s easy,” and then continues on with the lecture.
The version I heard used ‘obvious’ instead of ‘easy’. I wonder ‘trivial’ would be funnier yet.
Ah, you’re right, sorry. Yes, diffusion rate works the same in 1D for concentration gradient over time. The difference I was thinking of is that in 1D for an individual molecule a random walk returns to the origin with probability 1, even though avg distance rises over time, while in higher dimensions that isn’t true.
Before going into why the real answer is what it is, there is a flaw in your reasoning for your original answer. Your reasoning about density assumes the gas is free to expand and contract, which is not an assumption given in the problem. An ideal gas in a box of fixed size does not gain or lose mass or volume when heat or cooled, only T and P change. And the speed of sound in air does not change if you measure it inside with the doors and windows closed and sealed. So what variables can we rule out?
If I make the room bigger or smaller while holding T and P constant, v(sound) does not change. If it did, it would be very obvious in daily life.
If I increase pressure while holding V and T constant, I do it by adding more gas molecules. This increases density, but does not change the velocity of the gas molecules. In this scenario, your answer is correct. Speed of sound varies with P. However, this is only true because real gases are not ideal gases, and there are more complex interactions between molecules. In an ideal gas where all the interactions are perfectly elastic and instantaneous collisions, the density has no effect because mean free path is irrelevant to instantaneous or average speed of movement across the group of molecules as a whole, it just changes which molecules are moving. Side note: Remember sound is a pressure wave. So, if this were not true, then the speed of sound would depend on how loud it is. Also, it would be different at the peak and trough of each cycle of the pressure wave. Again, in real gases there are effects of this kind, and there is a maximum volume sound can be (in air at sea level, around 194 decibels the pressure at the trough hits zero; this is a shockwave and it can move faster than sound). But ideal gases assume all that away.
If we increase or decrease the amount of gas at fixed T, then either we’re just adding volume again, or we’re just increasing pressure again, and we’ve seen that those shouldn’t have an effect on an ideal gas’ speed of sound.
So now let’s increase T. It doesn’t matter what effect this has on P and V and n, as seen in the above. So what’s left? Increasing T linearly increases the average kinetic energy of the gas molecules (PV and NkT both have units of energy, this is why), and velocity increases as the sqrt of kinetic energy. So if gas molecule velocity is what determines v(sound), then it has to be that v(sound) increases as sqrt(T).
I’m not sure what level physics class you’re in, but I have two very general pieces of advice.
The first is that you can often get very far by figuring out what doesn’t or can’t matter, and what has to be held constant. This is why conservation laws are so important! It’s also why boundary conditions of a problem are important. They let you write down equations that constrain what the solution can possibly depend on. As an example, a string attached to fixed points on both ends can only vibrate at frequencies where those ends don’t move. Air vibrating in a tube that’s closed at one end can’t have net motion at the closed end, but has to be moving at the other end for any sound to come out at all. These kinds of considerations let you figure out what resonance frequencies are possible in principle for idealized versions of different types of musical instruments. (If you get to more advanced physics, this is why there’s so much discussion of symmetry. Noether’s theorem tells us that every symmetry of the laws of physics implies a conservation law, and vice versa. “Energy is conserved” is equivalent to saying “The laws of physics don’t change over time.” “Linear and angular momentum are conserved” is equivalent to saying “the laws of physics don’t depend on where you are or what direction you’re facing.” There’s a reason Einstein originally wanted Relativity to be called “The Theory of Invariants”: he arrived at it in large part by thinking about what couldn’t matter and what couldn’t change, then letting everything else vary however it needed to in order to accommodate that.)
The second is something a professor used to like to say: that if you really want an intuition for physics you need to feel it in your bones. Try things out. Look for places in your regular life where different concepts could apply, and try to reason out how they apply. It’s a slow process that doesn’t always match up to a semester-long class schedule, but it’s the kind of thing where a year or two later you’ll find yourself saying, “Oh yeah, that’s what they meant!” Sometimes this takes a long time and a lot more physics than you’d expect! I was in grad school for materials science when one professor wrote down an equation at the end of a handful of math-heavy lectures and said, “And that’s why metals are shiny.” Other times it turns out you can explain seemingly complicated things with high school physics and very careful reasoning.
This feels a bit too handwavy to me, I could say the same thing about temperature: if the speed of sound were affected by making a room hotter or colder, it would be very obvious in daily life, therefore the speed of sound doesn’t depend on temperature. But it isn’t obvious in daily life that the speed of sound changes based on temperature either.
I think this also falls short of justifying that v(sound) increases as T increases. Why does it have to be that v(sound) increases with gas molecule velocity and not decreases instead? Why is it the case that gas molecule velocity determines v(sound) at all?
To your first point: there is no scenario in daily life where we experience a change in absolute temperature spanning orders of magnitude (at most about 30%, from ~250K to ~325K, but we do experience room sizes that span multiple orders of magnitude (a closet vs. a concert hall vs outdoors in an open grassland). Similarly, our experiences of pressure variation almost never span more than about +/-30% from 1 atm. So I maintain that a dependence on volume would be much more obvious than a dependence on temperature or pressure unless it were something like log(V) or V^.1 (which would be hard to reconcile with ending up with units of m/s), and even then there would be scenarios that should be very odd, like a long, narrow cave with an open mouth where the speed of sound was several times faster along the short axis than the long axis, or vice versa. Or think about it this way. If you stand across a football field in a sealed, airtight indoor stadium bang cymbals together, I hear it about 1⁄4 of a second later. If we do the same thing without the stadium in a place where there’s nothing blocking sound propagation for many miles, I hear it about 1⁄4 of a second later. This happens even though the volume of gas is at least 6 orders of magnitude larger (say, if we call the distance to the horizon the new relevant room size) or much more than that (if we count the whole atmosphere as the room size).
To your second point—I agree this is not obvious. So, we have to dig deeper. What is sound? A pressure wave. What does that mean? Well it means you create a pattern of high and low pressure regions that propagates, for example by vibrating a membrane and imparting force to (initially randomly/thermally moving) molecules. Well why does it propagate? Because all the individual molecules move randomly, so the molecules in the high pressure regions tend to flow into the nearby low pressure regions more than the reverse (diffusion along concentration gradients). Now, could the initial forces creating the high and low pressure regions at the source impart some net kinetic energy in a particular direction or set of directions? After all, sound moves net away from the source (note: diffusion in 2D and 3D is also automatically net away from any point, but not in 1D, that’s another fun derivation). It takes a bit more investigation to figure out why that doesn’t let you make a sound wave that’s faster or slower, but basically that has to do with boundary conditions again. You can totally get sound to a distant point sooner if your source has net velocity in that direction (aka I can theoretically hear a bullet that passes a foot away from me before the sound from the gun itself reaches me), or if your sound is loud enough to make a shockwave, but the former isn’t propagation of the sound itself and the latter violates the ideal gas assumption. If you have a stationary source, though, then that boundary condition applies the constraint that each time you move right to provide rightward force must be balanced by moving left to apply leftward force (or less rightward force, if the source is the wall of your box). So you aren’t applying net force to the gas (that would violate F=ma=0 for reaction forces on the source in the frame of reference where the source is stationary). Therefore the already-existing thermal motion of the gas is the only velocity we have to work with. Now it doesn’t automatically have to be a simple relationship with that velocity. After all, there’s a wide distribution of velocities of individual molecules, and depending on how the source is behaving you can even get into subtle distinctions like group vs phase velocity (it’s the group velocity that matters for what we’re discussing here). But that’s the velocity distribution we’ve got. And somehow we have to end up with something measured in m/s. Those are pretty significant constraints on the form of the resulting equation.
You can also do all sorts of fancy things that make sound have different speeds based on direction and frequency, if you carefully combine materials with different densities and bulk moduli (see acoustic metamaterials). In this case you’re interrupting the movement of molecules on distance scales shorter than the wavelength of sound, which isn’t explicitly listed as an assumption of something forbidden in the problem, but it is one implicitly forbidden in the conventional understanding of what “sound” means at the level where teachers are asking questions about the behavior of ideal gases.
After writing all this I’m reminded of a story/joke where a professor says “It’s easy to show that [something.]” A student asks, “Um… is that easy?” The professor starts writing and checking his notes. He leaves, comes back an hour later, says “Yes, it’s easy,” and then continues on with the lecture.
What? The Green’s function for the 1D heat equation has the same dependance on distance as the 2D and 3D case.
The version I heard used ‘obvious’ instead of ‘easy’. I wonder ‘trivial’ would be funnier yet.
Ah, you’re right, sorry. Yes, diffusion rate works the same in 1D for concentration gradient over time. The difference I was thinking of is that in 1D for an individual molecule a random walk returns to the origin with probability 1, even though avg distance rises over time, while in higher dimensions that isn’t true.
Simple random walks return to the origin in 2D as well, but not 3D or higher. I don’t know if the continuous case is different, but I suspect not.
Then I’m probably just misremembering, it’s been about 15 years since I looked at that one. Thanks!