Are the number of microstates of a gas just equivalent to pressure

In summary, the concept of microstates is useful in some systems, but not all. In the case of gas samples with the same type and number of molecules, but different volumes, the temperature will be different, changing the number of available microstates. This is reflected in the Boltzmann equation, which gives the same result for both samples. Entropy is related to energy and temperature, and is not affected by volume. However, volume does play a role in the number of available microstates, which can impact entropy. The Sackur-Tetrode equation can be used to calculate entropy in ideal gas systems, taking into account volume, number of particles, internal energy, and other parameters.
  • #1
PiersNewberry
7
0
I am quite confused about this area.

First entropy does not contain any reference to volume. So if we can theoretically set the entropy of A and B gas samples as the same but in different volumes. If A is in a larger volume it would be able to exhibit a larger number of microstates? Yet the Boltzman equation gives the same result for both as it also ignores volume.

I would also be interested to know if the concept of microstates is actually at all useful, or is it just a bystander in real world physics.

Thanks
 
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  • #2
So if we can theoretically set the entropy of A and B gas samples as the same but in different volumes.
If both samples have the same type and number of molecules, their temperature will be different, which changes the number of available microstates as well.

I would also be interested to know if the concept of microstates is actually at all useful, or is it just a bystander in real world physics.
Depends on the system. If you consider systems where that number is easy to calculate (some spins, or whatever), it can be quite useful.
 
  • #3
mfb said:
If both samples have the same type and number of molecules, their temperature will be different, which changes the number of available microstates as well.
.

Sorry still confused. Let's say we have 100 molecules of gas in each of the differently sized boxes both with the same average molecular velocity I would have thought this would give the same temperature reading as interactions with a thermometer would be the same, though less frequent in the larger box, so it would give the same read out.

So E, S and T are all the same, but Boltzman states should be different.

(Lets assume radiatively reflective housing to eliminate infrared heat loss.)
 
  • #4
with the same average molecular velocity
Ok.
But then you get a different number of microstates, and a different entropy.
 
  • #5
PiersNewberry said:
Sorry still confused. Let's say we have 100 molecules of gas in each of the differently sized boxes both with the same average molecular velocity I would have thought this would give the same temperature reading as interactions with a thermometer would be the same, though less frequent in the larger box, so it would give the same read out.

So E, S and T are all the same, but Boltzman states should be different.

(Lets assume radiatively reflective housing to eliminate infrared heat loss.)

mfb said:
Ok.
But then you get a different number of microstates, and a different entropy.

Yes. You have different volumes, same temperature, same number of particles, so entropy is not the same. Also, what do you mean by "Boltzmann states"?
 
  • #6
Sorry for the basic confusion there. I has always imagined microstates visually - making the volume and locations of the particles an aspect of the calculations, whereas it is actually just about the distribution of energies according to one source I have just read and not about the volume. And it follows that this is true from:

Entropy = energy over temperature - nothing to do with volume
Entropy = k log.w - nothing to do with volume either

(Though not 100% sure how W is assessed)

(If we are being finicky about it, in reality, the volume does make a difference due to gravitation reducing the maximum energy probable away from the earth.)

An important exception is if we vapourise a few hunderd molecules of gold - there is the likelihood of, on average, a nice normal distribution of energies in the atoms in a confined space, however if we distribute these molecules in a sufficiently large space wherein they will not collide evidently they will retain their initial energies. So the law of entropy breaks down here.
 
  • #7
I have just been thinking about this a little more and it turns out that volume does make a difference to the number of available microstates as multiatom collisions become less likely in a less dense substance wherein the momentum of two atoms in vector x might add up to produce a high value otherwise rarely produced (?).
 
  • #8
PiersNewberry said:
I have just been thinking about this a little more and it turns out that volume does make a difference to the number of available microstates as multiatom collisions become less likely in a less dense substance wherein the momentum of two atoms in vector x might add up to produce a high value otherwise rarely produced (?).
I have no idea what you mean here. You don't have to consider any collisions here.

More volume -> more states for the particles at same energy -> more microstates for a given temperature.
It is as simple as that.

Entropy = energy over temperature - nothing to do with volume
They are related via derivatives, you do not get an absolute entropy value here.

Entropy = k log.w - nothing to do with volume either
Volume influences w.
 
  • #9
Brilliant; thanks; that is a lot clearer now.
 
  • #10
PiersNewberry said:
Brilliant; thanks; that is a lot clearer now.

A good "reality check" when thinking of entropy is the Sackur-Tetrode equation for the entropy of an ideal gas: [tex]S=kN\ln\left[\left(\frac{V}{N}\right)\left(\frac{Um}{N}\right)^c\phi\right][/tex] where V is volume, N is number of particles, U is internal energy, m is mass per particle, [itex]\phi[/itex] is a universal constant, and c is dimensionless specific heat at constant volume (e.g. 3/2 for a monatomic gas). You can use PV=NkT and U=cNkT to see how entropy varies with other parameters. For the case we were discussing, we were thinking of entropy as a function of N, T, and V, so using U=cNkT, the entropy is [tex]S=kN\ln\left[\frac{V}{N}\left(mckT\right)^c\phi\right][/tex]
 

1. How are microstates and pressure related?

The number of microstates of a gas is directly proportional to its pressure. As the number of microstates increases, so does the pressure of the gas. This is because an increase in the number of microstates means an increase in the number of ways the gas molecules can arrange themselves, leading to a higher probability of collisions and therefore, a higher pressure.

2. What exactly are microstates?

Microstates refer to the different possible arrangements of molecules in a gas at a given moment. These arrangements can include the position, momentum, and energy of each molecule. The more microstates a gas has, the more dispersed its molecules are, resulting in a higher entropy and pressure.

3. Can the number of microstates change?

The number of microstates of a gas can change as the gas undergoes physical or chemical changes. For example, if the gas expands into a larger volume, the number of microstates will increase. However, the total number of microstates for a given gas sample will remain constant as long as the temperature and number of molecules remain the same.

4. How is the number of microstates related to the Second Law of Thermodynamics?

The Second Law of Thermodynamics states that the total entropy of a closed system will always increase over time. The number of microstates is directly related to entropy, as an increase in the number of microstates leads to an increase in entropy. Therefore, the Second Law of Thermodynamics can be explained by the increase in the number of microstates in a system.

5. Are the number of microstates and temperature related?

The number of microstates of a gas is not directly related to temperature. However, temperature does affect the distribution of molecular energies, which in turn affects the number of microstates. As temperature increases, the distribution of molecular energies becomes broader, leading to an increase in the number of microstates and a higher pressure.

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