What is the correlation between entropy and velocity distribution?

[lschuh]

Hello,
The Boltzmann velocity distribution may show the curves for a low (T1) and a high (T2) temperature. The T2 curve is broader and flatter. A flatter curve means: a broad range of speed/KE. This means more micro states, which means a higher entropy S. This explains the increase of S with increasing T. Consent?
The transfer of heat Q increases T of an ideal gas, because the KE increases - The velocity v of the particles increases

Is it correct to state the following:
Q is represented by the average (root of v^2) in the Maxwell Boltzmann Distribution curve at T
S is represented by the width of the Maxwell Boltzmann Distribution curve at T

Regards
Lo

 

[Dale]

This explains the increase of S with increasing T. Consent?

A higher temperature has a lower entropy. (Edit: for a fixed energy) $$dS=\frac{\delta Q}{T}$$
For the same $\delta Q$ you get a smaller $dS$ for a bigger $T$

 

[256bits]

Is it correct to state the following:
Q is represented by the average (root of v^2) in the Maxwell Boltzmann Distribution curve at T
S is represented by the width of the Maxwell Boltzmann Distribution curve at T

Hi Ischuh,

I know what you mean, but,
Probably not best to describe the distribution that way, because technically:
1. Systems do not have heat Q. They have internal energy. Q is the heat transferred from one system to another.
2.The width of the distribution at any temperature is the same for all T, as it is a continuous function ( except at T=0 ). Even though it appears as if there is a larger width at higher temperatures, this is a result of the fraction of particles having a more probability of greater speed at higher temperature, even if at a lower temperature the probability may be essentially considered as being 0. The skewness, and flatness as you mentioned, is what changes with temperature.

As @Dale mentioned , by adding an amount of Q to the system results in diminishing returns in S. If a system is basically close to random at high T, adding some more Q changes the randomness not by much.

 

[Matterwave]

A higher temperature has a lower entropy. $$dS=\frac{\delta Q}{T}$$
For the same $\delta Q$ you get a smaller $dS$ for a bigger $T$

Your first sentence is a bit confusing isn't it? The equation and the second sentence is correct, but $dS$ is a change in entropy and your first sentence seems to equate that with the absolute entropy. I get what you mean, so I'm just talking in terms of how the sentence is phrased.

 

[Dale]

I am open to concise alternatives. But the fact is that the reason something is hot is precisely because it has a low entropy (Edit: for a fixed energy). Energy wants to flow from hot to cold to increase the entropy. It goes from a low entropy hot reservoir to a high entropy cold reservoir.

 

[Matterwave]

I am open to concise alternatives. But the fact is that the reason something is hot is precisely because it has a low entropy. Energy wants to flow from hot to cold to increase the entropy. It goes from a low entropy hot reservoir to a high entropy cold reservoir.

How about "the change in entropy is higher if you add heat to a lower temperature object"?

Let's say we have two systems A and B of same volume and amount of gas except A is high temperature while B is low temperature. When A and B are put in contact (no moveable interface or particle exchange) the heat flows from A to B.

The entropy of the joint A/B system is maximized by that entropy flow and is higher than summing the initial entropies of A and B separately. The temperature of the joint A/B system is lower than A's original temperature.

So your sentence compares A against the A/B joint system. And there it is true.

But if you compare A to B before contact, as a plain reading of your sentence suggests, I can't see how the sentence would be true.

Am I missing something basic?

 

[Dale]

if you compare A to B before contact

A and B of same volume and amount of gas

Good point. You always need to specify what you are keeping fixed in a comparison. I was keeping energy fixed. Either two systems with the same energy or a fixed amount of energy transferred.

 

[256bits]

the reason something is hot is precisely because it has a low entropy.

The above statement is problematic.
A system at a temperature of absolute zero is where it has the lowest entropy.
An addition of heat to increase the temperature, to make it hotter, increases the entropy.

 

[Dale]

A system at a temperature of absolute zero is where it has the lowest entropy.

One joule at absolute zero would have more entropy than a joule at any other temperature.

Roughly the same amount of radiated EM energy is leaving Earth as is entering it. The radiation that comes in is hot because it has low entropy while the radiation that leaves is cold because it has high entropy.

 

[256bits]

One joule at absolute zero would have more entropy than a joule at any other temperature.

Roughly the same amount of radiated EM energy is leaving Earth as is entering it. The radiation that comes in is hot because it has low entropy while the radiation that leaves is cold because it has high entropy.

Ok. Photon entropy.

 

[Demystifier]

You always need to specify what you are keeping fixed in a comparison.

Yes, that's particularly important in thermodynamics. No branch of physics is so profound in being precise about what is kept fixed in partial derivatives, as is thermodynamics. See e.g. https://en.wikipedia.org/wiki/Maxwell_relations

 

[Dale]

Yes, that's particularly important in thermodynamics. No branch of physics is so profound in being precise about what is kept fixed in partial derivatives, as is thermodynamics.

Yes, the confusion here is my fault for not originally specifying that I was keeping energy fixed in my mind. I think of entropy in terms of how energy is spread out, rather than in terms of "disorder". Keeping energy fixed is natural in that view, but nobody can read my mind.