Entropy increase of solid vs liquid

[thinkingcap81]

A hypothetical question. Heat Q is transferred from water to a metallic solid. Both have same heat capacities and the same initial temperature. Now since molecules in a liquid are more randomly oriented than a solid, will the entropy decrease of the liquid be more than the entropy increase of the solid? I know this cannot be true but do not know how to explain it. Is it related to phase space and probabilities of finding particles in a region of space? If so how?

 

[Dale]

It has nothing to do with the phase of matter, just the temperature. Temperature is defined by $$\frac{1}{T}=\frac{dS}{dE}$$ where T is the temperature, S is the entropy and E is the energy. So if you transfer a fixed amount of energy then the change in entropy will be larger the smaller the temperature. No other details of the material are relevant.

 

[thinkingcap81]

Hi Dale,

I know the equation that you have mentioned. I am seeking clarity with respect to the oft repeated explanations of entropy increase with regard to the increase in randomness of the molecules/atoms of matter. If the same amount of heat is transferred to a solid and gas/liquid at the same temperature, in which case is randomness increased more? In a solid the randomness is less in comparison to a gas, and yet the entropy change is the same when we use your equation. I would like to know the relation between randomness and entropy.

 

[Dale]

If the same amount of heat is transferred to a solid and gas/liquid at the same temperature, in which case is randomness increased more?

In neither case. That is what the equation means

I would like to know the relation between randomness and entropy.

Why do you think that they are different? I know what entropy is. What do you mean by the word “randomness” if not entropy?

 

[thinkingcap81]

Hi Dale,

I know that the entropy change is same for both cases, but am unable to explain it in terms of randomness.

How do we quantify randomness of a solid since the molecules have a well defined structure and do not shift from their positions except for vibrations about their mean position? In contrast the molecules of a gas move about randomly.

I suppose a suitable explanation would be that the increase in vibrations in the atoms/molecules of a solid give the same increase in randomness as that of the molecules of a gas under the same amount of heat transfer at the same temperature ranges.

I'd be happier if i knew of an equation that spoke about entropy in terms of the randomness (or phase space, if applicable). This might help me in understanding the situation better.

 

[thinkingcap81]

Why do you think that they are different? I know what entropy is. What do you mean by the word “randomness” if not entropy?

I know that you know what entropy is. I was asking for an equation defining entropy in terms of randomness. Or you can offer me a better explanation from the statistical POV in place of the classical equation in your reply.

Thanks.

 

[Chestermiller]

If they both have the same initial temperatures, how can the heat be transferred?

 

[thinkingcap81]

If they both have the same initial temperatures, how can the heat be transferred?

Hypothetical reversible heat transfer at infinitesimal temperature difference. I just wanted to make the comparison between the solid and gas/liquid as simple as possible, and so kept the same heat capacity and temperatures.

 

[Chestermiller]

Do you regard changes to the velocity distributions of the molecules in the solid and fluid as part of the "randomness?"

 

[Dale]

I was asking for an equation defining entropy in terms of randomness.

I don’t know what “randomness” means if not entropy. Can you explain what you mean? As far as I know the equation is $\text{entropy}=\text{randomness}$. If you think of them as different concepts then I need you to explain what you mean by “randomness” because I use them interchangeably.

 

[thinkingcap81]

Do you regard changes to the velocity distributions of the molecules in the solid and fluid as part of the "randomness?"

Yes, i do, in an intuitive sense. I cannot precisely explain it.

 

[thinkingcap81]

I don’t know what “randomness” means if not entropy. Can you explain what you mean? As far as I know the equation is $\text{entropy}=\text{randomness}$. If you think of them as different concepts then I need you to explain what you mean by “randomness” because I use them interchangeably.

I think that there's some miscommunication between us. I know that randomness is a measure of entropy. I am hoping for an equation that says ##\text{this much change in randomness gives rise to this much change in entropy}##

 

[Andrea Vironda]

##\text{this much change in randomness gives rise to this much change in entropy}##

Since randomness is described by entropy, i think you're asking something like: $\text{this much change in quantity of water in air gives rise to this much change in humidity}$

 

[Dale]

I think that there's some miscommunication between us.

Yes, there is. You keep refusing to explain what you mean by the word “randomness” despite being asked repeatedly. You are using a word that you refuse to define or even just generically explain what you mean. I can tell from context that you mean something different than I do when I use the word, but instead of explaining what you mean by the word you just keep using it.

Please don’t post again without explaining what you mean by the word “randomness” as precisely and explicitly as possible.

For example, here is an expression for the entropy of an ideal gas, can you give an expression for the randomness of an ideal gas: http://hyperphysics.phy-astr.gsu.edu/hbase/Therm/entropgas.html

 

[thinkingcap81]

Yes, there is. You keep refusing to explain what you mean by the word “randomness” despite being asked repeatedly. You are using a word that you refuse to define or even just generically explain what you mean. I can tell from context that you mean something different than I do when I use the word, but instead of explaining what you mean by the word you just keep using it.

Please don’t post again without explaining what you mean by the word “randomness” as precisely and explicitly as possible.

For example, here is an expression for the entropy of an ideal gas, can you give an expression for the randomness of an ideal gas: http://hyperphysics.phy-astr.gsu.edu/hbase/Therm/entropgas.html

If i knew how to precisely and explicitly explain randomness then it would be a lot better for me. Let me just say that to me it means, in a crude sense, how particles can be located in a region of space.

GAS: Identify a region of space containing some amount of a gas. A particle spends a certain amount of time in that region. This time changes with changes in temperature. So when temperature increases from T1 to T2 the time spent by the particle in than region of space decreases. So the probability of finding a particle in the region decreases. This is my crude understanding of randomness in a gas. The entropy thus increases. Of course there will be other factors to consider.

SOLID: A particle of the solid oscillates about its mean position in a particular region. When temperature increases from T1 to T2 the amplitude of oscillations increase and a bigger region is necessary to locate the particle. The entropy thus increases because of this increase in 'randomness'.

For both gas and solid, if T1 and T2, heat transferred and heat capacities are the same then the decrease in probability of finding a particle in an appropriate fixed region decreases by the same amount. This is my way of looking at entropy changes in a microscopic sense.

I don't know how better to explain my position. i hope that you get what i am trying to say.

Thanks.

 

[Chestermiller]

Have you consulted any references on Statistical Thermodynamics such as McQuarrie or Hill? The fundamentals are all explained in these books.

 

[Dale]

I don't know how better to explain my position. i hope that you get what i am trying to say.

Yes, this is very helpful. I would never have guessed that this is what you meant by randomness without this good explanation.

Unfortunately, I don’t think that this concept of randomness has any use in thermodynamics, so your best option will be to simply dispense with it entirely and focus only on the standard concepts like entropy. For example, the randomness of a gas does not increase with temperature.

So when temperature increases from T1 to T2 the time spent by the particle in than region of space decreases. So the probability of finding a particle in the region decreases.

Actually, the time spent in that region decreases, but the frequency of visiting that region increases such that the probability of finding the particle in that region is constant. If you have a volume of gas $V$ then the probability of finding a given gas particle in a region of volume $v$ is simply $v/V$. It does not depend in any way on temperature or pressure, just volume.

Imagine what would happen if the probability worked the way you think. Suppose we have a gas in a volume $V$ and we partition it into 1000 sub volumes $v_i$. At a given temperature the probability of a molecule of gas being in each $v_i$ is 1/1000 so the sum of the probabilities is 1. Now, let’s say that, as you suggest, upon heating the gas the probability of being in $v_i$ decreases to 0.9/1000. The problem is that now when you sum across all $v_i$ the probability that it is found anywhere in $V$ is only 0.9. We know that this is incorrect.

In a liquid the situation is similar except that $V$ no longer is the volume of the container, but the volume of the liquid itself. This volume does change as temperature changes as determined by the coefficient of thermal expansion.

In a solid the situation is a bit different. There the atom is highly localized so the probability of a given atom being at a given point does not change much over time. However, that volume of localization for an individual atom also increases with temperature according to the coefficient of thermal expansion.

So in the end, your idea of randomness is basically just directly proportional to $V$ which is not a function of temperature for a gas but is a function of temperature for solids and liquids where the derivative with respect to temperature is given by the coefficient of thermal expansion.

 

[Lord Jestocost]

SOLID: A particle of the solid oscillates about its mean position in a particular region. When temperature increases from T1 to T2 the amplitude of oscillations increase and a bigger region is necessary to locate the particle. The entropy thus increases because of this increase in 'randomness'.

Maybe, the following might be of help regarding the entropy change of a solid when it is heated up.

From “Entropy as a Measure of the Multiplicity of a System" (http://hyperphysics.phy-astr.gsu.edu/hbase/Therm/entrop2.html#c1)

"The probability of finding a system in a given state depends upon the multiplicity of that state. That is to say, it is proportional to the number of ways you can produce that state. Here a "state" is defined by some measurable property which would allow you to distinguish it from other states…………”

From “Einstein Model of a Solid" (http://hyperphysics.phy-astr.gsu.edu/hbase/Therm/einsol.html)

"The conceptual Einstein solid is useful for examining the idea of multiplicity in the distribution of energy among the available energy states of the system.“

 

[thinkingcap81]

Unfortunately, I don’t think that this concept of randomness has any use in thermodynamics, so your best option will be to simply dispense with it entirely and focus only on the standard concepts like entropy. For example, the randomness of a gas does not increase with temperature.

Actually, the time spent in that region decreases, but the frequency of visiting that region increases such that the probability of finding the particle in that region is constant. If you have a volume of gas $V$ then the probability of finding a given gas particle in a region of volume $v$ is simply $v/V$. It does not depend in any way on temperature or pressure, just volume.

Imagine what would happen if the probability worked the way you think. Suppose we have a gas in a volume $V$ and we partition it into 1000 sub volumes $v_i$. At a given temperature the probability of a molecule of gas being in each $v_i$ is 1/1000 so the sum of the probabilities is 1. Now, let’s say that, as you suggest, upon heating the gas the probability of being in $v_i$ decreases to 0.9/1000. The problem is that now when you sum across all $v_i$ the probability that it is found anywhere in $V$ is only 0.9. We know that this is incorrect.

In a liquid the situation is similar except that $V$ no longer is the volume of the container, but the volume of the liquid itself. This volume does change as temperature changes as determined by the coefficient of thermal expansion.

In a solid the situation is a bit different. There the atom is highly localized so the probability of a given atom being at a given point does not change much over time. However, that volume of localization for an individual atom also increases with temperature according to the coefficient of thermal expansion.

So in the end, your idea of randomness is basically just directly proportional to $V$ which is not a function of temperature for a gas but is a function of temperature for solids and liquids where the derivative with respect to temperature is given by the coefficient of thermal expansion.

Ah! Thanks about the insight about probabilities. It did cross my mind that the more energized particle will frequent the same region more number of times, but i didn't think of the math. I think that you kept the volume $V$ fixed for a gas, but of course, not for a solid.

Now let's say that $V$ of a gas is allowed to change due to temperature changes, keeping the pressure fixed. Let the gas lose heat to a solid kept at lower temperature. The entropy of the gas reduces but that of the solid increases. We know that the net entropy change should be positive. Since, as you said, the randomness of a gas does not decrease with temperature, and, for solids the volume of localization for an individual atom increases with temperature according to the coefficient of thermal expansion, is it possible to think of the entropy changes in the gas and solid in terms of the how the molecules move about? Or is it best to think of multiplicity of states as suggested by @Lord Jestocost ?

Again, not asking for a lot of math, just some good physical explanation.

Hope I'm not consuming too much of your time.

 

[Dale]

is it possible to think of the entropy changes in the gas and solid in terms of the how the molecules move about?

It might be possible, but if so I have never seen it done. So certainly it is not easy or straightforward to think of entropy in terms of how the molecules move about. Since there is not much previous work along these lines, to my knowledge, you would have to develop it yourself and would run a substantial risk of making serious mistakes. I would not recommend it.

Or is it best to think of multiplicity of states as suggested by @Lord Jestocost ?

That is certainly the easiest way and the most well developed. I would recommend learning it first, even if later you try an alternative. You have to know what is “in the box” before being able to “think outside the box” rationally.