Does the formula dS = dQ/T contradict the concept of entropy?

[Joe]

I saw another post about dS = dQ/T, but the subject of question was different - not related to the entropy of universe.

This is what i understand from this formula:

As the temperature goes down, the entropy goes up. Is this not the opposite (contradictory) to what entropy (disorder) is about? At a lower temperature, there should be more order (lower entropy). From what I was taught, the highest entropy would be when the temperature of a system reaches absolute zero - the third law of thermodynamics. Am I correct?

Also how does the energy of the system (in this case universe) explain this equation? A higher drop in the enthalpy - which is directly related to the energy of the system and its volume (space) - would result in more disorder? That would make sense to me. Is that correct?

 

[Dale]

As the temperature goes down, the entropy goes up. Is this not the opposite (contradictory) to what entropy (disorder) is about?

No, the equation is correct.

Consider two systems at temperatures $T_{hot}>T_{cold}$. If they exchange an amount of energy $dQ_{hot}=-dQ_{cold}$. Then the total change in entropy is $dS_{hot}+dS_{cold}=dQ_{hot}/T_{hot}+dQ_{cold}/T_{cold}$
$= dQ_{hot}/T_{hot}-dQ_{hot}/T_{cold} = dQ_{hot}(1/T_{hot}-1/T_{cold})$.

This term in parentheses is negative so this quantity is positive only if $dQ_{hot}$ is negative. That means that the hot system loses energy. Thus a given amount of energy has more entropy at low temperature. That is why heat goes from hot to cold.

 

[Joe]

No, the equation is correct.

Consider two systems at temperatures $T_{hot}>T_{cold}$. If they exchange an amount of energy $dQ_{hot}=-dQ_{cold}$. Then the total change in entropy is $dS_{hot}+dS_{cold}=dQ_{hot}/T_{hot}+dQ_{cold}/T_{cold}$
$= dQ_{hot}/T_{hot}-dQ_{hot}/T_{cold} = dQ_{hot}(1/T_{hot}-1/T_{cold})$.

This term in parentheses is negative so this quantity is positive only if $dQ_{hot}$ is negative. That means that the hot system loses energy. Thus a given amount of energy has more entropy at low temperature. That is why heat goes from hot to cold.

Thank you, Dale. So, the entropy of the universe does go up as it cools down. And yet, does this not contradict the third law of thermodynamics: "The entropy of a system approaches a constant value as its temperature approaches absolute zero. "?

 

[Dale]

And yet, does this not contradict the third law of thermodynamics: "The entropy of a system approaches a constant value as its temperature approaches absolute zero. "?

It does not contradict it. Why do you think it would?

 

[Joe]

It does not contradict it. Why do you think it would?

Because the most ordered state of the universe would be at the coldest temperature. If the universe is already expanding and getting colder, that would mean it is moving towards the most order (low entropy?). On the other hand, it's said that expansion of the universe, from the Big Bang till now, means higher disorder.

 

[Dale]

Because the most ordered state of the universe would be at the coldest temperature. If the universe is already expanding and getting colder, that would mean it is moving towards the most order (low entropy?).

You are mistaken here. Go back to the equation. For a fixed energy* the entropy is higher at a lower temperature. Note that the third law of thermodynamics is not about a fixed energy.

It seems like you are getting confused with the somewhat fuzzy idea of “order” vs the concrete idea of entropy. Focus on the quantitative definition of entropy only until you understand entropy on its own. Only then try to understand how it relates to “order”. In all likelihood you will have to change your idea of what “order” means in thermodynamics.

The initial state of the universe is a very low entropy state and the entropy is increasing as the temperature decreases.

*one caveat is that the total energy of the universe is not defined, so this requires some care to actually translate into cosmology

 

[Anand Sivaram]

Dale has given a very insightful answer. Actually I also had the exact doubt as the OP a few years
before. I thought over it a lot and here are some of my understandings (it could be wrong too)

>> As the temperature goes down, the entropy goes up.

What we calculate is the *change* in entropy, rather than the actual entropy, as per all of these equations. The change in entropy is much more prominent at lower temperatures as T is in the denominator. So at Absolute Zero, that is at 0K, there would be minimum kinetic energy, means maximum "order". But, when an object takes heat at 0K, the change in entropy would be infinite.

Consider this. A vehicle increases speed from 100kmph to 101kmph, then the change is 1 percent,
when speed is increased from 10kmph to 11kmph it is 10 percent. Whereas a standstill vehicle starts
crawling to a a speed of 1kmph, then the change of speed is infinite.

 

[Joe]

You are mistaken here. Go back to the equation. For a fixed energy* the entropy is higher at a lower temperature. Note that the third law of thermodynamics is not about a fixed energy.

It seems like you are getting confused with the somewhat fuzzy idea of “order” vs the concrete idea of entropy. Focus on the quantitative definition of entropy only until you understand entropy on its own. Only then try to understand how it relates to “order”. In all likelihood you will have to change your idea of what “order” means in thermodynamics.

The initial state of the universe is a very low entropy state and the entropy is increasing as the temperature decreases.

*one caveat is that the total energy of the universe is not defined, so this requires some care to actually translate into cosmology

Thank you. As you said, I have to get a better grasp of the entropy and thermodynamics.

 

[Joe]

Dale has given a very insightful answer. Actually I also had the exact doubt as the OP a few years
before. I thought over it a lot and here are some of my understandings (it could be wrong too)

>> As the temperature goes down, the entropy goes up.

What we calculate is the *change* in entropy, rather than the actual entropy, as per all of these equations. The change in entropy is much more prominent at lower temperatures as T is in the denominator. So at Absolute Zero, that is at 0K, there would be minimum kinetic energy, means maximum "order". But, when an object takes heat at 0K, the change in entropy would be infinite.

Consider this. A vehicle increases speed from 100kmph to 101kmph, then the change is 1 percent,
when speed is increased from 10kmph to 11kmph it is 10 percent. Whereas a standstill vehicle starts
crawling to a a speed of 1kmph, then the change of speed is infinite.

Thank you Anand. That was my understanding too. In the above equations, they consider the change in the entropy and its relation to temperature; whereas, in the Third Law, it's the value of the entropy - or lack of change in entropy - at a given temperature. I may still be wrong! I really have to review the whole thing, as Dale said.

 

[Philip Koeck]

I saw another post about dS = dQ/T, but the subject of question was different - not related to the entropy of universe.
This is what i understand from this formula:
As the temperature goes down, the entropy goes up. Is this not the opposite (contradictory) to what entropy (disorder) is about? At a lower temperature, there should be more order (lower entropy). From what I was taught, the highest entropy would be when the temperature of a system reaches absolute zero - the third law of thermodynamics. Am I correct?
Also how does the energy of the system (in this case universe) explain this equation? A higher drop in the enthalpy - which is directly related to the energy of the system and its volume (space) - would result in more disorder? That would make sense to me. Is that correct?

Hi Joe. The formula you give means the following: If you add heat dQ to something at constant temperature T the entropy of this something increases by dS. (This is strictly only true if you add the heat in a reversible way.) So adding heat always increases the entropy, and the increase of entropy is bigger the smaller temperature is, given the same amount of heat added.
The formula only applies to a change of entropy due to added heat at constant T.
It doesn't mean that entropy itself is smaller when T is higher
This also means that you can't apply this formula to the universe since you can't add heat to the universe. Where would you take it from? Hope that helps a bit.