# Why is entropy of an object inversely related to its temperature?

• I
Gold Member

## Summary:

Please explain why in thermodynamics the entropy (disorder) of an object is inversely related to its temperature.

## Main Question or Discussion Point

I read in a book "Quantum Space" by Jim Baggot, page 290, that the entropy of an object is inversely proportional to its temperature. (He was describing the temperature of a black hole. Does this statement only apply to black holes?) No doubt he is correct, but wouldn't an increase of energy within an object give more possible states? Some molecules within a gas could have higher velocities. The gas might expand into a larger volume.

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Staff Emeritus
2019 Award
Please explain why in thermodynamics the entropy (disorder) of an object is inversely related to its temperature.
In general it's not/

Does this statement only apply to black holes?
It's certainly a leap from "it applies to black holes" to "it applies everywhere". It is probably true for any system with negative heat capacity.

the entropy of an object is inversely proportional to its temperature
The entropy of the heat energy inside a rock is inversely proportional to the temperature of the rock. And proportional to the amount of the heat energy.

S = U/T

I wonder if we are allowed to say that the entropy of the heat energy inside a rock is inversely proportional to the temperature of the heat energy inside the rock? Like maybe the temperature T is the temperature of the heat energy U?

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Staff Emeritus
2019 Award
Do you understand heat capacity?

wouldn't an increase of energy within an object give more possible states? Some molecules within a gas could have higher velocities. The gas might expand into a larger volume.
Let's say the object is a mixture of steam and liquid water at constant pressure. When we heat this object, its heat energy increases, its entropy increases, and its temperature stays constant. And that's because when one Joule of heat energy is added to that object, the object changes so that it has one Joule more potential energy. In this case there is no correlation between temperature and entropy.

What would happen if adding one Joule of heat energy to some object caused the object to have two Joules more potential energy? Well, when that object is heated, its heat energy increases, its entropy increases, and its temperature decreases. In this case there is a negative correlation between temperature and entropy.

And there are many cases when there is a positive correlation between entropy and temperature.

hilbert2
Gold Member
If heat capacity ##C(T)## is a constant, entropy depends logarithmically on temperature. It's a result of integrating the infinitesimal definition

##\displaystyle dS=\frac{dQ}{T} = \frac{CdT}{T}##,

where there is the reciprocal ##T## proportionality. To get the entropy of something behave like ##S(T)\propto T^{-1}##, you'd have to set the function ##C(T)## just right for the integral to produce that result. What would it have to be?

• Livio Arshavin Leiva
If heat capacity ##C(T)## is a constant, entropy depends logarithmically on temperature. It's a result of integrating the infinitesimal definition

##\displaystyle dS=\frac{dQ}{T} = \frac{CdT}{T}##,

where there is the reciprocal ##T## proportionality. To get the entropy of something behave like ##S(T)\propto T^{-1}##, you'd have to set the function ##C(T)## just right for the integral to produce that result. What would it have to be?
It would have to be ##C(T)\propto-T^-1##, right? In case the entropy is positive, this would mean the heat capacity needs to be negative.

hilbert2
Gold Member
It would have to be ##C(T)\propto-T^-1##, right? In case the entropy is positive, this would mean the heat capacity needs to be negative.
Yes, that's true and I don't think there is any material with that property.

Staff Emeritus
2019 Award
Yes, that's true and I don't think there is any material with that property.
Every gravitationally bound system has this property.

It appears that the OP has lost interest, though.

• hilbert2 and etotheipi
hilbert2
Gold Member
Every gravitationally bound system has this property.
Like a large cloud of point particle ideal gas that doesn't expand indefinitely because gravitation holds it together? I think there's the problem of having to set both temperature and density at the exact correct values for it to not collapse either, but I don't know much about GR.