# Effect of temperature on Entropy

#### weng cheong

i'm a pre-U student, and i came across this when i study the topic on entropy

dS = dQ / T

i understand that as temperature increases, entropy increases as well, as there are more quanta of energy and more thermal states(energy levels) available.

however according to this equation, it seems to indicate that with a lower temperature, we can get a greater entropy?

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#### Bloodthunder

Re: how temperature affect entropy

S is entropy. dS is change in entropy.
So, for a given temperature T, the change in entropy is equal to the change in heat energy divided by whatever the temperature that system is currently in.

#### Mapes

Homework Helper
Gold Member
The equation you wrote describes the infinitesimal increase in entropy of a system at constant temperature when infinitesimal energy is added reversibly by heating. That's not the same as the change in entropy of a system with increasing temperature, which is $\partial S/\partial T$. This quantity is, indeed, always positive.

You can integrate your equation to get $\Delta S=Q\ln(T_2/T_1)$, which confirms that entropy increases with temperature when a system is heated.

Does this make sense?

#### Mike H

You get a larger change in entropy, that's correct.

ETA - I figured that constant temperature and reversibility were givens, considering the Clausius formalism used by the OP. (Although that dQ should be dQrev, which is how I recall seeing it written in textbooks.)

ETA 2 - One analogy that might be useful - if you scream at a sports game (add a tiny bit of energy to a high-temperature system), most likely you will not make that much of a dent in the general soundscape (a small change in entropy). If you yell in the middle of a wedding (adding a tiny bit of energy to a low-temperature system), it would be far more dramatic (a large change in entropy).

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#### Studiot

Re: how temperature affect entropy

So, for a given temperature T, the change in entropy is equal to the change in heat energy divided by whatever the temperature that system is currently in.
Careful!

The temperature of the the system at the point (moment) of heat exchange, currently is a bit vague and could mean after the exchange.

Note also it is a differential relationship so may be integrated from point to point.

#### thebiggerbang

Imagine this situation.
You provide X joules of heat to a system at
1)say 100 K and 2)say 300K

So, in the first case, what you will observe is the there will be a larger change in the randomness than in the second case, as due to the pre-existing higher temperature, there will be already a large amount of disorder in the system!

It's just like shuffling a deck of kinda properly arranged cards (analogous to a lower T) will give you a larger change in disorder than shuffling a pack of cards that are already random in order (higher temperature!).

Am I correct?

#### Studiot

Imagine this situation.
You provide X joules of heat to a system at
1)say 100 K and 2)say 300K
The entropy change in the system, in both cases, depends upon the conditions of energy input, which have not been specified.

#### thebiggerbang

The entropy change in the system, in both cases, depends upon the conditions of energy input, which have not been specified.

#### Studiot

If both additions of heat are reversible then the entropy change of the second case input is 1/3 of the first.

The addition of heat may become the work of expansion, which allows greater entropy in the form of freedom of space to occupy,

or

It may be taken up in a phase (state) change so the particles become more disordered.

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