# Dependence of change in entropy on temperature

The thermodynamic definition of entropy is expressed as $$dS = \frac{dq_{rev}}{T}$$ I understand that it depends on amount of heat transferred because heat transfer can be understood as the degree of dispersion of energy but what does the temperature dependence actually mean?

• By "what does [it]... actually mean", do you mean "how can we intuitively understand why there is a temperature dependence"? – ericksonla Apr 30 '15 at 18:37
• yes @ericksonla, I'm looking for an intuitive explanation. – Apoorv Apr 30 '15 at 19:13

An analogy that I like is one paraphrased from Peter Atkins' great book "The Laws of Thermodynamics - a VSI":

Imagine two rather different places in terms of noise - a silent library and a busy train station. Imagine that the temperature is the amount of noise at the given place - a measure of the "state of noise". Let's say that you laugh out loud, corresponding to supplying energy as heat.

In the library, where there is little background noise (low $T$) this would cause a large amount of disturbance and disorder - corresponding to a large increase in entropy. On the busy train station, on the other hand, you laughing would go largely unnoticed - there is lots of background noise already (high $T$). The (relative) amount of additional disturbance and disorder would be much smaller, corresponding to a smaller increase of entropy.

In a very cold system, even a tiny amount of supplied energy in terms of heat would cause a large change in entropy. On the other hand, supplying the same amount of energy to a really hot system - with lots of molecular motion going on - would cause a minuscule change in entropy.

Disclaimer: Note that the analogy and explanation above are two ways of convincing yourself that the entropy change due to added heat is inversely proportional to the system temperature. They do not claim to be correct or physical in any other sense, so read them with a (large) pinch of salt.

• that's a good way to think about it but i would also like to hear the physical side of it. – Apoorv May 1 '15 at 7:03

What this equation is saying essentially is that the level of disorder can be modeled as the number of configurations if you have a number of particles and a number of quanta of energy. It's like having three cups and three balls; all the balls can occupy one cup or one ball can occupy one cup and two can occupy another etc. Hence an increase in energy would result in an increase in entropy (more balls and more configurations). Temperature is defined as the average energy of the system. Hence if energy is concentrated (all the balls are in one cup) then the average temperature per particle will be lower and the entropy will be lower which makes sense since the system is more ordered with all the balls in one cup.

I hope this helps answer the question somewhat

• Thanks for answering but isnt this the statistical definition of entropy? I believe there might be an alternate explanation that does not involve talking about microstates – Apoorv May 1 '15 at 5:38