# Making sense of Entropy being energy over temperature

Over the internet, I found many resources that explain very well what entropy represents. However, the dimensions still appear hard for me to make sense of: energy, simply put, is the ability to do work. Temperature is a measure of the average kinetic energy of the particles inside the system. How can one then make intuitive sense of entropy being one over the other?

Intuitively, if I think of a system that has different temperatures across, I realize that energy (heat) tends to spread out as temperature becomes more uniform throughout the system. But if we assume the system to be isolated and that the entropy increases as the process happens (which it must), then by the law of conservation of energy the only possibility for entropy to increase is that the average temperature of the system decreases. So it should be that the energy is distributed more "evenly" across the molecules, i.e. first "only a few molecules were very excited", now "more molecules are somewhat excited" where by excited I mean high in kinetic energy.

At a physical level, is this what we mean by "there is more heat for every unit of temperature"?

• You have three questions here. You should pick one and focus on that
– Dale
Aug 29, 2021 at 0:50
• @Dale the first two questions are related to the last, most important one, in which they are an attempt to give an answer to it. I thought describing my thought process would help seeing where I've gone wrong to someone who knows the answer. Aug 29, 2021 at 10:01
• The problem is that a person who might answer the last question might be discouraged from answering because they cannot or do not want to answer all three. I didn’t respond because I would have written a lengthy answer to the first and then ran out of time for the others. You should remove the first two questions and explain your thought process without asking additional questions.
– Dale
Aug 29, 2021 at 11:51
• @Dale had complettely forgotten to rewrite this. It should be more addressable now. Oct 13, 2021 at 16:14

I have an analogy that is helpul (at least to me).

Replace the world energy by income, and energy levels by income ranges. In a country with no restriction for minimum or maximum incomes, and where luck to be born rich and/or gifted is the major factor to get money, there is an income distribution of equilibrium.

In that analogy, temperature is related to how spread is the curve, how big is the inequality. Supposing that the degree of inequality is the same (due to the randomic hypothesis), countries with higher average incomes have bigger absolute differences between top and bottom ranges. For example to be considered rich in a poor country means less income than in a rich country. So, higher energy correlates with higher temperature.

If for any historical reason, two regions were artificially separated, and had different average incomes, sometime after joining, instead of two different distributions there will be one. Supposing no increase of the total income (what means: keeping the same total energy), natural competition in the market leads the average income of the poorer region to increase and that of the richer region to decrease. In the process, if we divide the income increase of each year of the poorer region by the spread of its curve, the quotient is greater than to divide the decrease of income (same magnitude) of the richer region by its (bigger) curve spread. The conclusion is that the natural process of contacting 2 regions of different average incomes (energies) and range of inequality (temperatures) results in an increase of entropy, when the total income (energy) is conserved.

• +1 If I am not mistaken, there is even an RMP article on the statistical physics of incomes. Oct 15, 2021 at 4:54

So it should be that the energy is distributed more "evenly" across the molecules,

Yes, I that is a good way to describe it. Often entropy is described in terms of “order” and “disorder”, but my experience is that those terms are not understood correctly and lead to all manner of incorrect conclusions about entropy. Instead, if entropy is described as a measure of how “spread out” the energy is then it becomes easier for people to make correct conclusions about how it behaves.

Regarding the relationship between temperature, entropy, and heat, I prefer this form: $$T \ dS = \delta Q$$

For a fixed $$\delta Q$$ it is easy to see that as $$T$$ decreases $$dS$$ increases. So, by having heat flow from a high temperature reservoir to a low temperature reservoir the decrease in entropy in the high temperature reservoir is less than the increase in entropy in the low temperature reservoir.

But how does that tie into the idea of entropy as being how spread out the energy is?

Intuitively, if you have a lot of energy in a small space it will be hot and if it is spread out then it will be cool. And looking microscopically we find that the characteristic energy of a single microscopic degree of freedom is given by $$kT$$. So that means that at a given $$\delta Q$$ corresponds to a small number of degrees of freedom for large $$T$$ and a large number of degrees of freedom for a small $$T$$. These degrees of freedom are where the energy spreads out. So a large number of degrees of freedom is energy that is quite spread out.

Entropy measures how spread out the energy is, spread out energy occupies a lot of degrees of freedom, that means a small amount of energy per degree of freedom, which is the characteristic energy $$kT$$.