What do the units of entropy quantify?

Question

Entropy has the units $J\cdot K^{-1}$. Velocity has the units $m \cdot s^{-1}$. In the latter example, I know what the units are quantifying across all applications of the quantity of velocity. Velocity as a quantity is applied to a body, and thus the meters quantifies the distance travelled by the body, and the seconds quantify the time it took for the object to travel that distance.

For entropy however, I do not know how to fully do this. Entropy is a quantity that is applied to the state of a thermodynamic system (I think). Thusly, I assume the Kelvins quantify the average kinetic movement of the particles within (the temperature of) the thermodynamic system in-question. But what about the Joules? They quantify energy, but the energy of what?

I suspect that with entropy, the Joules are quantifying the energy transmission between the system and its surroundings. Likely, I think the Joules are specifically quantifying the heat exchanged between the system and its surroundings. At max entropy, all of the energy accessible from the system at any moment (as quantified by Kelvins) is being transmitted as heat by the system to its surroundings. At minimum entropy, as much as possible of the accessible energy accessible from the system is being transmitted to the surroundings via non-heat, like electricity or mechanical work. If something like this interpretation is correct, it would explain a lot of the descriptions of entropy, like unuseful energy and disorder.

Basically, I am asking what the units of entropy are quantifying. I know they quantify energy and temperature respectively, but the question is; the energy and temperature of what?

Answer 1

It kind of depends on how you are using it... for example, if you are looking at temperature as expressed from the thermodynamic identity as

$$\frac1T=\left(\frac{\partial S}{\partial U}\right)_{V,\,N}$$

then entropy is the thing whose rate of change with respect to internal energy gives us temperature. i.e. the energy in the units of entropy is the internal energy of the system.

Of course, at constant $V$ and $N$ we know that the change in internal energy is due to transfer of energy due to heat: $\text dS=\text dQ/T$. So you could also say it is heat energy.

And still then we can define things such as free energy $F=U-TS$ where entropy also pops up, which leads to things like

$$S=-\left(\frac{\partial F}{\partial T}\right)_{V,\,N}$$

I understand wanting to relate units to simpler things like velocity, but with entropy you have different ways it relates to different variables, and you have so many different types of systems, it's a little harder to nail down what is a useful way to view the entropy units.

Answer 2

From my answer to Which effect in everyday life is due to the Boltzmann constant?

The SI units of entropy occur because historically energy and temperature are measured in different units. The Boltzmann constant $k_B$ is essentially a conversion factor between energy and temperature.

The real physics in entropy $S=k_B \ln\Omega$ comes from the multiplicity $\Omega$, not from the Boltzmann constant $k_B$.

Answer 3

I assume the Kelvins quantify the average kinetic movement of the particles within (the temperature of) the thermodynamic system in-question. But what about the Joules? They quantify energy, but the energy of what?

I suspect that with entropy, the Joules are quantifying the energy transmission between the system and its surroundings.

Yes.

$ Change of Entropy = \frac{kinetic Energy Added To All Molecules}{kinetic Energy Of Average Molecule} $

The rest of what you said made no sense at all :)