# Why can entropy be measured in joules per kelvin?

I'm a little confused as to what the measurement associated with entropy represents. It's obviously the joules of energy per kelvin, but how does this measure the disorder of a system?

• I'm not 100% sure if the units can be directly related to the interpretation as a measure of disorder. The equation is $S = k \ln W$ and if the system is more "disordered", then $W$ increases, but $W$ and $\ln W$ are both dimensionless; the dimension of J/K arises from the Boltzmann constant $k$. That said, there's probably some explanation that I haven't heard of. – orthocresol Nov 16 '16 at 22:44
• btw, using 'disorder' related to entropy should be avoided as its difficult to distinguish in ones mind between a little and a lot of disorder. The 'number of configurations', ( 'W' or $\Omega$) is better but unclear also, so its better to think about the number of ways of placing particles into their several energy levels. Classically entropy is related to the heat absorbed from surroundings ($T\Delta S$) when an isothermal reaction runs reversibly. If positive the work done is greater than heat of reaction. Only from stat. mech. do we understand entropy fully. – porphyrin Nov 17 '16 at 11:49

The definition of entropy in 'Classical Thermodynamics' is $$\Delta S = \int\frac{\delta Q}{T}$$

The quantity on the left is the entropy change associated with a physical process. $\delta Q$ is an inexact differential, made exact by an integrating factor $\frac{1}{T}$ This notion was devolped by Kelvin and Cartheodory (I believe). This comes out of the analysis of certain kinds of differential equations called Pfaffian forms.

One notes the dimensions ("units") of this quantity, and observers them to be $\mathrm{J\ K^{-1}}$

It was only later, that the statistical definition of entropy (developed by Boltzman) came to be (and was found to be equivalent to the thermodynamic definition).

$$S = -k_B \sum_{i}p_i\ln(p_i)$$

In this framework entropy is a logarithmic measure of the number of states with significant probability of being occupied. In this expression, the the quantity is changed from dimensionless to dimensionful, by the Boltzmann Constat, which has the units, (you guesed it) $\mathrm{J\ K^{-1}}$

The measure of "disorder" is in the term $\sum_{i}p_i\ln(p_i)$, which has no units. The units are appended to it by the Boltzmann Constant.

Entropy is a derived quantity. It, therefore, takes on values [and units] from its definition. In this case, the SI base units of J and K (for heat and temperature respectively).

A change in entropy may be defined as being equal to a quantity of heat transferred in a reversible isothermal process divided by the absolute temperature like e.g. in some parts of a Carnot cycle:

$$\Delta{S}=\frac{Q_{rev}}{T}$$

It is simply a relationship that was given its own special symbol for convenience i.e. S is a function that describes the thermodynamic state of a system (which is what makes it important).