Reference: Daniel V. Schroeder, *An Introduction to Thermal Physics*, (Addison-Wesley, 2000) – Problem 3.1.

The concept of entropy as , where is the multiplicity of the macrostate in which the system is found, can be used to define the temperature of the system. Schroeder gives a good explanation in his section 3.1 so I’ll summarize the argument here.

We’ll use two interacting Einstein solids with and , with total energy quanta. The multiplicity function for each solid is

and the total multiplicity of the combined system is

The entropies are therefore

where we pick the subscript for the system we’re interested in. We can plot the three entropy curves as functions of (remember ), to get

Here the top red curve is , the violet curve is and the turquoise curve is . reaches a maximum at which is the macrostate where the energy is evenly distributed among all the oscillators. At this point, therefore

Since , this implies that

Since , so we can write this as

In the small scale system here, it’s not quite right to consider and as continuous variables, so the partial derivatives are an approximation. In very large systems, however, the quanta merge into a continuous energy variable , so we can write

As the units of entropy are and of are , this derivative has the dimensions of or the reciprocal of temperature. We can therefore *define* temperature to be

where the partial derivative implies that we hold everything else apart from the energy of the system constant while taking the derivative. Thus two systems that can exchange energy until they reach their most probable macrostate will end up with the same temperature, so they are in thermal equilibrium.

The relation 6 states that the slopes of the entropy-versus-energy curves are equal for the most probable macrostate. In the graph above, this means that the slope of the turquoise line is the negative of the slope of the violet line at , which looks about right if you eyeball the graph. We could prove it by taking the actual derivatives, but we’ll make do with a numerical example.

Suppose each energy quantum has a value of . We can then estimate the temperatures of the two solids at by calculating the slope of the line connecting the points for and . For solid , we use so the two energy points are and . We get

Thus the two temperatures are indeed equal at .

For we can use the slope between and for solid and and for solid . We get

Here, solid is much hotter than solid so if they are interacting, there would be a strong tendency for to transfer some of its energy to to bring the solids into thermal equilibrium.

Pingback: Zeroth law of thermodynamics | Physics pages

Pingback: Thermal equilibrium for entropy plots | Physics pages

Pingback: Temperature of an Einstein solid | Physics pages

Pingback: Energy of a system with quadratic degrees of freedom | Physics pages

Pingback: Temperature of a black hole | Physics pages

Pingback: Predicting heat capacity | Physics pages

Pingback: Third law of thermodynamics; residual entropy | Physics pages

Pingback: Two-state paramagnet: numerical solution | Physics pages

Pingback: Two-state paramagnet: analytic solution | Physics pages

Pingback: Einstein solid – numerical solution | Physics pages

Pingback: Einstein solid: analytic solution for heat capacity | Physics pages

Pingback: Pressure in terms of entropy; the thermodynamic identity | Physics pages

Pingback: Chemical potential; application to the Einstein solid | Physics pages

Pingback: Helmholtz energy as a function of volume | Physics pages