Reading a ln W vs E graph

Problem

The graph shows $\ln W(E)$ as a function of energy $E$ for two isolated systems, $1$ and $2$, which can exchange energy through a diathermic wall ($W$ is the number of microstates). Discuss how thermal equilibrium is established and under what condition the total entropy is maximum.

Solution

Setup: the total microstates multiply, so the entropies add up: $S_\text{tot}=k_B\ln(W_1 W_2)=k_B\big(\ln W_1+\ln W_2\big)$ The total energy is fixed, $E_1+E_2=\text{const}$, i.e. $dE_2=-dE_1$: by exchanging a bit of energy from $2$ to $1$, $S_\text{tot}$ varies until it reaches the maximum.

Maximum condition ($dS_\text{tot}/dE_1=0$): $\frac{\partial \ln W_1}{\partial E_1}=\frac{\partial \ln W_2}{\partial E_2}$ that is, the slopes of the two curves on the graph must coincide. Since by definition $\frac{\partial S}{\partial E}=\frac{1}{T}$ the equality of the slopes is equivalent to $1/T_1=1/T_2$, i.e. $\ev{T_1=T_2}$ Thermal equilibrium is therefore the point of maximum total entropy: energy spontaneously redistributes itself until the two temperatures become equal.

Linked atoms

Topics: Entropia e secondo principio Concepts: Entropia · Temperatura Skills: Interpretazione micro-macro · Lettura dei grafici