Equivalence of Boltzmann and Clausius entropy

[Bobhawke]

I can't figure out how to show that the Clausius entropy

$$\Delta S= \int_{T_1}^{T_2} \frac{dQ}{T}$$

is equivalent to the Boltzmann entropy

$$S=kln \Omega$$

at thermal equilibrium.

Does anyone know how to do this?

 

[flatmaster]

Might want to see how it works for monoatomic ideal gas first, then look for a generalization.

 

[astrokreng]

It is a common misconception that the Boltzmann and Clausius entropies are not equivalent. In fact, at thermal equilibrium, they are mathematically equivalent. This can be shown by considering the statistical mechanics approach to thermodynamics.

The Boltzmann entropy, S, is defined as the logarithm of the number of microstates, Ω, that are consistent with a given macrostate. This means that at thermal equilibrium, the system is in a state with the highest possible number of microstates. This is also known as the maximum entropy state.

On the other hand, the Clausius entropy, ΔS, is defined as the change in entropy of a system due to the transfer of heat, Q, from a reservoir at temperature T. At thermal equilibrium, there is no net transfer of heat between the system and the reservoir, meaning that ΔS=0.

Using the definition of the Clausius entropy, we can rewrite it as:

ΔS= \int_{T_1}^{T_2} \frac{dQ}{T}

Since at thermal equilibrium, ΔS=0, we can set this expression equal to zero:

0= \int_{T_1}^{T_2} \frac{dQ}{T}

Now, using the first law of thermodynamics, we can write dQ=TdS. Substituting this into the above equation, we get:

0= \int_{T_1}^{T_2} \frac{TdS}{T}

Simplifying, we get:

0= \int_{T_1}^{T_2} dS

Integrating both sides, we get:

0= S(T_2)-S(T_1)

Since S(T_2) and S(T_1) are both at thermal equilibrium, they are both equal to the maximum entropy state, S_{max}. Therefore, we can rewrite the above equation as:

0= S_{max}-S_{max}=0

This shows that at thermal equilibrium, the Clausius entropy and the Boltzmann entropy are mathematically equivalent, as they both represent the maximum entropy state of a system.