What Are the Implications of Choosing Different Paths in Entropy Calculations?

[Haorong Wu]

TL;DR

Are there any restrictions of choosing circles in Clausius theorem?

Hi, I am currently reading Introduction to statistical physics by Huang. In the section of entropy, it reads

Let $P$ be an arbitrary path from $A$ to $B$, reversible or not. Let $R$ be a reversible path with the same endpoints. Then the combined process $P-R$ is a closed cycle, and therefore by Clausius' theorem $\int_{P-R} dQ/T \leq 0$, or
$\int_{P} \frac {dQ} {T} \leq \int_{R} \frac {dQ} {T}$.
Since the right side is the definition of the entropy difference between the final state $B$ and the initial state $A$, we have $S \left ( B \right ) - S \left ( A \right ) \geq \int_{A}^{B} \frac {dQ} {T}$ where the equality holds if the process is reversible.

But what if I choose $R-P$ as a closed cycle? Then in a similar process, I should have $\int_{R} \frac {dQ} {T} \leq \int_{P} \frac {dQ} {T}$ and $S \left ( B \right ) - S \left ( A \right ) \leq \int_{A}^{B} \frac {dQ} {T}$, which are contradicted to the equations above. I am not sure what goes wrong. Maybe there are some restrictions when I choose a closed cycle, but I did not find any relevant context in the book.

 

[anuttarasammyak]

In your setting you should have
$$\int_R \frac{dQ}{T} \leq -\int_{-P}\frac{dQ}{T}$$ where
$$-\int_{-P} \frac{dQ}{T} \neq \int_{P}\frac{dQ}{T}$$
because P is not a reversible process.

 

[Haorong Wu]

In your setting you should have
$$\int_R \frac{dQ}{T} \leq -\int_{-P}\frac{dQ}{T}$$ where
$$-\int_{-P} \frac{dQ}{T} \neq \int_{P}\frac{dQ}{T}$$
because P is not a reversible process.

Thanks, anuttarasammyak. I understand it now.

 

[Chestermiller]

Summary:: Are there any restrictions of choosing circles in Clausius theorem?

Hi, I am currently reading Introduction to statistical physics by Huang. In the section of entropy, it reads
But what if I choose $R-P$ as a closed cycle? Then in a similar process, I should have $\int_{R} \frac {dQ} {T} \leq \int_{P} \frac {dQ} {T}$ and $S \left ( B \right ) - S \left ( A \right ) \leq \int_{A}^{B} \frac {dQ} {T}$, which are contradicted to the equations above. I am not sure what goes wrong. Maybe there are some restrictions when I choose a closed cycle, but I did not find any relevant context in the book.

For the irreversible paths, P can't be made the same for any opposite path and for the forward path. If it happens spontaneously for the forward path, it will not be spontaneous for the reverse path, and you can't even force it to follow the exact reverse path.

 

[Haorong Wu]

For the irreversible paths, P can't be made the same for any opposite path and for the forward path. If it happens spontaneously for the forward path, it will not be spontaneous for the reverse path, and you can't even force it to follow the exact reverse path.

Thanks, Chestermiller. I just start learning statistical physics, and thanks for pointing this important point for me.