Show that the entropy is non-negative

• patrickmoloney
In summary, the conversation discusses the change in entropy after two vessels containing N molecules of the same monatomic gas are brought into thermal contact at the same pressure but different temperatures. The solution shows that the total change in entropy is positive, indicating that entropy increases during this process. This is proven by using the first law of thermodynamics and determining the final temperature using the average of the initial temperatures.
patrickmoloney

Homework Statement

Two vessels $A$ and $B$ each contain $N$ molecules of the same perfect monatomic gas. Initially, the two vessels are thermally isolated from each other, with the two gases at the same pressure $P$ and at temperatures $T_A$ and $T_B$. The two vessels are now brought into thermal contact, with the pressure of the gases kept constant at the value $P$

Find the change in entropy after equilibrium and show that this change is non-negative.

Homework Equations

$$Q_{in} = -Q_{out}$$

$$\Delta S = \int \dfrac{dQ}{T}$$

The Attempt at a Solution

$$\Delta S_A = \int_A^f \dfrac{dQ_A}{T}= \int_A^f \dfrac{mcdT}{T}= mc \ln \Big{(}\dfrac{T_f}{T_A}\Big{)}$$
$$\Delta S_B = \int_B^f \dfrac{dQ_B}{T}= \int_B^f \dfrac{mcdT}{T}= mc \ln \Big{(}\dfrac{T_f}{T_B}\Big{)}$$

$$\Delta S_{tot} = \Delta S_B - \Delta S_A = mc \ln \Big{(}\dfrac{T_A}{T_B} \Big{)}$$

how can a prove that $\ln \Big{(}\dfrac{T_A}{T_B}\Big{)} > 0$

It should be the summation of the entropy changes, not the difference.

patrickmoloney
Oh yeah! Thanks $$\Delta S_{tot}= mc \ln \Bigg{(}\dfrac{T_{f}^2}{T_A T_B}\Bigg{)}$$

I know that $mc > 0$. How do I prove that $$\dfrac{T_{f}^2}{T_A T_B} >1$$ I suppose $T_{f}^2 \gg T_A T_B$ but that's not a proof.

Edit: However

$$T_f = \Big{(}\dfrac{T_A +T_B}{2}\Big{)}$$ so I guess that's it.

You need to determine Tf using the 1st law.

patrickmoloney
Chestermiller said:
You need to determine Tf using the 1st law.
Yeah I thought so! Thanks a lot.

1. What is entropy and why is it important?

Entropy is a measure of the disorder or randomness in a system. It is important because it helps us understand the efficiency of a process and the direction of energy flow.

2. How is entropy related to the second law of thermodynamics?

The second law of thermodynamics states that the total entropy of a closed system will always increase over time. This means that the entropy of a system is a measure of the system's tendency to move towards disorder.

3. How is the non-negativity of entropy proven?

The proof of the non-negativity of entropy involves using statistical mechanics and the concept of microstates and macrostates. It can be shown that in any closed system, the number of microstates that correspond to a given macrostate (state of order/disorder) will always be greater than or equal to the number of microstates in any other macrostate, resulting in a non-negative value for entropy.

4. Can entropy ever decrease?

In a closed system, the total entropy will always increase or remain constant. However, within the system, it is possible for entropy to decrease in one part while increasing in another, as long as the total entropy of the system as a whole increases.

5. How does entropy relate to information theory?

In information theory, entropy is a measure of the uncertainty or randomness in a system. It is used to quantify the amount of information contained in a message or signal. The higher the entropy, the more uncertain or random the message is, and the more information it contains.

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