An ideal gas in a cylinder is separated by a piston in such a way that the entropy of one part is S1

Q: Why don't we multiply the entropies?

Entropy is an extensive property, meaning it is additive, just like mass or volume. You add masses together, you don't multiply them.

Q: Does the condition $S_1 > S_2$ affect the final answer?

No, the condition $S_1 > S_2$ is just given to indicate that the two parts have different amounts of gas or are in different states, but the additive rule still applies perfectly.

Q: What about the entropy of mixing?

If the gases are identical and at the same temperature and pressure, the entropy of mixing is zero. The question implies a simple combination of state variables without additional irreversible generation.

Q: Is entropy a state function?

Yes, entropy is a state function. Its value depends only on the current state of the system, not on the path taken to reach it.

Q: What are examples of intensive properties?

Temperature, pressure, and density are intensive properties. They do not depend on the quantity of matter present.

Question

An ideal gas in a cylinder is separated by a piston in such a way that the entropy of one part is $S_1$ and that of the other part is $S_2$. Given that $S_1 > S_2$. If the piston is removed, then the total entropy of the system will be

Options:

(A) $S_1 	imes S_2$

(B) $S_1 - S_2$

(C) $\frac{S_1}{S_2}$

(D) $S_1 + S_2$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine a rigid cylinder filled with an ideal gas. Right in the middle, or perhaps slightly off-center, sits a piston that divides the cylinder into two distinct compartments. The problem tells us that the entropy of the gas in the first compartment is $$S_1$$, and the entropy of the gas in the second compartment is $$S_2$$. We are also given a condition that $$S_1 > S_2$$, which simply implies that the two compartments might have different volumes, different amounts of gas, or are in different states.

### The Master Concept: Extensive Properties

To solve this, we need to recall a fundamental concept from thermodynamics: the difference between **intensive** and **extensive** properties.

Intensive properties, like temperature and pressure, do not depend on the amount of matter. If you combine two identical glasses of water at $$25^\circ	ext{C}$$, the final temperature is still $$25^\circ	ext{C}$$, not $$50^\circ	ext{C}$$.

Extensive properties, on the other hand, depend directly on the quantity of matter. Mass and volume are classic examples. If you combine $$1	ext{ kg}$$ of water with another $$1	ext{ kg}$$ of water, you get $$2	ext{ kg}$$.

**Entropy ($$S$$)** is an extensive property. It is a measure of the total number of microstates available to the system, and naturally, more particles mean more microstates and more entropy.

### Final Calculation

When the piston is removed, the two compartments merge into a single, unified system. Because entropy is an extensive property, the total entropy of the combined system is simply the algebraic sum of the entropies of its individual parts.

$$S_{	ext{tot}} = S_1 + S_2$$

It really is that straightforward! The condition $$S_1 > S_2$$ is just a distractor to make you overthink. The additive nature of extensive state variables holds true regardless of their relative magnitudes.

Animated Solution for Physics - Thermodynamics: An ideal gas in a cylinder is separated by a piston in such a way that the entropy of one part is $S_{1}$ and that of the other part is $S_{2}$ . Given that $S_{1} > S_{2}$ . If the piston is removed, then the total entropy of the system will be

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Visualized Solution

Visual Anchor

Logic Bridge

Raw Setup

Final Answer

The Way Forward

The Sigma Insight: Second Law of Thermodynamics

Analyzing the Setup

The Master Concept

Final Calculation

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