$2\ ext{mol}$ of an ideal gas expands isothermally and reversibly from $10\ ext{L}$ to $20\ ext{L}$ at a constant temperature. Calculate the entropy change of the gas. ($R = 8.314\ ext{J K}^{-1} ext{mol}^{-1}$.)

Question

$2\ 	ext{mol}$ of an ideal gas expands isothermally and reversibly from $10\ 	ext{L}$ to $20\ 	ext{L}$ at a constant temperature. Calculate the entropy change of the gas. ($R = 8.314\ 	ext{J K}^{-1}	ext{mol}^{-1}$.)

Accepted Answer

Concept understanding — Entropy and Spontaneity

Entropy and Spontaneity: From Intuition to Precision

You already know that some things happen on their own — a hot cup of coffee cools down, a drop of ink spreads in water, a gas expands into a vacuum. These are spontaneous processes. They don't need any external push. But why do they happen? What decides the direction?

The old idea was that systems always try to lower their energy. That works for a ball rolling downhill, but it fails for the ink spreading in water — the ink doesn't lose energy, it just spreads out. So energy alone cannot be the whole story.

The Intuition: Nature Prefers Spreading Out

Think about a deck of cards. A brand-new deck is perfectly ordered — all hearts together, all spades together, and so on. If you shuffle it once, you get a slightly more mixed arrangement. Shuffle it a hundred times, and the cards are thoroughly random. Now ask: which arrangement is more likely?

The ordered deck is just one specific arrangement. The number of random, mixed arrangements is astronomically larger. So if you shuffle blindly, you are overwhelmingly more likely to end up with a mixed deck than the perfectly ordered one. The system naturally moves toward the state that has more possible arrangements — toward higher probability.

That is the core intuition behind entropy. Entropy ( $S$ ) is a measure of how many microscopic arrangements correspond to a given macroscopic state. More arrangements = higher entropy. Nature, left to itself, tends toward states with more arrangements — higher entropy.

Note

In a gas, the "arrangements" are the positions and velocities of each molecule. A gas confined to a corner has fewer possible arrangements than one spread evenly through the container. So the gas spontaneously expands — it moves to a state with more arrangements, higher entropy.

The Precise Statement: The Second Law of Thermodynamics

The Second Law of Thermodynamics gives the exact condition for spontaneity:

For any spontaneous process, the total entropy of the universe (system + surroundings) increases.

\Delta S_{\text{total}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} > 0

This is not a guess — it is a law of nature, verified by every spontaneous process ever observed.

Important

The Second Law does not say that the entropy of the system alone must increase. The system's entropy can decrease, as long as the surroundings' entropy increases enough to make the total positive. That is how life exists — living organisms are highly ordered (low entropy), but they dump waste heat into the surroundings, increasing the surroundings' entropy by more than the decrease in the organism.

How to Calculate Entropy Change

For a system exchanging heat reversibly at constant temperature:

\Delta S = \frac{q_{\text{rev}}}{T}

where $q_{\text{rev}}$ is the heat absorbed reversibly, and $T$ is the absolute temperature. For the surroundings, if the process happens at constant pressure (as most do in a lab), the heat exchanged by the surroundings is just the negative of the enthalpy change of the system:

\Delta S_{\text{surroundings}} = -\frac{\Delta H_{\text{system}}}{T}

So the total entropy change becomes:

\Delta S_{\text{total}} = \Delta S_{\text{system}} - \frac{\Delta H_{\text{system}}}{T}

Multiplying through by $-T$ gives a more familiar condition:

-T \Delta S_{\text{total}} = \Delta H_{\text{system}} - T \Delta S_{\text{system}}

The quantity $\Delta H - T \Delta S$ is called the Gibbs free energy change, $\Delta G$ .

\Delta G = \Delta H - T \Delta S

What This Means in Practice

A process is spontaneous when:

Exothermic ( $\Delta H < 0$ ) and entropy increases ( $\Delta S > 0$ ) — always spontaneous.
Endothermic ( $\Delta H > 0$ ) and entropy decreases ( $\Delta S < 0$ ) — never spontaneous. …

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For the isothermal, reversible expansion of $n$ moles of an ideal gas from volume $V_1$ to $V_2$ , the entropy change is given by $\Delta S = nR\ln\!\left(\dfrac{V_2}{V_1}\right)$ . Here $n=2\ \text{mol}$ , $R=8.314\ \text{J K}^{-1}\text{mol}^{-1}$ , $V_1=10\ \text{L}$ and $V_2=20\ \text{L}$ , so $\dfrac{V_2}{V_1}=2$ . Computing $\ln 2 \approx 0.6931$ : $\Delta S = 2 \times 8.314 \times 0.6931 \approx 11.53\ \text{J K}^{-1}$ . The positive entropy change reflects the gas molecules having access to a larger volu …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set ANNUAL1 markMCQ
$Q. Substance with minimum Entropy is -$
(a) Water
(b) Ice
(c) Water vapour
(d) None of these
›Reveal solutionSolution

Ice has the minimum entropy among water, ice, and water vapour, because solids are the most ordered state of matter.

Entropy (S) is a measure of the randomness or disorder of a system. For the same substance, entropy increases in the order solid < liquid < gas, because molecules in a solid are locked in a fixed, highly ordered lattice with only vibrational motion, molecules in a liquid have more freedom to move past one another (more disorder), and molecules in a gas move almost independently and randomly (maximum disorder).

…

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CBSE 2026Set ANNUAL1 mark
$Q. For an isolated system, ΔU = 0, the value of ΔS will be ____________.$
›Reveal solutionSolution

For a real (spontaneous) process in an isolated system with ΔU = 0, ΔS is positive (ΔS > 0).

By the second law of thermodynamics, the entropy of an isolated system can never decrease. For any spontaneous (irreversible) process it increases (ΔS > 0); only for a hypothetical reversible process does it stay constant (ΔS = 0). Since ΔU = 0 defines an isolated system, entropy …

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CBSE 2024Set ANNUAL1 markMCQ
$Q. Match the correct pair -- Entropy should be matched with the correct item from Column B:$
(a) 45 Neutron
(b) Octahedral
(c) mol L^-1
(d) ΔS = q(rev)/T
(e) (triangle symbol, drawn as a downward-pointing triangle -- representing a 3-membered ring shape)
›Reveal solutionSolution

Entropy (S) is a thermodynamic state function measuring disorder; for a reversible process its change is defined as $\Delta S = \frac{q_{rev}}{T}$ , which is option (d).

…

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CBSE 2023Set ANNUAL1 mark
$Q. Fill in the blank: The unit of Entropy is ____________.$
›Reveal solutionSolution

Entropy has the unit joule per kelvin (J K^-1), or J K^-1 mol^-1 on a molar basis.

Entropy (S) is defined thermodynamically by the relation dS = q_rev / T, where q_rev is heat absorbed reversibly (in joules) and T is the absolute temperature (in kelvin).

…

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CBSE 2023Set ANNUAL1 mark
$Q. Match the column: Column I item 'Entropy of universe' is to be matched with the correct term from Column II:$
(a) Sharing of electron,
(b) increasing,
(c) To find out molecular mass of organic compound,
(d) SF6,
(e) Transition element,
(f) Molarity,
(g) Mulliken,
(h) Heisenberg,
(i) Molality. Which Column II term matches this Column I item?
›Reveal solutionSolution

The Second Law of Thermodynamics states that the entropy of the universe is always increasing (option b) for spontaneous processes.

Entropy (S) is a measure of disorder/randomness. The Second Law of Thermodynamics states that for any spontaneous process, the total entropy of the universe (dS_universe = dS_system + dS_surroundings) always increases; it never decreases. Only for a perfectly reversible process does the total entropy remain unchanged ( …

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CBSE 2022Set ANNUAL1 mark
$Q. Write right or wrong: Entropy of substance in gaseous become zero.$
›Reveal solutionSolution

The statement is Wrong: it is a perfectly crystalline solid at absolute zero (0 K) whose entropy is zero - not a gaseous substance.

Entropy is a measure of disorder/randomness. Gas molecules move randomly and rapidly in all directions, so gases are the MOST disordered state of matter and have the HIGHEST entropy of the three states, not zero. The Third Law of Thermodynamics instead says that the entropy of a perfect crystalline substance (atoms/ions arranged in one single, perfectly ordered pattern) approaches zero as the temperature approaches absolute zero (0 K), because at that point there is essentially onl …

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CBSE 2022Set ANNUAL1 markMCQ
$Q. The entropy change of a spontaneous process always$
(a) increases
(b) decreases
(c) keeps changing
(d) remains unchanged
›Reveal solutionSolution

The entropy of the universe always increases for any spontaneous process — this is the Second Law of Thermodynamics.

Entropy (S) is a measure of the randomness or disorder of a system. The second law of thermodynamics states that for any spontaneous process, the total entropy change of the universe ( $\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings}$ ) is always positive, i.e., entropy alw …

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CBSE 2022Set ANNUAL1 mark
$Q. What is the unit of Entropy?$
›Reveal solutionSolution

Entropy (S) is defined via dS = q(rev)/T, so its unit is energy divided by temperature: joule per kelvin (J/K), or J K^-1 mol^-1 for molar entropy.

Entropy change is defined as deltaS = q(reversible)/T, where q is heat exchanged reversibly and T is the absolute temperature. Since heat is measured in joules and temperature in kelvin, the unit of e …

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CBSE 2019Set annual1 mark
$Q. Which has greater entropy at 400 K, N2 or NH3?$
›Reveal solutionSolution
NH3 has higher entropy than N2 because entropy depends on the number of ways energy can be distributed among a molecule's motions, and a more complex (more atoms, non-linear) molecule has more such ways.

Entropy (S) is a measure of the randomness or disorder of a system, which in molecular terms corresponds to the number of accessible microstates. For gas-phase molecules, this includes translational, rotational, and vibrational degrees of freedom.

N2 is a simple linear diatomic molecule: it has only 1 rotational degree of freedom (about an axis perpendicular to the bond) and 1 vibrational mode.

NH3 is a non-linear, pyramidal molecule with 4 atoms: it has 3 rotational degrees of freedom and 6 vibrational modes (3N-6 = 3x4-6 = 6). …
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CBSE 2018Set annual21 markMCQ
$Q. Randomness means:$
(a) Free energy
(b) Internal energy
(c) Entropy
(d) Work
›Reveal solutionSolution
Entropy is the thermodynamic quantity built specifically to measure how disordered/random a system's arrangement is.

Among thermodynamic quantities:

Free energy (Gibbs energy, G) measures the maximum useful (non-expansion) work obtainable from a process at constant T and P, and predicts spontaneity.

Internal energy (U) is the total energy content of a system (kinetic + potential energy of its particles). …
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CBSE 2018Set ANNUAL1 mark
$Q. Write answer in one sentence: Define "Entropy".$
›Reveal solutionSolution

Entropy (S) is a state function measuring the randomness/disorder of a system — it increases as disorder increases (e.g., solid → liquid → gas).

Entropy is defined thermodynamically via dS = q_rev/T (the reversible heat exchanged divided by temperature). Physically, it reflects the number of ways a system's particles can be arranged — greater disorder (more possible microscopic arrangements) means higher entropy. Gases have higher entropy than liquids, which have higher entropy than solids, because molecular motion becomes more random going from solid to ga …

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Q.2 mol2\ \text{mol}2 mol of an ideal gas expands isothermally and reversibly from 10 L10\ \text{L}10 L to 20 L20\ \text{L}20 L at a constant temperature. Calculate the entropy change of the gas. (R=8.314 J K−1mol−1R = 8.314\ \text{J K}^{-1}\text{mol}^{-1}R=8.314 J K−1mol−1.)

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Q. $2\ \text{mol}$ of an ideal gas expands isothermally and reversibly from $10\ \text{L}$ to $20\ \text{L}$ at a constant temperature. Calculate the entropy change of the gas. ( $R = 8.314\ \text{J K}^{-1}\text{mol}^{-1}$ .)

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