Concept Explanation: Entropy Change in Isothermal Reversible Expansion

Entropy is a measure of disorder or randomness in a system. For an ideal gas undergoing an isothermal (constant temperature) reversible expansion, the entropy change (ΔS) can be calculated using the formula derived from the second law of thermodynamics.

Step-by-Step Solution:

Step 1: Identify the given values

• Number of moles, n = 2 mol
• Initial volume, V_i = 10 dm³
• Final volume, V_f = 100 dm³
• Temperature, T = 27°C = 27 + 273 = 300 K (converted to Kelvin)
• Gas constant, R = 8.314 J mol^-1 K^-1

Step 2: Recall the formula for entropy change in isothermal reversible expansion

The entropy change for an ideal gas is given by:

Step 3: Substitute the values into the formula

Step 4: Simplify the expression

First, compute the volume ratio: V_f/V_i = 100/10 = 10

Then, ln(10) ≈ 2.302585

Now, calculate:

Step 5: Final Answer

The entropy change is approximately 38.3 J K^-1 (since it is for the entire system of 2 moles, the value is already total, not per mole). However, note the options are given per mole. So we need to find per mole entropy change.

But careful: The question asks for entropy change, and options are in J mol^-1 K^-1. So we should compute ΔS for the system and then divide by moles to get per mole? Actually, the formula gives total ΔS. Let's check the calculation:

Our calculated total ΔS = 38.288 J K^-1. For per mole, it would be 38.288 / 2 = 19.144 J mol^-1 K^-1, but this is not in options. Wait, perhaps the options are total entropy change? But they are labeled "J mol^-1 K^-1", which might be a mistake in the options, or perhaps we need to see.

Actually, the formula ΔS = nR ln(Vf/Vi) gives the total entropy change for the system. So for n=2, it is 38.3 J K^-1. But the options are given as "J mol^-1 K^-1", which might be implying the molar entropy change. However, 38.3 J mol^-1 K^-1 is one of the options, and it matches our total value? No, careful: Our total is 38.3 J K^-1, but if we express per mole, it would be half.

But looking at the options, they are all around 30-40, so likely the options are for total entropy change (even though written as per mole, it might be a mislabel). Since our calculation gives 38.3 J K^-1, and option is "38.3 J mol^-1 K^-1", it is probably the total, and the "mol^-1" is a mistake, or perhaps it is understood.

Given the calculation, the correct value is 38.3 J K^-1 for the system.

So the answer is 38.3 J mol^-1 K^-1 (considering the option as is).

Related Topics:

• Thermodynamics: Laws of thermodynamics, especially the second law.
• Ideal Gas Law: PV = nRT.
• Reversible Processes: Work done in reversible expansion.
• Entropy: Definition, significance, and calculations in various processes.

Formulae and Theory:

Key Formula:

For isothermal reversible expansion of an ideal gas:

Alternatively, using pressure:

Theory: Entropy is a state function, and its change depends only on initial and final states for a reversible process. In isothermal expansion, heat is absorbed reversibly, and the entropy increase quantifies the dispersal of energy.