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Question #44058

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The Question

$n$ moles of a real gas obeying the van der Waals equation of state undergoes a reversible isothermal expansion from an initial volume $V_1$ to a final volume $V_2$ at a constant temperature $T$ . Which of the following expressions correctly represents the change in internal energy ( $\Delta U$ ) and the heat absorbed ( $q$ ) by the gas? (Given: $a$ and $b$ are van der Waals constants)

Options

A.
$\Delta U = an^2 \left( \frac{1}{V_1} - \frac{1}{V_2} \right) ; \quad q = nRT \ln\left(\frac{V_2-nb}{V_1-nb}\right)$
B.
$\Delta U = 0 ; \quad q = nRT \ln\left(\frac{V_2-nb}{V_1-nb}\right) + an^2 \left( \frac{1}{V_1} - \frac{1}{V_2} \right)$
C.
$\Delta U = an^2 \left( \frac{1}{V_1} - \frac{1}{V_2} \right) ; \quad q = nRT \ln\left(\frac{V_2-nb}{V_1-nb}\right) + an^2 \left( \frac{1}{V_1} - \frac{1}{V_2} \right)$
D.
$\Delta U = an^2 \left( \frac{1}{V_2} - \frac{1}{V_1} \right) ; \quad q = nRT \ln\left(\frac{V_2}{V_1}\right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Numerical_Precision

Improve calculation stamina and avoid rounding errors in high-stakes competitive examinations.

Concepts Behind This Question

Internal Energy of Real Gases|Reversible Expansion Work|IUPAC Sign Convention

Why This Question Is Tricky

Common Mistakes

This problem tests the thermodynamic behavior of real gases, specifically the concept of internal pressure $\pi_T = (\partial U/\partial V)_T$ .

Students often fall into two traps: (1) Assuming $\Delta U = 0$ for any isothermal process (only true for ideal gases), or (2) Assuming $q = -w$ and incorrectly including the van der Waals attractive parameter ' $a$ ' in the heat absorbed.

The 'Hidden Variable' here is that the work done against internal attractive forces exactly equals the increase in internal energy.

Consequently, these terms cancel out in the First Law energy balance, making the reversible heat absorbed dependent only on the excluded volume parameter ' $b$ ' and completely independent of ' $a$ '.

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Thermodynamics - Class 11 Part 1
Concept	Internal Energy of Real Gases
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free