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Question #44209

Difficulty: 8/10 | 🟢 Free

The Question

Consider $1\text{ mol}$ of a real gas obeying the Van der Waals equation of state, $\left(P + \frac{a}{V^2}\right)(V - b) = RT$ , undergoing a reversible adiabatic expansion. Recognizing that the internal energy $U$ of a real gas is a function of both temperature $T$ and volume $V$ , which of the following expressions correctly represents the slope of the adiabatic curve, $\left(\frac{\partial P}{\partial V}\right)_S$ , on a $P-V$ indicator diagram? (Assume the molar heat capacity at constant volume, $C_V$ , is independent of temperature).

Options

A.
$-\frac{RT}{(V-b)^2} \left( 1 + \frac{R}{C_V} \right) + \frac{2a}{V^3}$
B.
$-\frac{RT}{(V-b)^2} \left( 1 + \frac{R}{C_V} \right) + \frac{aR}{C_V V^2 (V-b)} + \frac{2a}{V^3}$
C.
$-\gamma \frac{P}{V}$
D.
$-\frac{RT}{(V-b)^2} + \frac{2a}{V^3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Mechanism_Trap

Don't get tricked by the process! Learn the mechanical traps used in competitive exam questions.

Concepts Behind This Question

State vs Path Functions|Internal Energy of Real Gases|Graphical Slopes Analysis

Why This Question Is Tricky

Common Mistakes

The problem tests the fundamental definition of internal energy for real gases.

Students often blindly apply the ideal gas relation $dU = nC_V dT$ to all adiabatic processes.

For a Van der Waals gas, intermolecular forces dictate that internal energy also depends on volume (internal pressure, $(\partial U/\partial V)_T = a/V^2$ ).

If a student ignores this and incorrectly equates $C_V dT = -P dV$ , they will fail to cancel the $a/V^2$ term during the expansion work, leading to an incorrect differential $dT$ and the erroneous slope presented in Option B.

Option C is the standard ideal gas trap, and Option D is the isothermal slope.

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Thermodynamics - Class 11 Part 1
Concept	State vs Path Functions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free