Before You Begin

Question #42541

Difficulty: 8/10 | 🟢 Free

The Question

One mole of an ideal rigid diatomic gas $A_2$ is enclosed in a cylinder with a massless, frictionless piston. The external pressure is maintained at a constant value $P_0$ . The initial temperature of the gas is $T_0$ . Heat is slowly supplied to the system, causing the gas to expand. As the temperature increases, the gas $A_2$ partially dissociates into a monatomic gas $A$ . The degree of dissociation $\alpha$ (fraction of initial moles of $A_2$ that has dissociated) varies with temperature $T$ as $\alpha = \frac{T - T_0}{T_0}$ for $T_0 \le T \le 2T_0$ . Assuming the bond dissociation energy of $A_2$ is negligible, what is the total heat supplied to the system to raise its temperature from $T_0$ to $2T_0$ ?

Options

A.
$\frac{13}{2} RT_0$
B.
$\frac{17}{4} RT_0$
C.
$\frac{23}{4} RT_0$
D.
$\frac{7}{2} RT_0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Molar Heat Capacity (Cv vs Cp)|Equation of State for Mixtures|Algebraic/Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the application of the First Law of Thermodynamics to a reacting gas mixture.

A common lethal mistake is to use the standard differential form $dQ = n_{mix} C_{p,mix} dT$ and integrate it over the temperature range.

This approach is fundamentally flawed because it assumes a constant number of moles and fixed degrees of freedom.

When a gas dissociates, energy is required to populate the newly created translational degrees of freedom, which is mathematically captured by the $T d(n C_p)$ term in the exact differential of enthalpy.

By evaluating the state function enthalpy $H = U + PV$ at the initial and final states, the dynamic shifting of molar fractions and heat capacities is rigorously accounted for.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Molar Heat Capacity (Cv vs Cp)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free