Before You Begin

Question #44228

Difficulty: 8/10 | 🟢 Free

The Question

A non-linear polyatomic ideal gas \ce{X} undergoes a unimolecular isomerization \ce{X(g) -> Y(g)} following the Lindemann-Hinshelwood mechanism. At $300\text{ K}$ , the reaction operates strictly at the low-pressure limit. The gas is initially confined in a vessel of volume $V_0$ and is essentially pure \ce{X}. It then undergoes an adiabatic free expansion against zero external pressure until its volume becomes $8V_0$ .

Let $k_{eff}$ be the effective first-order rate constant defined by $\text{Rate} = k_{eff}[\ce{X}]$ . Given that the activation energy for the bimolecular energization step is $E_a = 300 R \ln(4)$ and assuming the Arrhenius pre-exponential factor is independent of temperature, what is the ratio of the effective first-order rate constant immediately after the expansion to that before the expansion, $\frac{k_{eff,2}}{k_{eff,1}}$ ?

Options

A.
1/8
B.
1
C.
1/32
D.
1/4

Check Your Answer

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

Adiabatic Free Expansion|Arrhenius Pre-exponential Factor

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of First Law thermodynamics and chemical kinetics.

The primary trap is the 'adiabatic' keyword.

Students often reflexively apply the reversible adiabatic relation $TV^{\gamma-1} = \text{constant}$ to find a new temperature (which would yield $150\text{ K}$ for a non-linear polyatomic gas where $\gamma = 4/3$ , leading to the distractors $1/32$ or $1/4$ ).

However, for an adiabatic free expansion, $w=0$ and $q=0$ , so $\Delta U = 0$ and temperature remains constant.

The secondary trap is assuming that since $T$ is constant, the effective rate constant $k_{eff}$ is also invariant (leading to distractor $1$ ).

In the Lindemann low-pressure limit, the reaction is effectively second-order, meaning the pseudo-first-order rate constant $k_{eff}$ is directly proportional to the concentration of the gas, which drops by a factor of 8 upon expansion.

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Thermodynamics - Class 11 Part 1
Concept	Adiabatic Free Expansion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free