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Question #44725

Difficulty: 8/10 | 🟢 Free

The Question

Di-tert-butyl peroxide, \ce{DTBP (g)}, undergoes a first-order decomposition reaction at 400 K:

\ce{C8H18O2 (g) -> 2 C3H6O (g) + C2H6 (g)}

The reaction is carried out in a cylindrical reactor fitted with a frictionless, massless piston, which maintains a constant external pressure of 1 atm. The temperature is maintained constant at 400 K.

Initially, the reactor contains a mixture of \ce{DTBP (g)} and an inert gas, \ce{Ar (g)}, in a 1:2 molar ratio ( $n_{\text{DTBP}} : n_{\text{Ar}} = 1:2$ ). The initial concentration of \ce{DTBP} is $C_0$ .

Let $k$ be the first-order rate constant for the decomposition of \ce{DTBP}. Which of the following expressions correctly represents the time, $t_{1/2, c}$ , required for the concentration of \ce{DTBP} to decrease to exactly $\frac{C_0}{2}$ ?

Options

A.
$t_{1/2, c} = \frac{\ln 2}{k}$
B.
$t_{1/2, c} = \frac{1}{k} \ln\left(\frac{4}{3}\right)$
C.
$t_{1/2, c} = \frac{1}{k} \ln\left(\frac{8}{5}\right)$
D.
$t_{1/2, c} = \frac{1}{k} \ln\left(\frac{7}{4}\right)$

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

Isotonic, Hypertonic, Hypotonic|First Order Half-Life Independence|Osmotic Pressure vs Diffusion

Why This Question Is Tricky

Common Mistakes

This problem tests the 'Kinetic Independence' and 'First-Order Half-Life' trap.

Students rote-memorize that the half-life of a first-order reaction is $\ln 2 / k$ , assuming an isochoric (constant volume) system.

However, in a non-isochoric system (constant pressure, variable volume), the volume expands as the reaction produces more moles of gas.

This expansion dilutes the reactant, causing its concentration to drop faster than its molar amount.

The hidden variable is the time-dependent volume $V(t)$ , which must be derived using reaction stoichiometry and the Ideal Gas Law.

Failing to account for the inert gas or the volume change leads to incorrect logarithmic arguments.

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Solutions - Class 12
Concept	Isotonic, Hypertonic, Hypotonic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free