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Question #44828

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The Question

The radioactive decay of a nuclide \ce{^A_Z X} follows first-order kinetics with a decay constant $\lambda = 8.64 \times 10^{-2}\text{ days}^{-1}$ at $300\text{ K}$ . An experimenter formally models this nuclear decay using the empirical Arrhenius equation $k = A e^{-E_a/RT}$ .

Simultaneously, a parallel first-order chemical degradation of the sample matrix occurs. This chemical reaction has the same rate constant as the nuclear decay at $300\text{ K}$ , but possesses an activation energy $E_{a,\text{chem}} = 1200 R \ln(10)\text{ J mol}^{-1}$ .

Assuming the Arrhenius equation applies to the chemical process, what are the respective values of the Arrhenius parameters ( $A$ in $\text{s}^{-1}$ and $E_a$ in $\text{J mol}^{-1}$ ) for the nuclear decay process, and at what temperature $T$ will the rate constant of the chemical degradation be exactly $10$ times that of the nuclear decay?

Options

A.
$A = 10^{-6}\text{ s}^{-1}$ , $E_a = 0\text{ J mol}^{-1}$ , $T = 400\text{ K}$
B.
$A = 8.64 \times 10^{-2}\text{ s}^{-1}$ , $E_a = 0\text{ J mol}^{-1}$ , $T = 400\text{ K}$
C.
$A = 10^{-6}\text{ s}^{-1}$ , $E_a = 0\text{ J mol}^{-1}$ , $T = 240\text{ K}$
D.
$A = 10^{-6}\text{ s}^{-1}$ , $E_a = 1200 R \ln(10)\text{ J mol}^{-1}$ , $T = 300\text{ K}$

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Correct Answer

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Explanation

How the Examiner Thinks

Numerical_Precision

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

Concepts Behind This Question

Arrhenius Pre-exponential Factor|Radioactive Decay as Kinetics

Why This Question Is Tricky

Common Mistakes

This problem tests the fundamental distinction between chemical kinetics and nuclear kinetics.

Students often blindly apply the Arrhenius equation to any first-order process.

The primary trap is failing to recognize that radioactive decay is independent of the Boltzmann distribution of thermal energy, meaning its activation energy ( $E_a$ ) is effectively zero, and the pre-exponential factor ( $A$ ) is simply the decay constant ( $\lambda$ ).

A secondary trap involves unit conversion ( $\text{days}^{-1}$ to $\text{s}^{-1}$ ) for the frequency factor, which is a common oversight.

Finally, it tests the correct application of the Arrhenius equation to a parallel chemical process to find the temperature at which the rates diverge by a specific factor.

Quick Facts About This Question

Question Information

Subject	Chemistry
Chapter	Chemical Kinetics - Class 12
Concept	Arrhenius Pre-exponential Factor
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free