Chapter 20: Problem 84

Peroxyacetyl nitrate (PAN) undergoes thermal decomposition as follows: $$ \mathrm{CH}_{3}(\mathrm{CO}) \mathrm{OONO}_{2} \longrightarrow \mathrm{CH}_{3}(\mathrm{CO}) \mathrm{OO}+\mathrm{NO}_{2} $$ The rate constant is $3.0 \times 10^{-4} \mathrm{~s}^{-1}$ at $25^{\circ} \mathrm{C}$. At the boundary between the troposphere and stratosphere, where the temperature is about $-40^{\circ} \mathrm{C},$ the rate constant is reduced to $2.6 \times 10^{-7} \mathrm{~s}^{-1} .$ (a) Calculate the activation energy for the decomposition of PAN. (b) What is the half-life of the reaction (in minutes) at $25^{\circ} \mathrm{C} ?$

Short Answer

Expert verified

The activation energy for decomposition of PAN is approximately $8.64 \times 10^4 J/mol$ and the half-life of the reaction at 25 degrees Celsius is approximately 38.5 minutes.

Step by step solution

Use the Arrhenius equation

To calculate the activation energy, we can use the Arrhenius equation: \[k = A e^{-E_a/RT} \] where k is the rate constant, A is the pre-exponential factor, E_a is the activation energy, R is the gas constant and T is the temperature in Kelvin. To find E_a, we can rearrange the equation to: \[ E_a = -RT \ln{(k/A)} \]. However, we need two sets of k and T to solve this equation, and in this case, we are given two temperatures and corresponding rate constants.

Convert temperatures to Kelvin

We firstly need to convert the given temperatures to Kelvin to use in the equation: \[T_1 = 25^{\circ}C = 25 + 273 = 298K \] and \[T_2 = -40^{\circ}C = -40 + 273 = 233K \]

Apply the two-point form of the Arrhenius Equation.

As we have two sets of T and k, we can use the two-point form of the Arrhenius Equation, known as the Arrhenius plot: \[ \ln{\left(\frac{k_1}{k_2}\right)} = \frac{E_a}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) \] Inserting the values we get: \[ \ln{\left(\frac{2.6 \times 10^{-7}}{3.0 \times 10^{-4}}\right)} = \frac{E_a}{8.314} \left(\frac{1}{233} - \frac{1}{298}\right) \] Now, solve this equation to find the value of activation energy, $E_a$.

Use the first-order half-life expression

To find the half-life at a temperature of 25 degree Celsius, we can use the expression for half-life (t1/2) of a first order reaction: \[ t_{1/2}= \frac{0.693}{k} \] where k is the rate constant. In our case, at 25 degree Celsius, k is given as $3.0 \times 10^{-4} s^{-1}$. Insert the value of k to find the half-life in seconds.

Convert the half-life from seconds to minutes

Once the half-life has been calculated in seconds, we need to simply divide it by 60 to convert it to minutes.

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Short Answer

Step by step solution

Use the Arrhenius equation

Convert temperatures to Kelvin

Apply the two-point form of the Arrhenius Equation.

Use the first-order half-life expression

Convert the half-life from seconds to minutes

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