Question

For the first-order thermal decomposition of ozone $$ \mathrm{O}_{3}(g) \longrightarrow \mathrm{O}_{2}(g)+\mathrm{O}(g) $$ \(k=3 \times 10^{-26} \mathrm{~s}^{-1}\) at \(25^{\circ} \mathrm{C}\). What is the half-life for this reaction in years? Comment on the likelihood that this reaction contributes to the depletion of the ozone layer.

Short answer

Answer: The half-life of the reaction is approximately 7.33 x 10¹⁷ years. This extremely long half-life suggests that the reaction is very slow and not efficient in decomposing ozone, meaning it is unlikely to contribute significantly to the depletion of the ozone layer. Other factors, such as chemical reactions involving chlorine and bromine atoms, are much more relevant for ozone depletion.

Step-by-step solution

Write down the given information

We are given the following information: - First-order reaction: \(\mathrm{O}_{3}(g) \longrightarrow \mathrm{O}_{2}(g)+\mathrm{O}(g)\) - Rate constant (k): \(3 \times 10^{-26} \mathrm{~s}^{-1}\) - Temperature: \(25^{\circ} \mathrm{C}\)

Use the half-life formula for a first-order reaction

To find the half-life (in seconds) of the reaction, we will use the formula for the half-life of a first-order reaction: $$ t_{1/2} = \frac{\ln 2}{k} $$ Substitute the given values into the equation: $$ t_{1/2} = \frac{\ln 2}{3 \times 10^{-26} \mathrm{~s}^{-1}} $$

Calculate the half-life in seconds

Now, let's calculate the half-life in seconds: $$ t_{1/2} = \frac{0.693}{3 \times 10^{-26} \mathrm{~s}^{-1}} $$ $$ t_{1/2} \approx 2.31 \times 10^{25} \mathrm{~s} $$

Convert the half-life from seconds to years

To convert the half-life from seconds to years, we will use the conversion factor of 1 year being equal to \(3.15 \times 10^{7}\mathrm{~s}\). $$ t_{1/2} \approx \frac{2.31 \times 10^{25} \mathrm{~s}}{3.15 \times 10^{7} \mathrm{~s/year}} $$ $$ t_{1/2} \approx 7.33 \times 10^{17} \mathrm{~years} $$

Comment on the likelihood of reaction and its impact on the ozone layer

The calculated half-life for the reaction is extremely long, approximately \(7.33 \times 10^{17}\) years. This indicates that the reaction is very slow and not efficient in decomposing ozone. Thus, it is unlikely that this reaction contributes significantly to the depletion of the ozone layer. Other factors, such as chemical reactions involving chlorine and bromine atoms, are much more relevant for ozone depletion.