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 decomposition reaction of ozone is approximately 7.3 × 10^{16} years. Given its extremely slow rate, this reaction is unlikely to contribute significantly to the depletion of the ozone layer in any relevant time scale. Other factors, such as human-made chemicals like chlorofluorocarbons (CFCs), have a much larger impact on the ozone layer compared to this reaction.

Step-by-step solution

Calculate the half-life of the reaction

Using the first-order reaction half-life formula, $$t_{1/2} = \frac{\ln 2}{k}$$, and the given rate constant, $$k = 3\times10^{-26}~s^{-1}$$, $$t_{1/2} = \frac{\ln 2}{3 \times 10^{-26}~\mathrm{s}^{-1}}$$

Convert the half-life from seconds to years

We can convert the half-life to years using the relationship between seconds and years: 1 year = 31,536,000 seconds Thus, $$t_{1/2} = \frac{\ln 2}{3 \times 10^{-26}~\mathrm{s}^{-1}} \times \frac{1~\mathrm{year}}{31,536,000~\mathrm{s}}$$

Evaluate the half-life

Carrying out the calculation, we find: $$t_{1/2} \approx 7.3 \times 10^{16}~\mathrm{years}$$

Comment on the reaction's contribution to ozone depletion

Given the calculated half-life of this reaction is about $$7.3 \times 10^{16}~\mathrm{years}$$, this reaction is extremely slow and unlikely to contribute significantly to the depletion of the ozone layer in any relevant time scale. There are other factors, such as human-made chemicals like chlorofluorocarbons (CFCs), which are known to have a much larger impact on the ozone layer compared to this reaction.

Key concepts

First-Order Reaction

Understanding the dynamics of chemical reactions is crucial for interpreting many processes, from industrial synthesis to environmental phenomena. A first-order reaction is one such type of chemical reaction where the rate of reaction is directly proportional to the concentration of one of the reactants. In simpler terms, if you were to double the amount of the reactant, the rate at which the products form would also double. This becomes particularly important when discussing the decomposition of ozone, as the rate at which it breaks down affects our atmosphere's health.

Mathematically, for a first-order reaction, the rate law can be expressed as rate = k[A], where 'k' is the rate constant and '[A]' is the concentration of the reactant. A characteristic feature of first-order reactions is their constant half-life, which does not change no matter what the starting concentration is. The half-life (\( t_{1/2} \) formula \( t_{1/2} = \frac{\text{ln} 2}{k} \) provides a clear and straightforward means to determine how long it takes for half of the reactant to convert into products, regardless of the initial amount present.

One classic example is the decay of radioactive isotopes, where the half-life is widely used to date archaeological finds and understand nuclear processes.

Rate Constant

At the core of the reaction kinetics is the rate constant, symbolized by 'k'. This crucial parameter defines the speed of a given chemical reaction. In the context of first-order reactions, it is the proportionality factor that links the reaction rate to the reactant concentration. A larger rate constant indicates a faster reaction. Factors such as temperature, pressure, and the presence of a catalyst can influence the value of the rate constant.

Units for the rate constant vary based on the order of the reaction. For a first-order reaction, the units are reciprocal seconds (\( s^{-1} \) because the rate of reaction is directly proportional to the frequency of the reactive species' collisions. The rate constant provided in the exercise, \( k = 3 \times 10^{-26} s^{-1} \) at \( 25^\text{o}C \) suggests a very slow reaction, which is further substantiated by calculating the half-life. Understanding and calculating the rate constant can provide invaluable insights into the mechanisms and conditions necessary for a certain chemical process to occur. This is of particular importance when evaluating chemical reactions happening in the atmosphere or in living organisms.

Ozone Layer Depletion

The ozone layer depletion is an environmental concern that has significant implications for life on Earth. Ozone (\( \text{O}_3 \) in the stratosphere acts as a protective barrier against the Sun's harmful ultraviolet radiation. Depletion of this layer increases the risk of skin cancers, cataracts, and can cause harm to ecosystems.

The decomposition of ozone can occur naturally, as in the thermal decomposition reaction discussed in the exercise. However, the calculated half-life of around \( 7.3 \times 10^{16} \) years highlights how this particular reaction is extraordinarily slow and therefore negligible in the contribution to current rates of ozone layer depletion.

Human activities have greatly accelerated ozone depletion, primarily through the emission of chlorofluorocarbons (CFCs) and other halogenated ozone-depleting substances. These chemicals have much faster reactions in the stratosphere leading to ozone destruction, compared to the natural decomposition of ozone. Measures like the Montreal Protocol, which was established to phase out these harmful substances, have been crucial in the effort to protect and restore the ozone layer. Understanding the chemistries, such as the half-life of reactions and the impact of human-made chemicals, is vital for developing strategies to prevent further damage to this essential shield.