91Ó°ÊÓ

In the stratosphere, the rate constant for the conversion of ozone to molecular oxygen by atomic chlorine is \(\mathrm{Cl} \cdot(g)+\mathrm{O}_{3}(g) \rightarrow \mathrm{ClO} \cdot(g)+\mathrm{O}_{2}(g)$$\left[\text { (half-life of } 5760 \text { years) } \mathrm{k}=\left(1.7 \times 10^{10} \mathrm{M}^{-1} \mathrm{s}^{-1}\right) e^{-260 K / T}\right]\) a. What is the rate of this reaction at \(20 \mathrm{km}\) where \\[\begin{array}{l}{[\mathrm{Cl}]=5 \times 10^{-17} \mathrm{M},\left[\mathrm{O}_{3}\right]=8 \times 10^{-9} \mathrm{M}, \text { and }} \\\T=220 \mathrm{K} ?\end{array}\\] b. The actual concentrations at \(45 \mathrm{km}\) are \([\mathrm{Cl}]=3 \times 10^{-15} \mathrm{M}\) and \(\left[\mathrm{O}_{3}\right]=8 \times 10^{-11} \mathrm{M} .\) What is the rate of the reaction at this altitude where \(T=270 \mathrm{K} ?\) c. (Optional) Given the concentrations in part (a), what would you expect the concentrations at \(20 . \mathrm{km}\) to be assuming that the gravity represents the operative force defining the potential energy?

Step by step solution

01

Calculate the rate constant k

Using the given expression for the rate constant k, we have: \[ k = (1.7 \times 10^{10}\ \mathrm{M^{-1}s^{-1}}) e^{-260\mathrm{K}/T} \] where T is the temperature at 20 km altitude, which is given as 220 K. Plugging in the given value of T, we get: \[ k = (1.7 \times 10^{10}\ \mathrm{M^{-1}s^{-1}}) e^{-260/220} \]

02

Calculate the reaction rate

The reaction rate is given by the rate constant multiplied by the concentrations of the reactants: \[ Rate = k[\mathrm{Cl}][\mathrm{O}_3] \] Using the given concentrations, [\(\mathrm{Cl}\)] = \(5 \times 10^{-17}\ \mathrm{M}\) and [\(\mathrm{O}_3\)] = \(8 \times 10^{-9}\ \mathrm{M}\), and the value of k from Step 1, we can find the reaction rate: \[ Rate = k(5 \times 10^{-17})(8 \times 10^{-9}) \] b. Reaction rate at 45 km altitude:

03

Calculate the rate constant k

Using the given expression for the rate constant k, we have: \[k = (1.7 \times 10^{10}\ \mathrm{M^{-1}s^{-1}}) e^{-260\mathrm{K}/T} \] where T is the temperature at 45 km altitude, which is given as 270 K. Plugging in the given value of T, we get: \[k = (1.7 \times 10^{10}\ \mathrm{M^{-1}s^{-1}}) e^{-260/270} \]

04

Calculate the reaction rate

The reaction rate is given by the rate constant multiplied by the concentrations of the reactants: \[Rate = k[\mathrm{Cl}][\mathrm{O}_3] \] Using the given concentrations, [\(\mathrm{Cl}\)] = \(3 \times 10^{-15}\ \mathrm{M}\) and [\(\mathrm{O}_3\)] = \(8 \times 10^{-11}\ \mathrm{M}\), and the value of k from Step 1, we can find the reaction rate: \[Rate = k(3 \times 10^{-15})(8 \times 10^{-11}) \] For part (c), it deals with the effects of gravity on the concentrations at 20 km. This part is optional and requires knowledge of atmospheric physics.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate Constant

In chemical reactions, the rate constant is a crucial factor that determines how fast the reaction proceeds. It is represented by the letter 'k' and is unique for each reaction at a given temperature. The rate constant provides insight: it reflects both the intrinsic speed of the reaction and the influence of temperature.

The rate constant in the given problem can be calculated using the Arrhenius equation: \[ k = Ae^{-E_a/RT} \] where:

• \( A \) is the pre-exponential factor.
• \( E_a \) is the activation energy.
• \( R \) is the gas constant.
• \( T \) is the temperature in Kelvin.

For the reaction occurring in the stratosphere, the expression is given as:\[ k = (1.7 \times 10^{10} \ \mathrm{M^{-1}s^{-1}}) e^{-260/T} \] This equation reveals the dependency of the rate constant on temperature, emphasizing how even a small change in temperature can lead to significant differences in reaction rates. Understanding this aspect allows us to predict how reactions might behave under different atmospheric conditions, which is vital for studying processes like ozone depletion.

Stratospheric Chemistry

Stratospheric chemistry deals with the chemical interactions and transformations occurring in the stratosphere, a layer of Earth's atmosphere. The stratosphere extends from about 10 km to 50 km above Earth's surface. Here, the concentration of various gases changes with altitude, affecting the chemical reactions that occur.

A key player in stratospheric chemistry is ozone (\( \mathrm{O}_3 \)), which absorbs and blocks harmful ultraviolet radiation from the sun. However, ozone can be broken down by reactions with other compounds, such as chlorine radicals. The reaction outlined in the problem demonstrates such a process:\[ \mathrm{Cl} \cdot(g) + \mathrm{O}_{3}(g) \rightarrow \mathrm{ClO} \cdot(g) + \mathrm{O}_{2}(g) \] The presence of reactive chlorine species, mainly from man-made chemicals like chlorofluorocarbons (CFCs), contributes to ozone depletion.

• CFCs release chlorine atoms when broken down by UV radiation.
• The released chlorine atoms act as a catalyst in ozone destruction.

Understanding these reactions is essential for predicting the environmental impact of various compounds and for devising strategies to protect the ozone layer.

Ozone Depletion

Ozone depletion refers to the reduction of ozone in the Earth's stratosphere, primarily due to human activities. This depletion of ozone leads to an increase in the amount of ultraviolet (UV) radiation reaching the Earth's surface, which can have harmful effects on living organisms.

The main cause of ozone depletion is the presence of chlorine and bromine compounds in the atmosphere, often originating from CFCs and halons. These substances are relatively inert in the lower atmosphere but break down in the stratosphere under UV radiation, releasing chlorine atoms.

Here’s what happens:

• Chlorine atoms react with ozone, forming chlorine monoxide (\( \mathrm{ClO} \cdot \)).
• Chlorine monoxide can then react with a free oxygen atom to regenerate the chlorine atom.

This cycle continues, allowing a single chlorine atom to destroy many ozone molecules before being deactivated.Efforts to combat ozone depletion include the Montreal Protocol, which globally restricted the use of ozone-depleting substances. Monitoring and understanding the chemical reactions involved in ozone depletion allow scientists to address this environmental challenge effectively, promoting measures and policies to restore and protect the ozone layer.