Question

the temperature between the top and bottom of the ozone layer varies from 270 k to 220 k. assuming other conditions are the same, how many times faster does ozone decompose at the top of the ozone layer compared to the bottom?

Answer 1

The ratio of rate constants using the Arrhenius equation determines how many times faster ozone decomposes at the top than the bottom of the ozone layer due to temperature variations.

The rate of a chemical reaction is influenced by temperature through the Arrhenius equation, which states that the rate constant (k)) of a reaction is exponentially dependent on temperature:

`\[ k = A \cdot e^{-(E_a)/(RT)} \]`

where:

• A is the pre-exponential factor (frequency factor),

• 
  `\( E_a \)`is the activation energy,

• R is the ideal gas constant, and

• T is the temperature in Kelvin.

Considering the given temperature range, we can analyze the ratio of the rate constants at the top
`(\(T_1 = 270 \, \text{K}\))` and bottom
`(\(T_2 = 220 \, \text{K}\))` of the ozone layer:

`\[ (k_1)/(k_2) = \frac{A \cdot e^{-(E_a)/(R \cdot T_1)}}{A \cdot e^{-(E_a)/(R \cdot T_2)}} \]`

Simplifying, we find:

`\[ (k_1)/(k_2) = e^{(E_a)/(R) \left((1)/(T_2) - (1)/(T_1)\right)} \]`

Given that
`\(E_a\)` and R are constants, we can evaluate the exponential term to find the ratio of rate constants.

This ratio represents how many times faster ozone decomposes at the top compared to the bottom of the ozone layer. As temperature increases, the rate of reaction generally increases, and since
`\(T_1 > T_2\)`, the exponential term will be positive, indicating a higher rate at the top.