What is the order of the reaction with respect to oxygen atoms?

Steps to Solve

1. Determine the change in concentration over time

Calculate the change in oxygen concentration, $[O]$, from the table provided. For example, at time $t = 0$, $[O] = 5.0 \times 10^9$, and at $t = 1.0 \times 10^{-2}$ s, $[O] = 1.9 \times 10^9$.

The change can be found as:

$$ \Delta [O] = [O]{\text{initial}} - [O]{\text{final}} $$

2. Calculate the rate of reaction

Using the formula for the rate, we can express it as:

$$ \text{Rate} = -\frac{\Delta [O]}{\Delta t} $$

Calculate the rate for each time interval based on the change in concentration calculated.

3. Set up a rate law expression

Assuming a general rate law of the form:

$$ \text{Rate} = k [O]^n $$

Where (n) is the order of the reaction with respect to $O$.

4. Plug in values and solve for (n)

Using the calculated rates and corresponding concentrations from each time interval, set up equations to solve for (n). Since we will have multiple rates, we can compare two time intervals to solve for (n):

$$ \frac{\text{Rate}_1}{\text{Rate}_2} = \frac{[O]_1^n}{[O]_2^n} $$

5. Take logarithms to simplify

To solve for (n), take the logarithm of both sides:

$$ \log \left( \frac{\text{Rate}_1}{\text{Rate}_2} \right) = n \log \left( \frac{[O]_1}{[O]_2} \right) $$

This will allow you to isolate (n).

6. Calculate (n) using your values

Substituting values into the logarithmic equation will yield the reaction order.