What is the half-life (in h) for the decomposition of O3 when the concentration of O3 is 2.37 ✕ 10−6 M? The rate constant for this second-order reaction is 50.4 L/mol/h.
t1/2) = 1/k(A)
To find the half-life (t1/2) for the decomposition of O3, we can use the formula for a second-order reaction:
t1/2 = 1 / (k * [O3])
Given:
Concentration of O3 ([O3]) = 2.37 × 10^(-6) M
Rate constant (k) = 50.4 L/mol/h
Substituting the values into the formula, we have:
t1/2 = 1 / (50.4 L/mol/h * 2.37 × 10^(-6) M)
Calculating this expression will give us the half-life. Let's perform the calculation:
t1/2 = 1 / (50.4 L/mol/h * 2.37 × 10^(-6) M)
= 1 / (50.4 × 2.37 × 10^(-6)) L/mol/h
= 1 / (119.488 × 10^(-6)) L/mol/h
= 1 / (1.19488 × 10^(-4)) L/mol/h
≈ 8366.41 hours
Therefore, the half-life for the decomposition of O3 when the concentration of O3 is 2.37 × 10^(-6) M is approximately 8366.41 hours.
To find the half-life for the decomposition of O3, we need to use the rate constant and concentration of O3 in the second-order reaction. The half-life can be determined using the integrated rate law for a second-order reaction.
The integrated rate law for a second-order reaction can be written as:
1/[A]t - 1/[A]0 = kt
Where:
- [A]t is the concentration of A at time t
- [A]0 is the initial concentration of A
- k is the rate constant
- t is the time
In this case, [A]t is 2.37 × 10^(-6) M and [A]0 is the initial concentration of O3. We need to solve for t (the half-life) using the given rate constant.
Let's substitute the values into the integrated rate law:
1/(2.37 × 10^-6) - 1/[A]0 = (50.4 L/mol/h) * t
The initial concentration ([A]0) cancels out because we are looking for the half-life. Now we can rearrange the equation to solve for t:
1/(2.37 × 10^-6) = (50.4 L/mol/h) * t + 1/(2.37 × 10^-6)
Simplifying the equation:
1/(2.37 × 10^-6) - 1/(2.37 × 10^-6) = (50.4 L/mol/h) * t
Multiplying both sides by (2.37 × 10^-6):
1 - 1 = (50.4 L/mol/h) * t * (2.37 × 10^-6)
0 = (50.4 L/mol/h) * t * (2.37 × 10^-6)
Now we can solve for t:
t = 0 / [(50.4 L/mol/h) * (2.37 × 10^-6)]
Since any value divided by zero is undefined, the half-life for the decomposition of O3 is not defined with the given information.
Note: It's important to check if the given information is sufficient to solve the problem. In this case, the concentration of O3 is given, but the initial concentration ([A]0) is missing. To find the half-life, we need both of these concentrations.