For a hypothetical second order reaction A arrow product rate law is Rate =[A] square How long is the thir half of reaction if [A] suffix 0 is 080M. andd first half life is 22min?

To determine the third half-life of a second-order reaction where the rate law is given as Rate = k[A]², we first need to understand how half-lives work for second-order reactions. Unlike first-order reactions, the half-life for second-order reactions is dependent on the initial concentration of the reactant.

Understanding Half-Life in Second-Order Reactions

For a second-order reaction, the half-life (t₁/₂) can be calculated using the formula:

• t₁/₂ = 1 / (k[A]₀)

Here, k is the rate constant, and [A]₀ is the initial concentration of the reactant. The key point to note is that as the concentration decreases with each half-life, the time taken for subsequent half-lives increases.

Calculating the Rate Constant

We know the first half-life (t₁/₂) is 22 minutes and the initial concentration [A]₀ is 0.80 M. We can rearrange the half-life formula to find the rate constant k:

• 22 min = 1 / (k * 0.80 M)

Rearranging gives:

• k = 1 / (22 min * 0.80 M)

Calculating this gives:

• k = 1 / 17.6 min·M = 0.05682 M⁻¹·min⁻¹

Finding Subsequent Half-Lives

Now that we have the rate constant, we can find the second and third half-lives. The second half-life (t₂/₁₂) can be calculated using the new concentration after the first half-life:

• [A] after first half-life = [A]₀ / 2 = 0.80 M / 2 = 0.40 M

Using the half-life formula again:

• t₂/₁₂ = 1 / (k * [A] after first half-life)
• t₂/₁₂ = 1 / (0.05682 M⁻¹·min⁻¹ * 0.40 M) = 1 / 0.022728 = 44 min

Next, we find the third half-life (t₃/₁₂) using the concentration after the second half-life:

• [A] after second half-life = 0.40 M / 2 = 0.20 M

Now, applying the half-life formula again:

• t₃/₁₂ = 1 / (k * [A] after second half-life)
• t₃/₁₂ = 1 / (0.05682 M⁻¹·min⁻¹ * 0.20 M) = 1 / 0.011364 = 88 min

Summary of Half-Lives

To summarize, the half-lives for the reaction are as follows:

• First half-life: 22 minutes
• Second half-life: 44 minutes
• Third half-life: 88 minutes

Thus, the third half-life of the reaction is 88 minutes. This illustrates how the half-life increases as the concentration of the reactant decreases in a second-order reaction.