# Problem: The decomposition of XY is second order in XY and has a rate constant of 7.02 x 10-3 M-1•s-1 at a certain temperature. a. What is the half life for this reaction at an initial concentration of 0.100M?b. How long will it take for the concentration of XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.100M? When the initial concentration is 0.200M?c. If the initial concentration of XY is 0.150 M, how long will it take for the concentration to decrease to 0.062 M?d. If the initial concentration of XY is 0.150 M, what is the concentration of XY after 5.0x101 s? After 5.50x102 s?

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###### FREE Expert Solution

The integrated rate law for a second-order reaction is as follows:

$\overline{)\frac{\mathbf{1}}{{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{t}}}{\mathbf{=}}{\mathbf{kt}}{\mathbf{+}}\frac{\mathbf{1}}{{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{0}}}}$

a. What is the half-life for this reaction at an initial concentration of 0.100M?

Recall that half-life (t1/2) is the time needed for the amount of a reactant to decrease by 50% or one-half

The half-life of a second-order reaction is given by:

$\overline{){{\mathbf{t}}}_{\mathbf{1}\mathbf{/}\mathbf{2}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{k}{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{0}}}}$

${\mathbf{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{k}{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{0}}}$

t1/2 = 1424.5 s

The half-life of the reaction is 1424.5 s–1.

b. How long will it take for the concentration of XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.100M? When the initial concentration is 0.200M?

[XY]0 = 0.100 M

[XY]t0.100 M × 12.5% = 0.0125 M

Solving for time:

$\frac{\mathbf{1}}{{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{t}}}\mathbf{=}\mathbf{kt}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{0}}}$

t = 9971.51 s

It would take 9971.51 s for XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.100 M.

[XY]0 = 0.200 M

[XY]t0.200 M × 12.5% = 0.025 M

Solving for time:

$\frac{\mathbf{1}}{{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{t}}}\mathbf{=}\mathbf{kt}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{\left[}\mathbf{A}\mathbf{\right]}}_{\mathbf{0}}}$

t = 4985.75 s

It would take 4985.75 s for XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.200 M.

c. If the initial concentration of XY is 0.150 M, how long will it take for the concentration to decrease to 0.062 M?

###### Problem Details

The decomposition of XY is second order in XY and has a rate constant of 7.02 x 10-3 M-1•s-1 at a certain temperature.

a. What is the half life for this reaction at an initial concentration of 0.100M?

b. How long will it take for the concentration of XY to decrease to 12.5% of its initial concentration when the initial concentration is 0.100M? When the initial concentration is 0.200M?

c. If the initial concentration of XY is 0.150 M, how long will it take for the concentration to decrease to 0.062 M?

d. If the initial concentration of XY is 0.150 M, what is the concentration of XY after 5.0x101 s? After 5.50x102 s?