# Question a6570

May 2, 2016

The time required is 35 days.

#### Explanation:

For a third order reaction of the type

$\text{A" → "P}$

the rate law is

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} - \frac{d \left[A\right]}{\mathrm{dt}} = k {\left[A\right]}^{3} \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ }$

and the integrated rate law is

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \frac{1}{A} ^ 2 = \frac{1}{A} _ {0}^{2} + 2 k t \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ }$

The time $t$ required for the concentration of $\text{A}$ to decrease from ${\text{[A]}}_{0}$ to $\text{[A]}$ is

$t = \frac{1}{2 k} \left(\frac{1}{A} ^ 2 - \frac{1}{A} _ {0}^{2}\right)$

t = 1/(2× 3.50×10^"-4"color(red)(cancel(color(black)("L"^2"mol"^"-2")))"s"^"-1")(1/(0.021 color(red)(cancel(color(black)("mol·L"^"-1"))))^2 - 1/(0.077 color(red)(cancel(color(black)("mol·L"^"-1"))))^2) = 1429×2099 = 3.00 × 10^6color(white)(l) "s" = 5.00 × 10^4 color(white)(l)"min" = "833 h" = "35 days"#