## Q. 14.4.2

The isomerization of methyl isonitrile to acetonitrile in the gas phase at 250 ºC is first order (k = 3.00 × 10$^{-3} s^{-1}$).
CH$_{3}$NC(g) → CH$_{3}$CN(g)
How much time is required for the concentration of CH$_{3}$NC to drop to 0.0142 M if its initial concentration was 0.107 M?

## Verified Solution

You are asked to determine the amount of time required for the concentration of a reactant to decrease by a given amount.
You are given the balanced equation for the reaction, the order of the reaction with respect to the reactant, the rate constant for the reaction, the initial concentration of the reactant, and the concentration of the reactant after a period of time.
Use the first order integrated rate equation.

$\ln \frac{\left[\text{CH}_{3}\text{NC}\right]_{t} }{\left[\text{CH}_{3}\text{NC}\right]_{0} }=-kt$      $\left[\text{CH}_{3}\text{NC}\right]_{t}$ = 0.0142 M

$\left[\text{CH}_{3}\text{NC}\right]_{0}$ = 0.107 M      k = 3.00 × 10$^{-3}s^{-1}$

$\ln \left(\frac{0.0142}{0.107} \right) = -\left(3.00\times 10^{-3}\text{s}^{-1}\right)t$

−2.02 = −(3.00 × 10$^{-3}$s$^{-1}$)t
t = 673 s