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# Methyl isonitrile, CH3NC, isomerizes when heated to give acetonitrile (methyl cyanide), CH3CN CH3NC (g) -> CH3CN (g) The reaction is first order. At 230 ºC, the rate constant for the isomerization is 6.3 x 10-4 /s. a) What is the half-life?

##"1100 s"##

Yes, the half-life for a first-order reaction is

##color(blue)( |bar( ul( t_"1/2" = ln(2)/k))|" "##, where

##k## - the rate constant of the reaction

Here's why that is the case.

You're dealing with a **first-order reaction**, so right from the start you know that the depends linearly on the concentration of the reactant, methyl isocyanide, ##"CH"_3"NC"##.

If you take the concentration of methyl isocyanide to be ##["CH"_3"NC"]##, you can say that rate of the reaction will be

##"rate" = -(d["CH"_3"NC"])/dt##

To bring the **rate constant**, ##k##, in the mix, you need to write the differential for this reaction

##"rate" = -(d["CH"_3"NC"])/dt = k * ["CH"_3"NC"]##

In order to be able to relate the rate of the reaction with time, you need to integrate the differential . This will get you the **integral rate law**, which for your reaction will look like this

##[-(d["CH"_3"NC"])/(["CH"_3"NC"]) = k * dt] -> int##

##-int(1/(["CH"_3"NC"]) * d["CH"_3"NC"]) = k * intdt##

This will get you

##ln(["CH"_3"NC"]) = -k * t + C" " " "color(red)("(*)")##

To get rid of the integration constant, use the fact that you have an initial concentration of methyl isocyanide, ##["CH"_3"NC"]_0##, at ##t=0##.

This will get you

##ln(["CH"_3"NC"]_0) = -k * 0 + C implies C = ln(["CH"_3"NC"]_0)##

Plug this into equation ##color(red)("(*)")## to get

##ln(["CH"_3"NC"]) - ln(["CH"_3"NC"]_0) = -k * t##

Finally, rearrange to get

##color(blue)(ln((["CH"_3"NC"])/(["CH"_3"NC"]_0)) = -k* t)##

In your case, the **half-life** of the reaction, ##t_"1/2"##, will be equal to the time needed for the concentration of methyl isocyanide to be reduced to **half** of its initial value. Therefore, at ##t = t_"1/2"##, you will have

##["CH"_3"NC"] = 1/2 * ["CH"_3"NC"]_0##

Plug this into the integrated rate law and solve for ##t_"1/2"##

##ln( (1/2color(red)(cancel(color(black)(["CH"_3"NC"]))))/(color(red)(cancel(color(black)(["CH"_3"NC"]))))) = -k * t_"1/2"##

Since

##ln(1/2) = ln(1) - ln(2) = - ln(2)##

this will get you

##-ln(2) = -k * t_"1/2" = color(blue)( |bar( ul( t_"1/2" = ln(2)/k))|##

Finally, plug in your values to get

##t_"1/2" = ln(2)/(6.3 * 10^(-4)"s"^(-1)) = 1.1 * 10^3"s" = color(green)(| bar( ul("1100 s"))|)##

The answer is rounded to two , the number of sig figs you have for the rate constant.