Question

For a certain first order reaction, the half-life is `"2 min"` and the frequency factor is `5000 xx 10^8` `"s"^(-1)` at a certain temperature. If its activation energy is `"100 kJ/mol"`, at what temperature is this reaction occurring?

Answer 1

This is a rather complex kinetics problem. Since the units of `A` are `"s"^(-1)`, the reaction is first order. Therefore, we'll need to use

`k = Ae^(-E_a//RT)`, and

`t_(1//2) = (0.693)/k`

Let's find `k`.

`2 cancel"min" xx ("60 s")/(cancel"min") = 0.693/k`

`therefore k approx 5.78*10^-3 "s"^-1`

Upon using the Arrhenius equation, we can derive the absolute temperature given your data. Hence,

`5.78*10^-3 "s"^-1 = (5000 xx 10^8" s"^-1)"exp"[-(10^5 "J"//"mol")/(("8.314 J")/("mol"cdot"K")*T)]`
`therefore T approx "374.8 K"`

I'm open to feedback if I made a mistake.