Question

Question #a2fab

Answer 1

`"25.4 kJ mol"^(-1)`

Explanation:

Your starting point here will be the Arrhenius equation, which shows how the rate constant of a given reaction depends on the absolute temperature at which the reaction takes place

`color(blue)(|bar(ul(k = A * exp(-E_a/(RT))))|)" "`, where

`k` - the rate constant
`A` - the pre-exponential factor, specific to a given reaction
`E_a` - the activation energy of the reaction
`R` - the universal gas constant, used as `"8.314 J mol"^(-1)"K"^(-1)`
`T` - the absolute temperature of the gas

To find the activation energy of your reaction, take the natural log of both sides of the Arrhenius equation. This will get you

`ln(k) = ln[A * exp(-E_a/(RT))]`

`ln(k) = ln(A) + ln[exp(-E_a/(RT))]`

`ln(k) = ln(A) - E_a/(RT)`

Now, you know the value of the rate constant at two different temperatures

• `k_1 = 3.2 * 10^(-6)"M s"^(-1) -> "at T"_1 = "298 K"`
• `k_2 = 4.2 * 10^(-5)"M s"^(-1) -> "at T"_2 = "398 K"`

This means that you can write

`ln(k_1) = ln(A) - E_a/(R * T_1)`

`ln(k_2) = ln(A) - E_a/(R * T_2)`

Divide these two equations to get rid of the `ln(A)` term

`ln(k_1) - ln(k_2) = color(red)(cancel(color(black)(ln(A)))) - color(red)(cancel(color(black)(ln(A)))) - E_a/(RT_1) + E_a/(RT_2)`

This will be equivalent to

`color(blue)(|bar(ul(ln(k_1/k_2) = E_a/R * (1/T_2 - 1/T_1)))|)`

This equation will allow you to find the value of `E_a` for this reaction. Rearrange and plug in your values to get

`E_a = ln(k_1/k_2) * R/(1/T_2 - 1/T_1)`

`E_a = ln((3.2 * 10^(-6)color(red)(cancel(color(black)("M s"^(-1)))))/(4.2 * 10^(-5)color(red)(cancel(color(black)("M s"^(-1)))))) * ("8.314 J mol"^(-1)color(red)(cancel(color(black)("K"^(-1)))))/((1/398 - 1/298)color(red)(cancel(color(black)("K"^(-1)))))`

`E_a = "25387 J mol"^(-1)`

I'll leave the answer rounded to three sig figs and expressed in kilojoules per mole

`E_a = color(green)(|bar(ul("25.4 kJ mol"^(-1)))|)`

Mind you, you should round this off to two sig figs, since that's how many sig figs you have for the two rate constants.