The rate constant for the first order decomposition of a certain reaction is described by the equation
`log k (s^(-1)) = 14.34 - (1.25 xx 10^(4)K)/(T)`
(a) What is the energy of activation for the reaction?
(b) At what temperature will its half-life period be `256 min`?

Explanation

Step I. Calculation of activation energy `(E_(a))`
According to Arrhenius equation , k=`Ae^(-Ea//RT)`.
log k `= log A - (E_(a))/(2.303 RT)`
The given equ. Is log k `= 14.34 - (1.25 xx 10^(4)(K))/(T)`
On comparison: `E_(a)/(2.303 RT) = (1.25 xx 10^(4)(K))/(T)`
`E_(a) = 1.25 xx 10^(4) xx 2.303 xx 8.314 (J mol^(-1))`
`=239,339 J mol^(-1) = 239.339 kJ mol^(-1)`
Step II. Calculation of desired temperature
For the first order reaction , k `= 0.693/t_(1//2) = 0.693/(256 min) = 0.693/(256 xx 60s) = 4.51 xx 10^(-5) s^(-1)`
According to Arrheneius theory, log k `= 14.34 - (1.25 xx 10^(4))/T` , `-4.35 = 14.34 -(1.25 xx 10^(4))/(18.69) = 669 K`