The rate constant for the first order decomposition of `H_(2)O_(2)` is given by the following equation `: log k =14.34-1.25 K//T.` Calculate `E_(a)` for this reactin and at what temperature will its half-life period be 256 minutes ?

Explanation

a) Calculation of activation energy `E_(a)`
According to Arrhenius equation: `k=Ae ^(-E_(a)//RT)`
`log k= log A-(E_(a))/(2.303RT)-(i)`
`log K =14.34-(1.25 xx10^(4)K)/(T)-(ii)`
on comparing both equations.
`(E_(a))/(2.303RT) =(1.25xx10^(4)K)/(T)`
`E_(a) =1.25 xx10^(4) K xx 2.303 xx8.314 (JK^(-1) mol ^(-1))`
`= 23.93 xx10^(4) J mol ^(-1) =239.3 kJ mol ^(-1)`
b) Calculation of required temperature
If `t_(1//2)=256 min. for 1^(st)` order reaction,
`k=(0.693)/(t_(1//2))=(0.693)/((256min))=(0.693)/(256xx60S)`
According to Arrhenius theory
`log k = 14.34 -(1.25 xx10^(4)k)/(T)`
`log (4.51xx10^(-5))=14.34 -(1.25xx10^(4)k)/(T)`
`(1.25xx10^(4))/(18.69)=669K`
`E_(a)=239 . 3 kJ mol^(-1)`
`T=669K`