1.

The rate of change of concentration of (A) for reaction A to B is given by -(d[A])/(dt)=k[A]^(1//3) Derive expression for half-life period of the reaction.

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Solution :`-(d[A])/(dt)=K[A]^(1//3):.-int(d[A])/([A]^(1//3))=k" dt or "-int[A]^(-1//3)d[A]=k" dt "-(3)/(2)[A]^(2//3)=KT+C[int X^(n)dx=(x^(n+1))/(n+1)]""...(i)`
At `t=0,[A]=[A_(0)]:. -(3)/(2)[A_(0)]^(2//3)=C`
Substituting the value of C in eqn. (i),
`-(3)/(2)[A]^(2//3)=kt-(3)/(2)[A_(0)]^(2//3)" or "kt=(3)/(2)[A_(0)]^(2//3)-(3)/(2)[A]^(2//3)=(3)/(2){[A_(0)]^(2//3)-[A]^(2//3)}`
At `t=t_(1//2),[A]=([A_(0)])/(2)`
`:.kt_(1//2)=(3)/(2){[A_(0)]^(2//3)-[(A_(0))/(2)]^(2//3)}`
or `t_(1//2)=(3)/(2k)[A_(0)]^(2//3)[1-(1)/((2)^(2//3))]=(3[A_(0)]^(2//3))/(2k)[((2)^(2//3)-1)/((2)^(2//3))]=(3[A_(0)]^(2//3))/(k)[((2)^(2//3)-1)/(2^(5//3))]`


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