The first order reaction:
Sucrose `rarr` Glucose + Fructose takes place at `308 K` in `0.5 N HCl`. At time zero the initial total rotation of the mixture is `32.4^(@)`. After `10 min`, the total rotation is `28.8^(@)`. If the rotation of sucrose per mole is `85^(@)`, that of glucose is `7.4^(@)`, and of fructose is `-86.04^(@)`, calculate the half life of the reaction.

Explanation

Suppose rotation per mole of sucrose, glucose and fructose are `r_(1),r_(2)` and `r_(3)` respectively
Sucrose `to` Glucose + Fructose
`{:("At "t=0,,,a,,,0,,,0),("At time "t,,,a-x,,,x,,,x):}`
`ar_(1)=32.4" or "a=(32.4)/(r_(1))=(32.4)/(85)=0.38`
`(a-x)r_(1)+xr_(2)-xr_(3)=28.8`
`ar_(1)+x(r_(2)+r_(3)-r_(1))=28.8`
or `x(r_(2)+r_(3)-r_(1))=28.8-32.4=-3.6`
`x(74-86.04-85)=-3.6" or "x=0.037`
For a first order reaction
`k=(2.303)/(t)log""(a)/(a-x)=(2.303)/(10)log""(0.38)/(0.38-0.037)=9.21xx10^(-3)min^(-1)`
`t_(1//2)=(0.693)/(k)=(0.693)/(9.21xx10^(-3))=75.2" min"`