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

Let `r_(1) , r_(2) , - r_(3)` be rotation per mole of sucrose , glucose the fructose respectively.
Sucrose `rarr` Glucose `+` Fructose
a 0 0 initial
a-x x x after time t
`a r_(1) = 32.4 rArr a = ( 32.4)/( 85) = 0.38`
`(a-x) r_(1) + xr_(2) - xr_(3) = 28.8`
`x( r_(2) - r_(3) -r_(1)) = 28.8 - 32.4 = - 3.6`
`x ( 7.4 - 86.04 - 85 ) = - 3.6`
`x = 0.022`
`k = ( 2.303)/( 10) log ( ( 0.38)/( 0.38 - 0.022))`
`k = 5.96 xx 10^(-3) "min"^(-1)`
`:. t_(1//2) = 116.2 ` min