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, and fructose, respectively.
`{:(,Sucrose,rarr,"Glucose+", "Fructose"),("inital",a,,0,0),("after time t",a-x,,x,x):}`
`ar_(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)) + ar_(1) = 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 = (2.3)/(10)log.((038)/(0.38 - 0.022))`
`= 5.9 xx 10^(-3) min^(-1)`
`:. t_(1//2) = (0.693)/(5.9 xx 10^(-3)) = 116.9 min`