Inverison of sucrose is studied by measuring the angle of rotation at any time `t`.
`underset(("Sucrose"))(C_(12)H_(22)O_(11))underset(("Glucose"))(+H_(2)O) overset(H^(o+))rarrunderset(("Fructose"))(C_(6)H_(12))O_(6)+C_(6)H_(12)O_(6)`
It is found that
`(r_(oo) - r_(0)) prop a` and `(r_(oo) - r_(t)) prop, (a-x)`, where `r_(0), r_(t)`, and `r_(oo)` are the angle of rotation at the start, at the time `t`, and at the end of the reaction, respectively. Form the following values calculate the rate constant and the time at which the solution is optically inactive.
`|{:("Time (min)",0.0,46.0,oo),("Rotation of polarized light (degree)",24.1,10.0,-10.7):}|`

Explanation

When have been given
`r_(0) = 24.1^(@), r_(t) = 10^(@), r_(oo) = -10.7^(@)`
Thus, `a = (r_(oo) - r_(0)) = -10.7-24.1 = -34.8^(@)`
`(a-x) = (r_(oo) - r_(t)) = -10.7 - 10.0 = -20.7`
`:. k = (2.303)/(46)log.((-34.8)/(-20.7)) = 0.011 mm^(-1)`
Solution is optically inactive when `r_(t) = 0` after time `t`
Thus, `r_(oo) - r_(0) = -34.8^(@)` and `r_(oo) - r_(t) = 10.7^(@)`
Thus, `0.011 = (2.303)/(t)log.((-34.8)/(-10.7))`
Thus gives `t = 107.2 min`.