Inversion of sucrose $\left({\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}\right)$ is first-order reaction and is studied by measuring angle of rotation at different instants of time

${\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}+{\mathrm{H}}_2\mathrm{O}\stackrel{{\mathrm{H}}^{+}}{⟶}{\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6+{\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6$

$\begin{array}{ccc}\text{ Sucrose }& \text{ Glucose }& \text{ Fructose }\\ d& d& l\end{array}$

If $(r_{\infty }-r_0)=a$ and $(r_{\infty }-r_t)=(a-x)$ (where $r_0,r_t$ and $r_{\infty }$) are the angle of rotation at the start, at the time $\mathrm{t}$ and at the end of the reaction respectively, then there is $50\%$ inversion when :