When sucrose, ${\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}$, is dissolved in ${\mathrm{H}}_2\mathrm{O}$ in the presence of an acid catalyst it reacts according to the equation ${\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}+{\mathrm{H}}_2\mathrm{O}⟶2{\mathrm{C}}_6{\mathrm{H}}_{12}{\mathrm{O}}_6$ with a rate law of Rate $=k\left[{\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}\right]$. If 3.00 g of sucrose decreases to 2.70 g in 2.50 hours in the presence of a certain concentration of an acid catalyst, what is the half-life for this reaction under these same conditions?

$\text{ Rate }=k{\left[{\mathrm{C}}_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}\right]}^{\mathrm{\prime }}$[First order reaction]

$3\mathrm{gm}⟶2.7\mathrm{gm},$[T=5 hours]

Total value V(L) $C_{12}{\mathrm{H}}_{22}{\mathrm{O}}_{11}$ molar mass

$\frac{3}{m.v}⟶\frac{2.7}{mv}$ 2.5 hours 

general equation of 1st order

reaction = $A=A_0{e}^{-kt}$

${\mathrm{A}}_0 =$ Initial conventration of Reactent

A = Concentration of Reactent after time 

$\begin{array}{c}{A}_0=\frac{3}{mv},A=\frac{2.7}{mv},t=2.5\text{ hrs }\\ \frac{{2.7}^A}{m=A_0{e}^{-kt}}=\frac{3}{m+e^{-k}}\left(2.5\right)\\ e^{-k\left(2.5\right)}=0.9\\ -k\left(2.5\right)=\ln \left(0.9\right)=-0.106\\ k=0.042i\text{ hour }\end{array}$

$k=0.0421\text{ hour }-1$

Ist order $\begin{array}{r}{t}_{1/2}=\frac{\ln 2}{k}\\ =\frac{0.693}{0.0421\text{ hour }}=\underset{\_}{16.4\text{ hours }}\end{array}$