Question

The rate expression for a reaction is $-\frac{dC}{dt}=\frac{\alpha +\beta C}{\alpha \beta C}$, where a, b are constants and C is the concentration of reactant at time ‘t’. At $t=0$ and $t=t_{1/2}$, the concentration of reactant is $Co$ and $Co/2$, respectively. The half-life for this reaction is

• A. $\alpha +\frac{\beta Co}{2}$
• B. $\frac{\alpha Co}{2}+\ln \frac{\beta Co}{2}$
• C. $\frac{\alpha }{\beta }\ln \frac{\alpha +\beta Co/2}{\alpha +\beta Co}+\frac{\beta Co}{2}$
  
• D. $\frac{{\alpha }^2}{\beta }\ln \frac{\alpha +\beta Co/2}{\alpha +\beta Co}+\frac{\alpha Co}{2}$

Answer 1

The correct option is D $\frac{{\alpha }^2}{\beta }\ln \frac{\alpha +\beta Co/2}{\alpha +\beta Co}+\frac{\alpha Co}{2}$

$-\frac{dC}{dt}=\frac{\alpha +\beta C}{\alpha \beta C}$
$-{\int }_{Co}^{Co/2}\frac{\alpha \beta C}{\alpha +\beta C}dC={\int }_0^{t/2}dt$
$-\alpha {\int }_{Co}^{Co/2}\frac{\alpha +\beta C}{\alpha +\beta C}-\frac{\alpha }{\alpha +\beta C}dC=t_{1/2}$
$-\alpha {\int }_{Co}^{Co/2}(1-\frac{\alpha }{\alpha +\beta C})dC=t_{1/2}$
$-\alpha \left[{\int }_{Co}^{Co/2}dC-\alpha {\int }_{Co}^{Co/2}\frac{1}{\alpha +\beta C}dC\right]=t_{1/2}$
$-\alpha [\frac{-Co}{2}-\frac{\alpha }{\beta }\ln \frac{\alpha +\beta Co/2}{\alpha +\beta C_0}]=t_{1/2}$
$t_{1/2}=\frac{{\alpha }^2}{\beta }\ln \frac{\alpha +\beta Co/2}{\alpha +\beta Co}+\frac{\alpha Co}{2}$
Option D is correct.