Question

For a first order reaction $R\to P$, the rate constant is $k$ , if the initial concentration of $R$ is$[R_n]$. The concentration of $R$ at any time $t$ is given by the expression.

• A. $[R_0]e^{kt}$
• B. $[R_0]e^{-kt}$
  
• C. $[R_0](1-e^{-kt})$
  
• D. $[R_0](1-e^{-kt})$

Answer 1

Answer: (B)

For the first order reaction,
$R\to P$
Rate $=-\frac{d\left[R\right]}{dt}=k\left[R\right]$
or $\frac{d\left[R\right]}{\left[R\right]}=-kdt$
On integration, we get
$\ln R=-kt+I....\left(i\right)$
When $t=0,R=R_0$
$\therefore In{\left[R\right]}_0=-k\times 0+I$
$\ln {\left[R\right]}_0=I....\left(ii\right)$
Substituting (ii) in (i) we get
$\ln R=-kt+In\left[R_0\right]$
$\frac{InR}{\left[R_0\right]}=-kt$
Taking exponential on both sides, we get
$\frac{R}{R_0}=e^{-kt}$
$\Rightarrow R=R_0{e}^{-kt}.$