Question

For a first order reaction $R \rightarrow P$, 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,
$R\rightarrow P$
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 $t=0, R=R_{0}$
$\therefore In \left[R\right]_{0}=-k\times0+I$
$\ln\left[R\right]_{0}=I ....\left(ii\right)$
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
$\frac{R}{R_{0}}=e^{-kt}$
$\Rightarrow R=R_{0}e^{-kt}.$