Question

The rate of a first order reaction is $0.04$ mole ${\ell }^{-1}{S}^{-1}$ at $10$ seconds and $0.03$ mol ${\ell }^{-1}{S}^{-1}$ at $20$ seconds after initiation of the reaction. The half-life period of the reaction is:

• A. $48.1$ s
• B. $24.1$ s
• C. $34.1$ s
• D. $44.1$ s

Answer 1

The correct option is A $48.1$ s

$\vartheta =kA\to$first order

$\vartheta =k{A}_0{e}^{-kt}$

$\vartheta ={\vartheta }_0{e}^{-kt}\to$first order wrt $\vartheta$

so,

${\vartheta }_{10}=0.04={\vartheta }_0{e}^{-10k}\to \left(1\right)$

${\vartheta }_{30}=0.03={\vartheta }_0{e}^{-30k}\to \left(2\right)$

now using (1)/(2)

$+l_n\left(1.33\right)=20k$

$k=0.0144$

now,$t_{\frac{1}{2}}=\frac{0.693}{k}$

$=48.125$

$\cong 50$ seconds