Question

The relationship between rate constant and half-life for first order reaction is:

• A. $t_{1/2}=\frac{0.693}{k}$
• B. $k=\frac{-t_{1/2}}{0.693}$
• C. $t_{1/2}=0.693k$
• D. $t_{1/2}=\frac{k}{0.693}$

Answer 1

The correct option is A $t_{1/2}=\frac{0.693}{k}$

Rate of a reaction,

$R\to P$

is given by the expression: $Rate=\frac{-d\left[R\right]}{dt}=k[R{]}^x$

where $\frac{-d\left[R\right]}{dt}$ = change in reactant concentration with time $dt$, $k$ = rate constant, $x$ = order of the reaction.

For first order reaction, $x$ = $1$, therefore

$\frac{-d\left[R\right]}{dt}=k[R{]}^{1}or,\frac{1}{\left[R\right]}d\left[R\right]=-kdt$

Integrating both sides, we get:

$ln\left[R_t\right]-ln\left[R_0\right]=-k(t-0)$

or, $ln\left(\frac{\left[R_0\right]}{\left[R_t\right]}\right)=kt$

For, half life $t=t_{1/2}and\left[R_t\right]=\frac{\left[R_0\right]}{2}$

Upon substitution we get,

$ln\left(2\right)=k{t}_{1/2}=0.693t_{1/2}=\frac{0.693}{k}$

Option $A$ is the correct answer.