Question

The relation between $\lambda$ and $T_{1/2}$ is given as ?
$(T_{1/2}\to halflife,\lambda \to decayconstant)$

• A. $T_{1/2}=\frac{\ell n2}{\lambda }$
• B. $T_{1/2}\ell n2=\lambda$
• C. $T_{1/2}=\frac{1}{\lambda }$
• D. $(\lambda +T_{1/2})=\frac{\ell n2}{2}$

Answer 1

The correct option is A $T_{1/2}=\frac{\ell n2}{\lambda }$

$\text{Relation between}\lambda \text{(decay constant) and}{T}_{1{/}_2}\text{(half life)}$

$T_{1/2}=\frac{\ln 2}{\lambda }$

$\begin{array}{c}\text{Lets proof}\\ \text{consider initial no of atom = No}\\ \text{at a time}{T}_{1{/}_2}\text{atom left will be =}\frac{\text{No}}{2}\end{array}$

$\begin{array}{c}{N}_t=N_0{e}^{-\lambda t}\\ t=T_{1/2}\\ \begin{array}{c}\frac{N_0}{2}=N_0{e}^{-\lambda T_{1/2}}\\ \frac{1}{2}=e^{-\lambda {T_{1/}}_2}\\ \ln 2=\lambda T_{1/2}\end{array}\end{array}$

$T_{1/2}=\frac{\ln 2}{\lambda }$