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The relation between $\lambda$ and $T_{1/2}$ is : ($T_{1/2}$ = half life, $\lambda$ = decay constant)
A
$T_{1/2} = \dfrac{\ln 2}{\lambda}$
B
$T_{1/2}\ln 2 = \lambda$
C
$T_{1/2} = \dfrac{1}{\lambda}$
D
$(\lambda + T_{1/2}) = \dfrac{\ln}{2}$
Detailed Solution
Decay law: $N = N_0e^{-\lambda t}$
At $t = T_{1/2}$, $N = \dfrac{N_0}{2}$: $\dfrac{1}{2} = e^{-\lambda T_{1/2}}$
$\Rightarrow \lambda T_{1/2} = \ln 2$
$T_{1/2} = \dfrac{\ln 2}{\lambda} = \dfrac{0.693}{\lambda}$
($\dfrac{1}{\lambda}$ is the mean life.)
At $t = T_{1/2}$, $N = \dfrac{N_0}{2}$: $\dfrac{1}{2} = e^{-\lambda T_{1/2}}$
$\Rightarrow \lambda T_{1/2} = \ln 2$
$T_{1/2} = \dfrac{\ln 2}{\lambda} = \dfrac{0.693}{\lambda}$
($\dfrac{1}{\lambda}$ is the mean life.)