CBSE Class 12 · 2004 · Modern PhysicsShort Answer

Define the terms 'half-life period' and 'decay constant' of a radioactive sample. Derive the relation between these terms.

Answer: $T_{1/2} = 0.693/ λ$ .

Solution

Half-life

T_{1/2}

: the time in which the number of undecayed nuclei falls to half its initial value.
Decay constant

λ

: the fraction of nuclei that decay per unit time; it is the reciprocal of the mean life.
From

N = N_{0} e^{- λ t}

, putting

N = N_{0} /2

t = T_{1/2}

\frac{N _{0}}{2} = N_{0} e^{- λ T_{1/2}} \Rightarrow e^{λ T_{1/2}} = 2 \Rightarrow λ T_{1/2} = ln 2.

T_{1/2} = \frac{ln 2}{λ} = \frac{0.693}{λ}

CBSE Class 12 Board · 2004 · Subjective