# In a sample of radioactive substance what fraction of initial no . of nuclei will remain undecayed after time $\;t=\large\frac{T}{2}\;$ , where T = half - life of substance

$(a)\;\sqrt{2}\qquad(b)\;\large\frac{1}{ \sqrt{2}}\qquad(c)\;\large\frac{1}{3 \sqrt{2}}\qquad(d)\;None$

Answer : (b) $\;\large\frac{1}{ \sqrt{2}}$
Explanation :
Fraction of nuclei which remain undecayed is
$f=\large\frac{N}{N_{0}}=\large\frac{N_{0} e^{- \lambda t}}{N_{0}}$
$=e^{- \lambda t}$
$=e^{-(\large\frac{ln 2}{T})\;(\large\frac{T}{2})}$
$=e^{\large\frac{1}{ln \sqrt{2}}}=\large\frac{1}{\sqrt{2}}\;.$