GATE-2018 (24) :Wyllie – Time average equation

GATE_2018

80) The ratio of the number of daughter nuclides to the number of parent nuclides after a decay period of 3 half-lives is _______.

(Thanks to Chandra Sekhar, ANU)

Solution:

$D= P_0-P----(1)$

$P=P_0*e^{-\lambda t}$

$P_0=P*e^{\lambda t}---(2)$

Substitute the above eq(2) in eq(1)

$ D = P(e^{\lambda t} -1) $--------------------------(3)

Givent that $t= 3 t_{\frac{1}{2}}$

$\frac{D}{P}$ = ?

By substituting the given values in the equation (3)

$D = P(e^{\lambda t} -1)$

$\frac{D}{P}= e^{\lambda t} -1$

here,

 

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

$\lambda =\frac{0.693}{t_{1/2}}$

Substitute the $\lambda$ value in above equation

=$e^{\frac{0.693}{t_{\frac{1}{2}}} \times 3t_{\frac{1}{2}} } -1$

=$e^{0.693\times 3} -1$

= $e^{2.079} -1$

=7.99 – 1

$\frac{D}{P} =6.99$

Reference : Fundamental of Geophysics, William Lowrie.