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  4. The half-life of thorium X is 3.64 days. After how many days will 0.1 of the mass of a sample of the substance remain undecayed?

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Q. The half-life of thorium X is 3.64 days. After how many days will 0.1 of the mass of a sample of the substance remain undecayed?

KEAMKEAM 2006Nuclei

Solution:

The decay constant is
$ \lambda = \frac{0.693}{T} = \frac{0.693}{3.64} = 0.1904 / day$
Also, $N = N_0 e^{- \lambda t}$
Given, $\frac{N}{N_0} = 0.1 = 10^{-1}$
$\therefore 10^{-1} = e ^{- \lambda t}$
$\Rightarrow e^{\lambda t} = 10$
$\ \lambda t = log_e 10$
$ = 2.3026 \times log_{10} 10$
$ = 2.3026 \times 1$
$\therefore t = \frac{2.3026 \times 1}{\lambda} = \frac{2.3026 \times 1}{0.1904} = 12.1 \, days$