The activity of a radioactive sample is measured as N_0 counts per minute at t=0 and N_0 / e counts per minute at t=5 ~min . The time (in minute) at which the activity reduces to half its value is

Correct answer: D. 5 _ e 2 Fraction remains after n half lives N N_0 =((1)/(2))^ n =((1)/(2))^ t / T Given N= N_0 e N_0 eN_0 =((1)/(2))^ 5 / T or 1 e =((1)/(2))^ 5 / T Taking on both sides, we get aligned ₁⁻ e& = 5 T (1)/(2) -1 & = 5 T (- 2) T& =5 _ e 2 aligned Now, let t^ be the time after which activity reduces to half aligned ((1)/(2))& =((1)/(2))^ t^ / 5 _ e 2 t^ & =5 _ e 2 aligned

NEET 2010

Exam: NEET
Year: 2010
Subject: Physics
Chapter: Nuclear Physics

NEET 2010 · Physics · Nuclear Physics

The activity of a radioactive sample is measured as N_0 counts per...

The activity of a radioactive sample is measured as

\mathrm{N}_0

counts per minute at

\mathrm{t}=0

and

\mathrm{N}_0 / \mathrm{e}

counts per minute at

\mathrm{t}=5 \mathrm{~min}

. The time (in minute) at which the activity reduces to half its value is

Options

A $\log _e 2 / 5$
B $\frac{5}{\log _e 2}$
C $5 \log _{10} 2$
D $5 \log _{\mathrm{e}} 2$

Correct answer

D. $5 \log _{\mathrm{e}} 2$

Step-by-step solution

Fraction remains after

n

half lives

\frac{\mathrm{N}}{\mathrm{N}_0}=\left(\frac{1}{2}\right)^{\mathrm{n}}=\left(\frac{1}{2}\right)^{\mathrm{t} / \mathrm{T}}Given \mathrm{N}=\frac{\mathrm{N}_0}{\mathrm{e}} \Rightarrow \frac{\mathrm{N}_0}{\mathrm{eN}_0}=\left(\frac{1}{2}\right)^{5 / \mathrm{T}}

\frac{1}{\mathrm{e}}=\left(\frac{1}{2}\right)^{5 / T}

Taking

\log

on both sides, we get

\begin{aligned} \log₁⁻\log \mathrm{e}& =\frac{5}{\mathrm{ T}} \log \frac{1}{2}\\ -1 & =\frac{5}{\mathrm{ T}}(-\log2) \\ \Rightarrow \quad \mathrm{T}& =5 \log _{\mathrm{e}} 2 \end{aligned}

Now, let \mathrm{t}^{\prime}

be the time after which activity reduces to half

\begin{aligned} \Rightarrow \quad\left(\frac{1}{2}\right)& =\left(\frac{1}{2}\right)^{\mathrm{t}^{\prime} / 5 \log _{\mathrm{e}} 2}\\ \Rightarrow \quad \mathrm{t}^{\prime} & =5 \log _{\mathrm{e}} 2 \end{aligned}

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