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BITSAT 2015 Paper

Section: Physics

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Question : 25 of 150

Marks: +1, -0

In an ore containing uranium, the ratio of

\mathrm{U}^{238}

\mathrm{Pb}^{206}

is 3. Calculate the age of the ore, assuming that all the lead present in the ore is the final stable product of

\mathrm{U}^{238}

. Take the half-life of

\mathrm{U}^{238}

10^{9}

yr.

1.6 × $19^{3}$ yr
1.5 × $10^{4}$ yr
1.867 × $10^{9}$ yr
2 × $10^{5}$ yr

Solution:

Let the initial mass of uranium be

M_{0}

Final mass of uranium after time t,

M =

\frac{3}{4} M_{0}

According to the law of radioactive disintegration

\frac{M}{M_{0}}

=

\left(\frac{1}{2}\right)^{\frac{t}{T}}

⇒

\frac{M_{0}}{M}

=

\left(2\right)^{\frac{t}{T}}

∴

\log_{10}\left(\frac{M_{0}}{M}\right)

=

\frac{t}{T} \log_{10}

(2)

t = T

\frac{\log_{10}\left(\frac{M_{0}}{M}\right)}{\log_{10}(2)}

=

\frac{T\log_{10}\left(\frac{4}{3}\right)}{\log_{10}(2)}

=

\frac{T\log_{10}(1.333)}{\log_{10}(2)}

= 4.5 ×

10^{9} \left(\frac{0.1249}{0.3010}\right)

⇒ t = 1.867 ×

10^{9}

yr.

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