Q

# BITSAT The half-life of a certain radioactive element is T. How long will it take for 75% of a sample of the element to decay?

1. 2T

2. T

3. 3/2T

4. 3T

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Radioactive decay, $N(t) = N_{0} e^{- \frac{0.693t}{t_{1/2}}}$

Where N(t) is the amount of sample which remains after time t.

Given that 75% of the sample has decayed, hence the remaining will be 25% of initial, $N(t) =0.25 N_{0}$

Also $t_{1/2} =T$

$0.25 N_{0} =N_{0}e^{- \frac{0.693t}{T}}$

$\Rightarrow 0.25 =e^{- \frac{0.693t}{T}}$

Take loge both sides and solve

$t = 2T$

Option (1) is correct

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