Question

One gram of a radioactive substance disintegrates at the rate of 3.7X10 disintegrations per second. The atomic mass of the substance is 226. Calculate its mean life.

Answer 1

Answer:

In order to solve this problem first calculate the number of nuclei

i.e., $N = moles \times {N_A}$

Where

${N_A} = $ Avogadro number

$ = 6.023 \times {10^{23}}$ per mole

After then by putting the value of activity in mean life formula we get desire solution i.e.,

$T = \dfrac{N}{A}$

Where

T $ = $ Mean life

A $ = $ Activity

N $ = $ Number of nuclei

Complete step by step answer:

We know that activity of any substance is the disintegrates rate of substance which is given as

$A = 3.7 \times {10^{10}}$ disintegration per second …..(1)

Let the number of nuclei is N.

So,

N $ = $ Number of moles $ \times {N_A}$

Where

${N_A} = 6.02 \times {10^{23}}$ per mole

Moles $ = $ 1 gram $/$ 226 gram per mole

Moles $ = 0.00442$

So, $N = 0.00442 \times 6.02 \times {10^{23}}$

$N = 0.0266 \times {10^{23}}$ …..(2)

The mean life of radioactive substance is given by following expression

$T = \dfrac{N}{A}$

From equation 1 and 2, putting the values of A and N we get

$T = \dfrac{{0.0266 \times {{10}^{23}}}}{{3.7 \times {{10}^{10}}}}$

$\implies T = 0.007189 \times {10^{23}} \times {10^{ - 10}}$

$\implies T = 0.00719 \times {10^{13}}\sec $

$\therefore T = 7.19 \times {10^{10}}\sec $

Hence, the mean life of substance is $7.19 \times {10^{10}}s$

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