Question

The spontaneous decomposition of radio nuclei is a first order rate process.U-238 disintegrates with the emission of $\alpha$-particles and has a half-life of $4.5\times {10}^9$ years. If at time t=0, 1 mole of U-238 is present, what will be the number of nuclei left after 1 billion years?

• A. $5.16\times {10}^{23}$
• B. $5.16\times {10}^{20}$
• C. $5.16\times {10}^{19}$
• D. None of these

Answer 1

The correct option is D None of these

We know that,

Radioactive disintegration follows first order kinetics.

Now, half life period of a reaction is the time taken by reaction to be half completed.

So, amount of reactant left after one half life period= $\frac{[A{]}_0}{2}$ ; $A_0$ is the amount present at $t=0$

Amount present after $2$ half lines= $\frac{[A{]}_0}{2^2}$

In general, amount of reactant left after $n$ half lives=$\frac{[A{]}_0}{2^n}$

Now, Half life of $U-238=4.5\times {10}^9$ yrs

Given time, $t=1$ billion years

$={10}^9$ years

So, no. of half lives=$\frac{{10}^9}{4.5\times {10}^9}=\frac{1}{4.5}$

Amount of reactant left= $\frac{[A{]}_0}{2^{4.5}}$

=$\frac{[A{]}_0}{22.68}$

Now, at $t=0$, amount of $U-238$ present= $1$ mole=$6.022\times {10}^{23}$ molecules (nuclei)

$\therefore$ No. of nuclei left after $1$ billion years= $\frac{6.022\times {10}^{23}}{22.68}$

$=0.27\times {10}^{23}$