Question

The half life of Uranium - 233 is 160000 years, i.e., Uranium 233 decays at a constant rate in such a way that it reduces to 50% in 160000 years. In how many years will it reduce to 25% ? 

(a) 80000 years 

(b) 240000 years 

(c) 320000 years 

(d) 40000 years

Answer 1

(c) 320000 years

Let the rate of decay of Uranium be R per cent per year. Also, let the initial amount of Uranium be 1 unit. Since, the half life of Uranium - 233 is 160000 years, therefore

\(\big(1-\frac{R}{100}\big)^{160000}\) = \(\frac{1}{2}\) ..........(i)

Suppose Uranium - 233 reduces to 25% in t years. Then,

\(\big(1-\frac{R}{100}\big)^t\) = \(\frac{25}{100}\) = \(\frac{1}{4}\) = \(\big(\frac{1}{2}\big)^2\)

= \(\Big(\big(1-\frac{R}{100}\big)^{160000}\Big)^2\) = \(\big(1-\frac{R}{100}\big)^{320000}\)

\(\Rightarrow\) t = 320000 years.