Two radioactive elements X and Y have half-lives of 100 and 50 years respectively. Samples of X and Y initially contain equal number of atoms. What is the ratio of the number of the remaining atoms of X to that of Y after 200 years?
A
4:1
B
3:1
C
1:1
D
1:2
E
1:4
Verified Correct Answer: A.
Physical Principles & Numerical Solution
Fraction remaining of X = (\(\frac{1}{2}\))\(^{\frac{t}{T_{\frac{1}{2}}}}\)
BUT \(\frac{t}{T_{\frac{1}{2}}}\) = \(\frac{200}{100}\) = 2
= (\(\frac{1}{2})^2\) = \(\frac{1}{4}\)
Fraction remaining of Y = (\(\frac{1}{2}\))\(^{\frac{t}{T_{\frac{1}{2}}}}\)
BUT \(\frac{t}{T_{\frac{1}{2}}}\) = \(\frac{200}{50}\) = 4
= (\(\frac{1}{2})^4\) = \(\frac{1}{16}\)
the ratio of the remaining atoms of X to Y = \(\frac{X}{Y}\) = \(\frac{\frac{1}{4}}{\frac{1}{16}}\) = \(\frac{1}{4}\) x \(\frac{16}{1}\) = 4
Therefore, the ratio of the number of remaining atoms of X to that of Y after 200 years is 4:1
Source: Myschool Past Questions Archive (Record #33700)