Here are two multiple choice questions which appeared in All India Pre-Medical/Pre-Dental Entrance Examination (AIPMT) 2008:

(1) Two radioactive materials X_{1} and X_{2} have decay constants 5λ and λ respectively. If initially they have the same number of nuclei then the ratio of the number of nuclei of X_{1} to that of X_{2} will be 1/e after a time

(1) λ/2

(2) 1/(4λ)

(3) e/λ

(4) λ

If the initial number of nuclei is *N*_{0} we have

*N*_{1} = *N*_{0}e^{–5}^{λ}* ^{ t}* and

*N*_{2} = *N*_{0}e^{–}^{λ}* ^{ t}* where

*N*

_{1}and

*N*

_{2}are the number of nuclei of X

_{1}and X

_{12}at time

*t*.

Therefore N_{1}/*N*_{2} = e^{–4}^{λt}

This will be equal to 1/e when *t =***1/(4λ)**.* *

(2) Two nuclei have their mass numbers in the ratio of 1:3. The ratio of their nuclear densities would be

(1) 3:1

(2) (3)^{1/3}:1

(3) 1:1

(4) 1:3

The mass of a nucleus is directly proportional to the number (A) of the nucleons. The volume of the nucleus is (4/3)πR^{3} where R is the nuclear radius. But, R = R_{0}A^{⅓} where R_{0} is a constant (equal to 1.1 ×10^{-15}m). So, the volume of the nucleus also is directly proportional to the nucleon number A. Since the density is the ratio of mass to volume, it follows that the *density of nuclear matter is independent of the nucleon number A*** **so that the correct option is (c).

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