To find: Number of nuclei remain after time ($t$ = 30 seconds) i.e, $N$.
To solve this problem we have to use the concept of half life and also the relation of the number of nuclei decayed to the original number of nuclei present in the sample of a radioactive element.
Now using the concept of half live, we can write the relation between the number of nuclei decayed ($N$) to the original number of nuclei(${N}_{o}$) as,
$\frac{N}{{N}_{o}}}=({\displaystyle \frac{1}{2}}{)}^{t/{t}_{1/2}$
Now after putting the values of the quantities in the above relation, we get;
$\frac{N}{{10}^{10}}}=({\displaystyle \frac{1}{2}}{)}^{30/60$
$\frac{N}{{10}^{10}}}=({\displaystyle \frac{1}{2}}{)}^{1/2$
$N={\displaystyle \frac{{10}^{10}}{\sqrt{2}}}\approx 7\times {10}^{9}$
Hence, the number of nuclei that remain after 30 seconds is $7\times {10}^{9}$.
Therefore, option a is correct.
Trick: The application of the half life concept and the relation between the number of nuclei decayed to the original number of nuclei in a sample is important to solve this problem.
The half-lives of radioisotopes P32 and P33 are 14 days and 25 days respectively. These radioisotopes are mixed in the ratio of 4 : 1 of their atoms. If the initial activity of the mixed sample is 3.0 mCi, find the activity of the mixed isotopes after 60 years.(Ans: 0.205 mCi)
A stable nuclei C is formed from two radioactive nuclei A and B with decay constant of 𝜆1 and 𝜆2 respectively. Initially, the number of nuclei of A is N0 and that of B is zero. Nuclei B are produced at a constant rate of P. Find the number of the nuclei of C after time t (Ans:${N}_{c}={N}_{0}(1-{e}^{-{\lambda}_{1}t})+P\u27eet+{\displaystyle \frac{{e}^{-{\lambda}_{2}t}-1}{{\lambda}_{2}}}\u27ef$)
In this article we have discussed the Nuclei chapter with respect to the NEET exam. In this article we have provided solved examples, previous year questions, important formulae list, etc..
1. What is the weightage of the motion in a plane in NEET?
Nearly 1-2 questions arise in the exam from this chapter.
2. What are the key points that need to be practiced for solving questions from Nuclei?
Students should practice numericals based on radioactivity and binding energy
3. Are previous year questions enough for NEET?
To score 640+ in NEET, NCERT (both 11th and 12th) and previous year NEET Mains papers are sufficient. Solving previous 10-year NEET examinations offers us a tremendous advantage because 6-7 questions with the identical alternatives are guaranteed to be repeated every year.