The half-life period of a radioactive element 'X' is same as the mean life of another radioactive element Y. Initially both of them have the same no. of atoms, then:
A. X and Y have the same decay rate initially.
B. X and Y decay at the same rate always.
C. Y will decay at a faster rate than X.
D. X will decay at a faster rate than Y.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → C Only
How fast a substance decays is fixed by its decay constant \( \lambda \). It is related to the half-life and the mean life by
\( \displaystyle T_{1/2} = \frac{0.693}{\lambda}, \qquad \tau = \frac{1}{\lambda} \)
Given \( T_{1/2}(X) = \tau(Y) = T \):
\( \displaystyle \lambda_X = \frac{0.693}{T}, \qquad \lambda_Y = \frac{1}{T} \)
So \( \lambda_Y \gt \lambda_X \). The rate of decay is \( R = \lambda N \), and both start with the same \( N_0 \).
A. X and Y have the same decay rate initially. — Incorrect. With equal \( N_0 \), the initial rate of Y is larger, since \( \lambda_Y N_0 \gt \lambda_X N_0 \).
B. X and Y decay at the same rate always. — Incorrect. The decay constants differ, so the two substances do not decay alike; their rates are not equal even at the start.
C. Y will decay at a faster rate than X. — Correct. Y has the larger decay constant, so a larger fraction of its atoms decays per second and its number of atoms falls faster.
D. X will decay at a faster rate than Y. — Incorrect. X has the smaller decay constant (its half-life is the longer of the two half-lives), so it decays more slowly.
Note: This question is from the pre-rationalised NCERT book (2022-23). The topic was removed from the latest NCERT book (2026-27), and it also appears to be out of the CUET 2026 syllabus.
NCERT (Pre-rationalised edition, 2022-23):
Law of Radioactive Decay – “… We can now write Eq. (13.10) in terms of the decay rate R of the sample as
\( \displaystyle R = \lambda N \qquad (13.16) \)”
Law of Radioactive Decay – “… One measure is the half-life T1/2 of a radionuclide, which is the time at which both N and R have been reduced to one-half their initial values. The other measure is the mean life \( \tau \), which is the time at which both N and R have been reduced to e–1 of their initial values.”
Law of Radioactive Decay – “… We summarise these results with the following:
\( \displaystyle T_{1/2} = \frac{\ln 2}{\lambda} = \tau \ln 2 \qquad (13.18) \)”