Half-life And Decay Law

Question

A radioactive isotope used in medical imaging has a half-life of 6 hours, and a technician needs its decay constant so the activity can be predicted accurately.

RankGuru · Accepted Answer

NCERT relates the decay constant $\lambda$, the probability per unit time that a nucleus decays, to the half-life through $T_{1/2}=\frac{\ln 2}{\lambda}=\frac{0.693}{\lambda}$. Rearranging gives $\lambda=\frac{0.693}{T_{1/2}}$. Substituting $T_{1/2}=6$ hours, $\lambda=\frac{0.693}{6}=0.1155$ per hour. The value 0.693 per hour is wrong because it uses $\ln 2$ directly without dividing by the half-life. The value 6.0 per hour is wrong because it merely repeats the half-life with an inverted unit and ignores the logarithm factor. The value 0.0192 per hour is wrong because it divides 0.1155 by 6 again, applying the half-life twice. As stated in NCERT Class 12, Chapter 13 (Nuclei), a shorter half-life corresponds to a larger decay constant, meaning faster decay, so isotopes chosen for medical imaging are deliberately short-lived to limit patient exposure. The decay constant is the intrinsic probability per unit time that any single nucleus disintegrates, and it does not depend on how many nuclei are present. A plausibility check: multiplying $\lambda=0.1155$ per hour by the 6-hour half-life returns 0.693, recovering $\ln 2$ as required, so the value is internally consistent with the defining relation between half-life and decay constant.

Half-life And Decay Law

Select the correct option:

🔒 Solution Hidden from View

More atoms and nuclei Practice Questions

A wooden artefact recovered from an excavation shows a carbon-14 activity that is one-fourth of the ...

The Sun generates its enormous power by fusing hydrogen nuclei into helium at its core, and a studen...

A neutron-rich nucleus undergoes beta-minus decay inside a detector, and a student is asked what par...

In a nuclear reactor, a uranium-235 nucleus captures a slow neutron and splits into lighter fragment...

A researcher considers a singly ionised helium ion with atomic number two behaving like a hydrogen-l...

Following an alpha or beta emission, a daughter nucleus is often left in an excited state and subseq...

Looking at the binding-energy-per-nucleon curve, a student notices it peaks near a certain mass numb...

A radioactive sample initially contains 8 billion nuclei of an unstable isotope, and after three suc...

A freshly prepared source contains a large number of identical unstable nuclei, and a researcher obs...

Given that the mass defect of a helium-4 nucleus is about 0.0304 u, an examiner asks candidates to f...

When a heavy parent nucleus undergoes alpha decay, it emits a helium nucleus, and a student must sta...

A lab worksheet asks students to compute the energy equivalent of one atomic mass unit of matter usi...

About This Question

Solution