Half-life And Decay Law

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.