MHTCET 15 May 2024 Shift 1 Paper

Concept:Radioactive decay follows first-order kinetics, so the rate of decay is directly proportional to the instantaneous mass of the substance.Explanation:Let $M$ be the mass of the substance at time $t$.The decay law is given by $\frac{dM}{dt} = -kM$, where $k > 0$ is the decay constant.Separating the variables: $\frac{dM}{M} = -k\,dt$.Integrating both sides, we get $\ln M = -kt + c$.At $t = 0$, $M = m_0$, so $c = \ln m_0$.Therefore, $\ln\left(\frac{M}{m_0}\right) = -kt$.At the half-life $t = h$, the mass becomes $M = \frac{m_0}{2}$.Substituting this gives $\ln\left(\frac{1}{2}\right) = -kh$, which simplifies to $\ln 2 = kh$.Hence, the decay constant is $k = \frac{\ln 2}{h}$.The initial decay rate at $t = 0$ is $\left.\frac{dM}{dt}\right|_{t=0} = -k m_0 = -\frac{m_0}{h}\ln 2$.Using $\log$ notation to match the options, this becomes $-\frac{m_0}{h}(\log 2)$.Answer:The initial decay rate is $-\frac{m_0}{h}(\log 2)$, which corresponds to option C.