A radioactive substance has a half-life of h | vedclass

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(C) Let $m$ be the mass of the substance at time $t$. The rate of decay is given by the differential equation: $\frac{dm}{dt} = -km$,where $k > 0$ is

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$-\frac{m_0}{h}(\log 2)$

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(C) Let $m$ be the mass of the substance at time $t$. The rate of decay is given by the differential equation: $\frac{dm}{dt} = -km$,where $k > 0$ is the decay constant. Separating variables,we get: $\frac{dm}{m} = -k dt$ Integrating both sides: $\ln m = -kt + C$ At $t = 0$,$m = m_0$,so $C = \ln m_0$. Thus,$\ln \left(\frac{m}{m_0}\right) = -kt$. Given the half-life is $h$,at $t = h$,$m = \frac{m_0}{2}$. Substituting these values: $\ln \left(\frac{m_0/2}{m_0}\right) = -kh$ $\ln \left(\frac{1}{2}\right) = -kh$ $-\ln 2 = -kh \Rightarrow k = \frac{\ln 2}{h}$. The initial decay rate is the value of $\frac{dm}{dt}$ at $t = 0$: $\left(\frac{dm}{dt}\right)_{t=0} = -k m_0 = -\left(\frac{\ln 2}{h}\right) m_0 = -\frac{m_0}{h} \ln 2$.

$A$ radioactive substance has a half-life of $h$ days. Its initial decay rate is given by (where the radioactive substance has an initial mass $m_0$).

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