The rate of disintegration of a radioactive e | vedclass

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(B) The rate of disintegration is given by $\frac{dM}{dt} = -kM$, where $k > 0$.Separating the variables, we get $\frac{dM}{M} = -k dt$.Integrating bo

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$\frac{1}{k} \log 3$

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(B) The rate of disintegration is given by $\frac{dM}{dt} = -kM$, where $k > 0$.Separating the variables, we get $\frac{dM}{M} = -k dt$.Integrating both sides, $\int \frac{dM}{M} = \int -k dt$, which gives $\ln M = -kt + C$.At $t = 0$, $M = M_0 = 1.5 	ext{ g}$, so $C = \ln(1.5)$.Thus, $\ln M = -kt + \ln(1.5)$, or $\ln(\frac{M}{1.5}) = -kt$.We want to find $t$ when $M = 0.5 	ext{ g}$.Substituting the values, $\ln(\frac{0.5}{1.5}) = -kt$.$\ln(\frac{1}{3}) = -kt$, which implies $-\ln 3 = -kt$.Therefore, $t = \frac{1}{k} \ln 3$ or $\frac{1}{k} \log_e 3$.

The rate of disintegration of a radioactive element at any time $t$ is proportional to its mass $M$ at that time. If $k (k > 0)$ is the constant of proportionality, find the time required for an initial mass of $1.5 \text{ g}$ to disintegrate to a mass of $0.5 \text{ g}$.

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