State the radioactive decay law. Derive mathematically the exponential decay law expression $N(t) = N0 e^{-\lambda t}$ for a radioactive substance.

Q: State the radioactive decay law. Derive mathematically the exponential decay law expression $N(t) = N0 e^{-\lambda t}$ for a radioactive substance.

Statement of Radioactive Decay Law:\nAt any instant, the rate of decay of radioactive nuclei (i.e., the number of nuclei decaying per unit time) is directly proportional to the number of undecayed radioactive nuclei present at that instant. Derivation:\nLet $N$ be the number of undecayed radioactive nuclei present in a sample at any time $t$. According to the radioactive decay law, the rate of change of $N$ with respect to time $t$ is given by: $$\frac{dN}{dt} \propto -N$$ \nwhere the negative sign indicates that the number of nuclei decreases with time. Introducing a radioactive decay constant $\lambda$ (disintegration constant), we can write the equation as: $$\frac{dN}{dt} = -\lambda N$$ \nRearranging the terms to separate variables $N$ and $t$: $$\frac{dN}{N} = -\lambda \, dt$$ \nIntegrating both sides within appropriate limits—let $N0$ be the initial number of nuclei at time $t = 0$, and $N$ be the number of nuclei at a later time $t$: $$\int{N0}^{N} \frac{dN}{N} = -\lambda \int{0}^{t} dt$$ \nUsing standard logarithmic integration, $\ln(N) \Big|{N0}^{N} = -\lambda t \Big|{0}^{t}$: $$\ln\left(\frac{N}{N0}\right) = -\lambda t$$ \nTaking the exponential of both sides: $$\frac{N}{N0} = e^{-\lambda t}$| $$N(t) = N0 e^{-\lambda t}$| \nThis shows that radioactive decay follows an exponential decrease over time.

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Statement of Radioactive Decay Law:
At any instant, the rate of decay of radioactive nuclei (i.e., the number of nuclei decaying per unit time) is directly proportional to the number of undecayed radioactive nuclei present at that instant. Derivation:
Let $N$ be the number of undecayed radioactive nuclei present in a sample at any time $t$. According to the radioactive decay law, the rate of change of $N$ with respect to time $t$ is given by: $$\frac{dN}{dt} \propto -N$$ 
where the negative sign indicates that the number of nuclei decreases with time. Introducing a radioactive decay constant $\lambda$ (disintegration constant), we can write the equation as: $$\frac{dN}{dt} = -\lambda N$$ 
Rearranging the terms to separate variables $N$ and $t$: $$\frac{dN}{N} = -\lambda \, dt$$ 
Integrating both sides within appropriate limits—let $N0$ be the initial number of nuclei at time $t = 0$, and $N$ be the number of nuclei at a later time $t$: $$\int{N0}^{N} \frac{dN}{N} = -\lambda \int{0}^{t} dt$$ 
Using standard logarithmic integration, $\ln(N) \Big|{N0}^{N} = -\lambda t \Big|{0}^{t}$: $$\ln\left(\frac{N}{N0}ight) = -\lambda t$$ 
Taking the exponential of both sides: $$\frac{N}{N0} = e^{-\lambda t}$| $$N(t) = N0 e^{-\lambda t}$| 
This shows that radioactive decay follows an exponential decrease over time.

MP Board · Class 12 · Physics · NucleiState the radioactive decay law. Derive mathematically the exponential decay law expression $N(t) = N0 e^{-\lambda t}$ for a radioactive substance.