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# Suppose that a radioactive substance decays according to the model $N=N_{0} e^{-\lambda t}$. (a) Show that affer a period of $T_{\lambda}=1 / \lambda$, the material has decreased to $e^{-1}$ of its original value. $T_{\lambda}$ is called the time constant and it is defined by this property. (b) A certain radioactive substance has a half-life of 12 hours. Compute the time constant for this substance. (c) If there are originally 1000 mg of this radioactive substance present, plot the amount of substance remaining over four time periods $T_{\lambda}$.

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Radioactive substance decays accoriding to the next model:

$N(t)=N_0\mathrm{e}^{-\lambda t}$

$\textbf{a)}$

$T_{\lambda}=\frac{1}{\lambda}-\text{ time constant}$

Let's look what happend with our radioactive substance after

$\text{period of T_{\lambda}:}$

$N(T_{\lambda})=N_0\mathrm{e}^{-\lambda T_{\lambda}}$

$N(\frac{1}{\lambda})=N_0\mathrm{e}^{-\lambda \frac{1}{\lambda}}$

$N(\frac{1}{\lambda})=N_0\mathrm{e}^{-1}$

$\text{Hence, we have that after period of T_{\lambda}, ammount of material}$

$\text{ is reduced to }N_0\mathrm{e}^{-1}.\text{ So, radioactive material is decreased }$

$\text{for \mathrm{e}^{-1} of its initial value N_0.}$

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