Question:
Find the half life for the process of radioactive decay. Assume that \lambda is the decay constant.
Answer:
The radioactive decay is understood as the comparison of the numbers of nucleus at time t and time t+\Delta t. The difference of the numbers can be expressed as N(t)-N(t+\Delta t), and this is proportional to a rate, \lambda.
N(t)-N(t+\Delta t) = -\lambda N(t) \Delta t
The right hand side indicates how the number decreases for \Delta t. We can rearrange it as
\frac{N(t)-N(t+\Delta t)}{\Delta t}=\frac{dN}{dt}=-\lambda N(t)
This is a first order of differential equation with respect to time. Again, rearrange and integrate it as follows:
\int^N_{N_0} \frac{dN}{N}=\int^{t}_{0} -\lambda dt
This gives
\ln \frac{N}{N_0} = -\lambda t \quad \rightarrow \quad N=N_0e^{-\lambda t}
In order to find the half life, let N=\frac{N_0}{2}. Then derive the time, t_{0.5}.
\begin{eqnarray*}
& & \frac{N_0}{2}=N_0e^{-\lambda t_{0.5}} \\
\rightarrow & & \frac{1}{2} = e^{-\lambda t_{0.5}} \\
\rightarrow & & 2 = e^{\lambda t_{0.5}} \\
\rightarrow & & \lambda t_{0.5} = \ln 2 \\
\rightarrow & & t_{0.5} = \frac{\ln 2}{\lambda}
\end{eqnarray*}
Thus, the half life is given by \frac{0.693}{\lambda}.
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