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The formula for Activity of a radioactive substance is

$ \frac{dN}{dt}=A=λN $.

If we have an initial number $N(0)$ of some Radionuclide, which has a halflife of, say, 12 hours, is there any particular reason why our time interval for measuring the number of remaining Radionuclides should be close to the halflife of the substance?

Is there any reason, for example, that I shouldn't take measurements of the number of remaining nuclei after every 10 seconds or 10 minutes? Is there a concrete reason that I should really measure the number of remaining nuclei every few hours or so?

The reason I ask is that as we experimentally can't measure the change in number of Nuclei after an infinitesimally small time interval dt, the above equation becomes:

$\frac{ΔN}{Δt} = λN = \frac {ln(2)}{T_{1/2}}N$

Where $Δt$ is the time interval after which each measurement of $N$ is taken.

Does this in some way say that $T_{1/2}$ should be close to $Δt$?

To my mind the obvious answer is no. I've been told, however, that the answer is yes. I don't understand why.

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There is no reason why you can't measure the rate frequently. However, in order to estimate the half life, you need to see a change in the rate of decay. How long you need to measure for, and how far apart you need to change your measurements, depends on the number of decays per second that you observe as well as the required accuracy.

For example, if you have 10,000 decays per second, the standard deviation of a one second measurement will be 100 ($\sqrt{N}$) - so you get the answer to 1% accuracy. If you measure for just one second, and measure again for a second one hour later, you would see (for your huypothetical 12 hour half life) a change of about 6%. With an error of 1% for both the initial and final measurement, you would have an uncertainty of about 1.4% in a value of 6% - which is almost a 25% uncertainty in your estimate of the half life.

By waiting longer, and measuring the individual time points with greater precision, you can improve your estimate. It is a good rule of thumb, when making any measurement, to do a simple error propagation analysis like the above in order to understand the accuracy of your result.

If your activity measurement is more precise (you count more decays - either because the initial activity level is higher, or you count for longer) then it becomes possible to space your measurement intervals more closely. But whatever you do, your choices will affect your uncertainty. The rule of thumb to wait "roughly as long as the half life" takes some of the guess work out. It is not a substitute for a formal error analysis, but a shortcut.

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you should resolve the differential equation of N. If you do that you'll get that $$ A = A_0 e^{\lambda (t-t_0)}, $$ where $A_0$ is the activity at time $t_0$. From there you can obtain the value of $\lambda$ from two measurements of the activity whatever the time interval.

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Firstly, the activity formula is in fact:

$$-\frac{dN}{dt}=A=λN,$$

because $\frac{dN}{dt}<0$.

[...] is there any particular reason why our time interval for measuring the number of remaining Radionuclides should be close to the half-life of the substance?

No and that's not how it's done in practice. $\lambda$ and the half-life are determined by measuring activity a number of times using appropriate time intervals. Of course these intervals have to be in reasonable proportion to the expected half-life. For an expected half-life of about $12\:\mathrm{h}$, $12$ activity measurements at hourly intervals would be a good starting point.

Bear in mind that $\lambda$ and the half-life are determined by statistical analysis and as a general rule a larger number of data points will lead to higher confidence levels $P$ of the calculated observables.

The activity differential equation solves to:

$$\ln\frac{N(t)}{N_0}=-\lambda t,$$

Or:

$$\ln\frac{A(t)}{A_0}=-\lambda t.$$

So from a logarithmic plot of $\frac{A(t)}{A_0}$ versus $t$, $\lambda$ is derived from the gradient of the plot.

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