Question

The activity of a sample of radioactive material was measured over 12 h. The following net count rates were obtained at the times indicated:

Time (h)	Counting Rate (counts/min)
1	3100
2	2450
4	1480
6	910
8	545
10	330
12	200

1a) Plot the activity curve on semilog paper.

1b) Determine the disintegration constant and the half-life of the radioactive nuclei in the sample.

Answer 1

1a) Activity curve is the curve between Counting Rate and Time with Counting Rate on y-axis and Time on the x-axis. The activity curve on semilog paper (y-axis is a log scale and x-axis is a linear scale) is shown below.

1b) In a radioactive decay process, the number of decay events -dN expected to occur in a small interval of time dt is proportional to the number of atoms present N, that is

Thus

where is the decay rate or disintegration constant. The solution to above first-order differential equation is

or

Therefore graph of   is linear with slope and intercept .

Half-life is the time when . Putting this in above equation we have,

Therefore, if we have decay rate or disintegration constant we can find the half-life.

In the present question, we are given the counting rate and time. They follow the exponential decay as indicated by eq.1 . Using eq2, when plotted on a semilog graph (part 1a of the question), slope of this curve is the decay rate or disintegration constant. So after fitting the above curve, we will have the slope. The fitted curve is shown below (plotting and fitting is done in the standard scientific graphing and data analysis software - Origin)

Thus disintegration constant is and half-life is given as

Now intercept is 8.29902 which implies

Thus the equation represents (approximately) the process that results in the above table as its data points.

For a check, put t=10 hr we have