The activity of a radioactive sample was measured over 12 h , with the net count rates shown in

The activity of a radioactive sample was measured over 12 h...

The activity of a radioactive sample was measured over 12 $\mathrm{h}$ , with the net count rates shown in the accompanying table. (a) Plot the logarithm of the counting rate as a function of time. (b) Determine the decay constant and half-life of the radioactive nuclei in the sample. (c) What counting rate would you expect for the sample at $t=0$ ? (d) Assuming the efficiency of the counting instrument is $10.0 \%,$ calculate the number of radioactive atoms in the sample at $t=0$ . (TABLE NOT COPY)

Key Concepts

Measurement of Radioactivity

Counting rates measured over time provide empirical data on the radioactive decay process. These measurements are used to plot decay curves, determine decay constants, estimate half-lives, and, when combined with detector efficiency, calculate the actual number of radioactive atoms present in the sample.

Detector Efficiency

Detector efficiency accounts for the fact that a measuring instrument might not register every radioactive decay event. If the efficiency is less than 100%, the observed counting rate must be corrected to estimate the true activity or number of decays occurring in the sample.

Logarithmic Transformation

Transforming the exponential decay data using the natural logarithm linearizes the relationship, turning a curve into a straight line. By plotting ln(N) versus time, the slope of the line corresponds to -?, making it easier to determine the decay constant and subsequently the half-life.

Decay Constant

The decay constant (?) represents the probability per unit time that a given nucleus will decay. It is a key parameter in the exponential decay law and directly determines the rate at which the radioactive material decreases. A larger decay constant means a faster decay rate.

Exponential Decay

Exponential decay describes how the quantity of a radioactive substance decreases over time. It is characterized mathematically by a function of the form N(t) = N0 * exp(-?t), where N0 is the initial amount, ? is the decay constant, and t is time. This principle is fundamental in understanding and modeling radioactive processes.

Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is inversely related to the decay constant through the relation t1/2 = ln(2)/?. This concept provides an intuitive measure of the rate of decay and is widely used to compare the stabilities of different radioisotopes.

Transcript

00:01 For this particular question we are given a table of time and the count rate on the activity count rates per minute at a particular time interval right for this particular radioactive sample we have this in terms of units of 1 2 2 4 6 8 etc so on so forth and the count rates what you want to do is to plot the logarithm of the count rate with the against time so what we have to do is to take the lawn of the count per minute of these values into here and then we're going to plot this as the y -axis and time as the x -axis so very simple you just take lawn of the first value is 3 ,100 so long 1 -200 it's about 8 .04, then so on and so far until the last value, which is actually about 5 .3.

01:18 So with all these values, we can then plot the graph x and y coordinates are given over here.

01:26 You should get a graph that looks similar to something like this.

01:30 And then we're going to draw a best fit line across.

01:32 The red line is the best fit line.

01:36 Entire graph now what is exactly are we plotting we know that activity is given as initial activity times exponential negative lambda t right so when we take the natural lock of the activity we have is lon of a comes on a nought minus lambda t right so plotting activity loan of activity against time what we have is in fact negative of lambda as the gradient and we have lon of a knot as d initial s d y intercept right we're going to use this uh properties to solve the next quality question right we we want to determine the decay constant in the half -life, right? so to get the decay constant, we just need to look at the gradient over here.

02:51 To get the gradient, we're going to take two points as far apart as possible.

02:54 So we're going to consider this point over here at 12 minutes.

02:59 This point in particular is about 5 .3.

03:03 In terms of the y value, right, this 5 .3.

03:07 And we're going to consider the final point over here.

03:11 When it cuts the y intercept.

03:13 This is approximately 8 .3.

03:16 At time it goes to 0.

03:21 So the gradient just take rise over run...