Using the decay curve on the graph, find the half-life of the sample.
The nucleus of an atom contains protons and neutrons. If the nucleus contains too many protons, too few neutrons, or too much energy, then it can become unstable. Unstable nuclei can become more stable by emitting radiation. This spontaneous emission of radiation from an unstable nucleus is called radioactive decay. This is why the graph shows a decay curve. A Geiger–Müller tube is used for the detection of this radiation. The tube is attached to a Geiger counter. When the radiation enters the tube, an electrical pulse is sent to the counter. The counter displays the count rate, which is the number of radiation detection events, such as electrical pulses, per second or per minute.
In the graph given, the count rate is in units of counts per minute. The 𝑥-axis of this graph is labeled with time in hours. As time passes, the unstable nuclei decay in order to become more stable. So, the number of unstable nuclei decreases. This means that the amount of radiation decreases; thus, the count rate will decrease. We can see this in the graph. As time passes, the count rate decreases.
So, now, we understand how the decay curve works, but we want to find the half-life of the sample. The half-life is the amount of time required for one-half of the radioactive nuclei to decay. So, it is a measure of radioactive decay. An example of this is that if we start with 100 unstable nuclei and it takes one hour for half of the sample or 50 unstable nuclei to decay into 50 stable nuclei, then the half-life of this sample is one hour.
In this question, we haven’t been given the values of how many nuclei we have. But we’ve been given the count rate, which as established previously directly correlates to the number of nuclei. So, for this calculation, we can change the definition of half-life from the amount of time required for one-half of the radioactive nuclei to decay to the amount of time required for the count rate to halve.
In the graph given, at time zero, the count rate is 1100 counts per minute. To find the half-life, we first need to halve the count rate. 1100 counts per minute, abbreviated to cpm, divided by two is 550 counts per minute. We then need to find 550 counts per minute on the graph and calculate how much time it took to go from 1100 counts per minute to 550 counts per minute. Since we started at time zero and ended up at a time of two hours, then we can see that it took two hours for the count rate to halve.
Therefore, using the decay curve on the graph, we have found that the half-life of the sample is two hours.