Question

Radioisotopes are used in a variety of manufacturing and testing techniques. Wear measurements can be made using the following method. An automobile engine is produced using piston rings with a total mass of 100 g, which includes 9.4 m Ci of whose half-life is 45 days. The engine is test-run for 1000 hours, after which the oil is drained and its activity is measured. If the activity of the engine oil is 84 decays/s, how much mass was worn from the piston rings per hour of operation?

Short answer

The relation between the activity R of a sample and the number of radioactive nuclei N in the sample is given by:

..............(i)

The number N of remaining nuclei after time t is given by:

..............(ii)

Where is the number of nuclei at t = 0.

Multiplying equation (ii) by A and substituting equation (i), we get

.............(iii)

Where R is the activity after time t and is the activity at t = 0.

The relation between the half-life and the decay constant is given by:

...............(iv)

Step-by-step solution

Known

The relation between the activity R of a sample and the number of radioactive nuclei N in the sample is given by:

..............(i)

The number N of remaining nuclei after time t is given by:

..............(ii)

Where is the number of nuclei at t = 0.

Multiplying equation (ii) by A and substituting equation (i), we get

.............(iii)

Where R is the activity after time t and is the activity at t = 0.

The relation between the half-life and the decay constant is given by:

...............(iv)

Given

The initial activity is , the half-life of is , the time of the test-run is t = 1000 h and the final activity of the lost mass in the oil is R= 84.0 Bq.

Concept

Let us introduce a new quantity called the specific activity, which means the activity per unit mass of the sample

Here, the sample does not mean only the radioactive sample, it means the sample that contains a radioactive isotope.

We can use equation (iv) to calculate the specific activity at t simply by putting the specific activity instead of the activity,

Thus,

..................(v)

Calculation

Assuming that contained in the piston is uniformly distributed, we can calculate the initial specific activity of the piston as follows:

So, initially, we have a number of nuclei with activity per kilogram of the mass of the piston. Then, we plug our value for into equation (iv), so we get the decay constant of :

Now, we plug our values for and t into equation (v), so we get the specific activity after that time:

But the activity of the mass of the piston worn in the oil has an activity of R = 84 Bq, so its mass is:

Over a total time interval of 1000h, the average rate at which the piston loses mass is: