• Sanosuke Sagara
In summary, the problem is that the activity (and specific activity) decreases with time due to the wear of the piston. However, the rate of mass loss must offset the rate of decrease of the specific activity in order for the activity to remain constant.
Sanosuke Sagara
I have my doubt,solution and question in the attachment that followed.Thanks for anybody that spend some time on this question.

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The problem is a bit difficult to follow.

One is given an initial activity, at t=0. The piston is installed on day 30, and the engine tested for 60 days (to day 90).

The equation for activity A(t) is A(t) = Ao exp(-$\lambda$t), where Ao is the initial activity and $\lambda$ is the decay constant = ln 2/t1/2. So at the beginning of the text t=30, and at end of test t=90. Also A(t) = $\lambda$ N(t) where N is the number of atoms of the radionuclide.

I find part 'd' a bit confusing, but I need to look at it again.

Thanks for your spending time in this question and I will try to understand better for the question before doing calculation.

Ah, I think I understand part 'd'.

First of all, activity (A) is a product of specific activity (As) and mass M. Specific activity is simply activity A divided by the mass M.

Remember that activity is simply $\lambda$*N, where N is the number of radionuclide atoms. Well the total N has a mass M = N*m where m is the mass of the radionuclide atom. This relationship (M = N * m) would work for a pure (100%) mass of the radionuclide, but in this case, the radioactive nuclide is disolved into the piston head.

However, on is given an initial activity 5.7 x 107Bq and a piston mass of 1.6 kg, and assuming that the radionuclide is uniformly distributed, this gives a specific activity of 3.5625 x 107Bq/kg. Assuming the mass of the piston to be constant, the specific activity decreases as the same rate as the activity. However, we know that the piston is losing mass to wear! On the other hand, the wear rate is very small, m(loss) << 1.6 kg. The activity of lost material at anytime is 620 Bq compared to the initial activity of 5.7 x 107Bq.

Now the problem states "During testing, any worn-off metal is found to have an activity of 620 Bq," i.e. the activity of the lost metal is constant during the test. But how can this be if the activity (and specific activity) is decreasing with time? Remember A = As * M or expressed as a function of time, A = As(t) * M(t). In order for A to be constant, M(t) must have a form such that the rate of loss of M (conversely, the rate of accumulation of metal lost) must offset the rate of decrease of the specific activity.

The total metal loss, is simply the integral over time (with appropriate limits 30-90 days) of the metal loss rate, i.e.
m(loss) = $\int_{30}^{90} \dot{m} dt$.

So to solve part 'd', one must determine the expression for the rate of mass loss.

Thanks for the intereseting problem. This represents a practical application of a tracer isotope, although the constant activity during the test is not realistic - hopefully the wear of a real piston does not increase exponentially.

Thanks for your help ,Astronuc for your detail explanation that make me understand better with the question and I will try to solve the question by myself after given the 'important clue' to this question.

## Related to Question regarding Radioactive

Radioactivity is the process by which an unstable atom releases energy in the form of particles or waves. This process is known as radioactive decay and it can occur in natural elements or in man-made isotopes.

Radioactivity is measured in units of becquerels (Bq) or curies (Ci). These units represent the rate at which a radioactive material emits radiation. One becquerel is equal to one radioactive decay per second.

## What are the different types of radiation?

There are three main types of radiation: alpha particles, beta particles, and gamma rays. Alpha particles are positively charged and can be stopped by a sheet of paper. Beta particles are negatively charged and can be stopped by a few millimeters of aluminum. Gamma rays are neutral and require several centimeters of lead or concrete to be stopped.

## What are the dangers of exposure to radioactivity?

Exposure to high levels of radioactivity can be harmful to living organisms. It can cause damage to cells and DNA, leading to various health issues such as cancer. However, low levels of exposure are generally considered safe and are even used in medical treatments.

## How is radioactivity used in everyday life?

Radioactivity has a variety of uses in everyday life. It is used in medical treatments such as cancer therapy and diagnostic imaging. It is also used in smoke detectors, power plants, and food preservation. Radioactive isotopes are also used in research and scientific experiments.

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