Animated Solution for Physics - Atoms and Nuclei: The activity of a freshly prepared radioactive sample is $10^{10}$ disintegrations per second, whose mean life is $10^9$ s. The mass of an atom of this radioisotope is $10^{-25}$ kg. The mass (in mg) of the radioactive sample is

Imagine you are holding a freshly prepared radioactive sample. It is not just sitting there; it is a bustling microscopic metropolis. Every single second, billions of unstable nuclei are transforming, emitting radiation in the process. This rate of transformation is what physicists call Activity.

In our specific problem, we are given a sample with an immense activity of $10^{10}$ disintegrations per second. We are also given the mean life of the radioisotope, which is $10^9$ seconds, and the mass of a single atom, which is a minuscule $10^{-25}$ kg. Our ultimate quest is to determine the total mass of this radioactive sample in milligrams.

Activity and Mean Life

To bridge the gap between the activity we observe and the number of atoms actually present, we rely on the fundamental law of radioactive decay. The activity $A$ is directly proportional to the total number of active nuclei $N$. The constant of proportionality is the decay constant $\lambda$:

But we aren't given the decay constant directly; we are given the mean life $\tau$. Fortunately, the relationship between the two is beautifully simple. The decay constant is exactly the reciprocal of the mean life:

By substituting this relationship into our activity equation, we get a highly useful formula that directly connects our knowns and unknowns:

Our immediate goal is to find $N$, the total number of atoms. By rearranging the equation we just derived, we can isolate $N$:

Now, we simply substitute the values provided in the problem. The activity $A$ is $10^{10}{\text{ s}}^{-1}$, and the mean life $\tau$ is $10^9\text{ s}$:

When multiplying numbers with the same base, we just add their exponents. Ten plus nine gives us nineteen. Therefore, the total number of active atoms in our sample is:

We now know exactly how many atoms are in our sample. To find the total mass $M$, we just need to multiply the number of atoms $N$ by the mass of a single atom $m$:

We know $N=10^{19}$, and the problem states that $m=10^{-25}\text{ kg}$. Let's plug these in:

Once again, we add the exponents. Nineteen plus negative twenty-five yields negative six. Thus, the total mass of the sample in kilograms is:

We have the mass, but the question specifically asks for the answer in milligrams (mg). This requires a quick unit conversion.

We know that $1\text{ kg}=10^3\text{ g}$, and $1\text{ g}=10^3\text{ mg}$. Therefore, $1\text{ kg}=10^6\text{ mg}$. To convert our mass from kilograms to milligrams, we multiply by $10^6$:

Adding the exponents negative six and positive six gives us zero, and $10^0$ is exactly $1$.

And there we have it! Despite the astronomical number of atoms and the intense activity, the total mass of the radioactive sample is a mere $1\text{ mg}$.