Question

If \(10^{30}\) atoms of a radioactive sample remain after 10 half-lives, how many atoms remain after 20 half-lives?

Short answer

Answer: Approximately \(9.77 \times 10^{28}\) atoms remain in the radioactive sample after 20 half-lives.

Step-by-step solution

Understand the concept of half-life

A half-life is the time it takes for half of a radioactive sample to decay. At the end of each half-life, the number of radioactive atoms remaining is halved.

Define the given values

After 10 half-lives, there are \(10^{30}\) atoms remaining. We will call this number R and the initial number of atoms as I. We are asked to find the number of atoms remaining after 20 half-lives.

Calculate the initial number of atoms

After 10 half-lives, the remaining atoms are given by the relation \(R = I(\frac{1}{2})^{10}\). To find the initial number of atoms, we can rearrange the equation to have I on one side: \(I = R(\frac{1}{2})^{-10} \). Plug the value of R into the formula. \(I = (10^{30})(\frac{1}{2})^{-10}\).

Calculate the remaining atoms after 20 half-lives

Now we need to find the remaining atoms after 20 half-lives. Again, we will use the same formula but now with an exponent of 20: \(R_{20} = I(\frac{1}{2})^{20}\). Replace I with the previously calculated expression. \(R_{20}=((10^{30})(\frac{1}{2})^{-10})(\frac{1}{2})^{20}\).

Simplify the expression

We can rewrite the expression as follows: \(R_{20} = 10^{30}(\frac{1}{2})^{-10}(\frac{1}{2})^{20}\). Further simplification gives: \(R_{20} = 10^{30}(\frac{1}{2})^{10}\).

Calculate the final answer

Now, compute the value of \(R_{20}\): \(R_{20} = 10^{30}(\frac{1}{2})^{10}\approx 9.77 \times 10^{28}\). So, after 20 half-lives, approximately \(9.77 \times 10^{28}\) atoms remain in the radioactive sample.

Key concepts

Radioactive Decay

Radioactive decay is a spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. This phenomenon is a cornerstone of nuclear physics and drives the constant change in the nucleus of atoms. During this process, a nucleus can emit particles such as alpha particles, beta particles, or gamma rays, which transforms it into a new element or a different isotope of the same element.

In our exercise scenario, billions of atoms are emitting particles simultaneously, which leads to the overall reduction in the number of radioactive atoms over time. It's essential to grasp that this decay process is random for each atom, which means we cannot predict exactly when a specific atom will decay. However, we can calculate the average behavior of a large sample of identical atoms using statistical methods.

Exponential Decay

Exponential decay is a concept that describes the process of decreasing at a rate that is proportional to its current value. Mathematically, this type of decay can be represented by an exponential function. In the context of radioactive decay, this implies that the quantity of radioactive atoms decreases exponentially over time.

The formula we use in our calculation demonstrates exponential decay: after each half-life, we apply a consistent factor that reduces the sample in half. That's why a plot of the number of radioactive atoms remaining versus time would show a characteristic steep decline that gradually flattens out, a hallmark of exponential functions.

Nuclear Physics

Nuclear physics is the branch of physics that studies the constituents and interactions of atomic nuclei. This field explores phenomena such as radioactive decay and nuclear reactions, which include the principles of fusion and fission, and applications such as nuclear power generation and medical imaging techniques.

The example in our exercise touches on nuclear physics by dealing with the number of atoms in a radioactive sample over time. Through such exercises, students gain insights into how nuclear physics governs the behavior of matter at its most fundamental level.

Half-Life Formula

The half-life formula is crucial in determining the time it takes for half of a given quantity of radioactive material to decay. Mathematically, it is represented as:
\[ R = I \left(\frac{1}{2}\right)^{t/T} \]
where \( R \) is the remaining quantity of the substance, \( I \) is the initial quantity, \( t \) is the elapsed time, and \( T \) is the half-life period. In our exercise, we calculate what happens after double the original amount of half-lives. By replacing \( t \) with 20 half-lives and \( T \) with 10 half-lives, we get to see how the initial amount of radioactive atoms diminishes significantly after 20 half-lives due to exponential decay.