Question

The radioactive decay of \(\mathrm{T} 1-206\) to \(\mathrm{Pb}-206\) has a half- life of 4.20 min. Starting with \(5.00 \times 10^{22}\) atoms of \(\mathrm{Tl}-206,\) calculate the number of such atoms left after 42.0 min.

Short answer

After 42.0 min, there are about \(4.88 \times 10^{17}\) atoms of Thallium-206 left.

Step-by-step solution

Determine the number of half-lives

First, let's calculate the number of half-lives that pass in 42 min. Given that each half-life is 4.20 min, the number of half-lives is 42.0 min divided by 4.20 min/half-life, which equals 10 half-lives.

Apply the decay formula

We know that after each half-life, half of the substance remains. Therefore, after 10 half-lives, the fraction of Thallium-206 that remains is \( (1/2)^{10} \).

Calculate the remaining amount

To find the remaining amount of Thallium-206, we multiply the starting amount by the fraction that remains: \(5.00 \times 10^{22} \times (1/2)^{10}\).

Key concepts

Half-Life

Understanding the concept of half-life is fundamental when studying the decay of radioactive materials. It is defined as the time taken for half the atoms of a radioactive substance to decay. It's important to note that every radioactive isotope has its own unique half-life, which can vary from fractions of a second to millions of years.

Using half-life, scientists can predict how much of a radioactive substance will remain after a given period. The calculation usually involves dividing the time elapsed by the half-life of the substance, which helps in determining the number of half-lives that have passed. As seen in the exercise solution, by knowing the number of half-lives, we can figure out the remaining amount of a radioactive element through a simple mathematical operation.

Exponential Decay

The process of exponential decay describes how the quantity of a radioactive substance decreases over time. This decay follows an exponential function, signifying that the rate of decay is proportional to the current amount of the substance. Mathematically, exponential decay is represented by the formula \( N(t) = N_0 \times (1/2)^{t/T} \) where \( N(t) \) is the quantity at time \( t \) , \( N_0 \) is the initial quantity, and \( T \) is the half-life.

To visualize how rapidly the substance diminishes, you might imagine a graph where the y-axis represents the remaining quantity and the x-axis represents time. The curve steeply drops at first and then gradually flattens, but it never reaches zero. This characteristic curve demonstrates the fascinating property of exponential decay in radioactive substances.

Nuclear Chemistry

The field of nuclear chemistry is focused on the reactions and transformations of atomic nuclei. This includes understanding how unstable atoms undergo radioactive decay. The nucleus of an atom holds protons and neutrons, and when it has an excess of either, it can become unstable. To reach stability, it might emit particles like alpha particles (helium nuclei), beta particles (electrons), or gamma rays (high-energy photons), over a series of decays, until it forms a stable nucleus.

Radioactive decay is a random process at the level of single atoms, meaning that the decay of one atom does not affect that of another. However, nuclear chemistry allows us to predict the behavior of a large number of atoms using statistical methods, such as those used in half-life and exponential decay calculations to elucidate the predictability of the decay process on a large scale.