Question

In close analog to the half-lives of \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\), let's say two 80 elements have half lives of \(4.5\) billion years and 750 million years. If we start out having the same number of each (1:1 ratio), what will the ratio be after \(4.5\) billion years? Express as \(x: 1\), where \(x\) is the larger of the two.

Short answer

Answer: The ratio between the remaining amounts of the two elements after 4.5 billion years is 1:32.

Step-by-step solution

Determine the number of half-lives

First, we should determine how many half-lives have passed for each element after 4.5 billion years. To find this, we will divide the total elapsed time by each element's half-life. For the first element (half-life of 4.5 billion years): Number of half-lives = 4.5 billion years / 4.5 billion years = 1 For the second element (half-life of 750 million years): Number of half-lives = 4.5 billion years / 750 million years = 6 So after 4.5 billion years, 1 half-life has passed for the first element, and 6 half-lives have passed for the second element.

Calculate the remaining amounts of each element

Now we need to calculate how much of each element remains after the respective half-lives. For the first element: After 1 half-life, the remaining amount is (1/2) * 1, since the initial ratio of the first element is 1. For the second element: After 6 half-lives, the remaining amount is (1/2)^6 * 1, since the initial ratio of the second element is 1.

Find the ratio

Now we need to find the ratio between the remaining amounts of the elements. Remaining amount of the first element = (1/2) * 1 = 1/2 Remaining amount of the second element = (1/2)^6 * 1 = 1/64 Since the ratio must be expressed in terms of the larger quantity to 1: Ratio = (1/64) : (1/2) The required ratio can be found by multiplying both sides of the ratio by 64: x:1 = 1:32 So the ratio between the remaining amounts of the two 80 elements after 4.5 billion years is 1:32.

Key concepts

Radioactive Decay

Radioactive decay is a fundamental concept in nuclear chemistry. It describes the process where unstable atomic nuclei lose energy by emitting radiation. This often results in a change into a different element or a different isotope of the same element.
Each radioactive element has a characteristic average decay time known as the half-life. This is the time required for half of the radioactive atoms in a sample to decay.
Understanding half-life is crucial for calculations related to radioactive decay:

• A longer half-life means the element is more stable and decays slower.
• A shorter half-life indicates quicker decay, releasing its radioactive particles faster.

In the problem we explored, elements with different half-lives decay at varying rates over time, leading to changes in their relative quantities.

Isotopes

Isotopes are variations of the same chemical element that have the same number of protons but different numbers of neutrons. This difference in neutron number gives isotopes distinct masses.
The mass number of an isotope is the sum of its protons and neutrons.
Isotopes can be stable or unstable. Unstable isotopes are radioactive and decay over time, potentially transforming into another element or isotope.

• Stable isotopes do not change over time, maintaining their form and not emitting radiation.
• Unstable isotopes possess excess energy that makes them radioactive, leading to decay.

In our calculation, understanding isotopes is important because each isotope's decay and resulting ratio depend on its specific properties, including half-life.

Nuclear Chemistry

Nuclear chemistry is a branch of chemistry that deals with the reactions, behavior, and properties of atomic nuclei. At its core, it explores how nuclear reactions can transform elements and isotopes, including their decay processes.
Several key concepts in nuclear chemistry are vital for understanding half-life calculations:

• Nuclear reactions involve changes in an atom's nucleus and usually result in the transformation of one element into another.
• Radioactive decay is a natural nuclear reaction where unstable nuclei lose energy by emitting radiation.
• Fission and fusion are processes that release energy by splitting or combining nuclei.

In the exercise, the principles of nuclear chemistry were used to predict how the ratio of two elements changes over time due to their decay.