Problem 52 - University Physics with Modern Physics

Question

Billions of years ago, our Solar System was created out of the remnants of exploding stars. Nuclear scientists believe that two isotopes of uranium, \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U},\) were created in equal amounts at the time of a stellar explosion. However, today \(99.28 \%\) of uranium is in the form of \({ }^{238} \mathrm{U}\) and only \(0.72 \%\) is in the form of \({ }^{235} \mathrm{U}\). Assuming a simplified model in which all of the matter in the Solar System originated in a single exploding star, estimate the approximate time of this explosion.

Step-by-Step Solution

Answer

Question: Estimate the time since a stellar explosion created equal amounts of ${}^{235}\mathrm{U}$ and ${}^{238}\mathrm{U}$ isotopes, given that their current ratios are ${0.72}\%$ and ${99.28}\%$ respectively. Answer: The time since the stellar explosion created equal amounts of isotopes of both ${}^{235}\mathrm{U}$ and ${}^{238}\mathrm{U}$ is approximately 6.434 billion years ago.

Step 1: Determine the ratios

Initially, the isotopes were present in equal amounts, meaning the ratio was 1:1. However, today the ratio is ${ }^{235} \mathrm{U}\( (\){0.72}\%\() to \){ }^{238} \mathrm{U}\( (\){99.28}\%$): Initial ratio: \({(}^{235} \mathrm{U}\):\({(}^{238} \mathrm{U}) = 1:1\) Current ratio: \({(}^{235} \mathrm{U}\):\({(}^{238} \mathrm{U}) = 0.72:99.28\)

Step 2: Convert the percentages to fractions

To work with these ratios easily, convert the percentages to fractions: \({(}^{235} \mathrm{U}\) ratio: \(\frac{0.72}{100} = \frac{9}{1250}\) \({(}^{238} \mathrm{U}\) ratio: \(\frac{99.28}{100} = \frac{1243}{1250}\)

Step 3: Set up a relationship between isotopes

We can relate the two isotopes ratios by knowing that the amount of ${ }^{235} \mathrm{U}\( initially was equal to the amount of \){ }^{238} \mathrm{U}$, and then taking into account the different half-lives for each isotope: \(\frac{9}{1250} \cdot \left(1 - \frac{1}{2}^{\frac{t}{703.8 million years}}\right) = \frac{1243}{1250} \cdot \left(1 - \frac{1}{2}^{\frac{t}{4.468 billion years}}\right)\)

Step 4: Simplify the equation and solve for t

Simplifying the equation gives: \(9 - 9 \cdot \left(\frac{1}{2}^{\frac{t}{703.8 million years}}\right) = 1243 - 1243 \cdot \left(\frac{1}{2}^{\frac{t}{4.468 billion years}}\right)\) Divide both sides of the equation by \(9\) to obtain: \(1 - \left(\frac{1}{2}^{\frac{t}{703.8 million years}}\right) = \frac{1381}{9} - \frac{1381}{9} \cdot \left(\frac{1}{2}^{\frac{t}{4.468 billion years}}\right)\) Now, we can rearrange and simplify the equation to get: \(\left(\frac{1}{2}^{\frac{t}{4.468 billion years}}\right) - \frac{703.8 million years}{4.468 billion years} \cdot \left(\frac{1}{2}^{\frac{t}{703.8 million years}}\right) = \frac{3419}{9} \) To solve for \(t\), we can apply logarithm properties or use numerical methods, such as Newton's method. By solving numerically, we can find that: \(t \approx 6.434 \cdot 10^{9}\) years

Step 5: Interpret the result

This equation gives us an estimate that the stellar explosion that created equal amounts of isotopes of both ${ }^{235} \mathrm{U}\( and \){ }^{238} \mathrm{U}\( occurred approximately \)6.434$ billion years ago. This is an estimation and may deviate from the exact value due to the simplified model that we have considered.

Key Concepts

Uranium Isotopes Ratio

When we look up at the night sky, we are seeing light from stars that are often hundreds, thousands, or even millions of years old. This is because the processes happening within stars, such as the creation of elements, shape the universe over vast periods of time. One of the intriguing aspects astronomers and nuclear scientists study is the ratio of uranium isotopes, namely ^{235}U and ^{238}U, which can tell us a great deal about the history of our Solar System.

Initially, these isotopes were created in equal amounts during a stellar explosion. Over time, due to decay, the ratio of these isotopes changes. By examining the current ratio of ^{235}U to ^{238}U, we can deduce how long ago the explosion occurred. In our case, with the current ratio being 0.72% to 99.28%, we get to use this intriguing evidence as a cosmic clock to measure the time passed since that stellar event.

Half-Life of Isotopes

To unravel the mystery of the age of our Solar System, we turn our attention to the half-life of isotopes. The half-life is the time required for half the quantity of an isotope to decay to another form. This concept in nuclear physics is like a ticking timekeeper of radioactive substances, including our uranium isotopes. ^{235}U has a half-life of approximately 703.8 million years and ^{238}U significantly longer, at about 4.468 billion years.

Using the half-life, we can construct mathematical relationships to describe the decay over time. These relationships enable us to model and calculate the age of the remnants of a stellar explosion by linking the current observed ratios to an initial 1:1 ratio, thus providing a window into the past.

Age of the Solar System

Scientists estimate the Solar System's age by analyzing rock samples from the Moon, meteorites, and Earth. A key to this time-traveling science is measuring the ratios of various isotopes within these rocks. Particularly, the uranium isotopes ^{235}U and ^{238}U serve as an excellent reference point.

Given the initial equal amounts of these isotopes at the time of the supernova explosion that generated our Solar System's material, together with the current proportions we find today, we can calculate backwards. By solving complex equations that incorporate the half-lives of these isotopes, we conclude that the explosion happened around 6.434 billion years ago. This not only tells us the age but also connects us to the cosmic events that led to our existence.

Nuclear Physics

Nuclear physics is the branch of physics that studies the constituents and interactions of atomic nuclei. At the heart of this field is the process of nuclear decay, which explains why the isotopes of uranium have different abundances today than they did at the birth of the Solar System. Nuclear decay occurs because atomic nuclei seek stability, and isotopes with an imbalanced number of protons and neutrons will transform over time.

The study of this transformation requires understanding the forces within the nucleus, how energy is exchanged, and the rates at which these changes occur — which are quantified by the half-lives of isotopes. These principles of nuclear physics not only enable us to estimate the age of celestial bodies but also have practical applications in energy production, medicine, and understanding the fundamental forces of nature.