Problem 105 - A Complete Resource Book in Chemistry for JEE Main

Question

Assuming the age of the earth to be \(10^{10}\) years, what fraction of the original amount of \({ }_{92} \mathrm{U}^{238}\) is still in existence on earth \(\left(\mathrm{t}_{1 / 2}\right.\) of \({ }_{92} \mathrm{U}^{238}=4.51 \times 10^{9}\) years \()\) ? (a) \(10 \%\) (b) \(20 \%\) (c) \(30 \%\) (d) \(40 \%\)

Step-by-Step Solution

Answer

(b) 20%

Step 1: Understand the Problem

We need to determine the fraction of the original amount of \( { }_{92} \mathrm{U}^{238} \) that remains after \( 10^{10} \) years, given that its half-life \( \left(\mathrm{t}_{1 / 2}\right) \) is \( 4.51 \times 10^9 \) years.

Step 2: Calculate Number of Half-Lives

Calculate how many half-lives have passed in \( 10^{10} \) years by dividing the age of the earth by the half-life of uranium-238: \[ \text{Number of half-lives} = \frac{10^{10}}{4.51 \times 10^9} \approx 2.21. \]

Step 3: Apply the Half-Life Formula

Use the formula for exponential decay to find the remaining fraction: \[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^{2.21}. \] Evaluate this using a calculator to get the precise decimal value.

Step 4: Calculate the Remaining Fraction

Using the result from Step 3, \( \left( \frac{1}{2} \right)^{2.21} \approx 0.203 \). This indicates that approximately 20.3% of the original amount of \( { }_{92} \mathrm{U}^{238} \) remains.

Step 5: Choose the Correct Answer

Among the given options, we find that 20% (option b) is closest to our calculated value of 20.3%. So, the answer is (b) 20%.

Key Concepts

Half-life calculation

Half-life is an essential concept in understanding radioactive decay. It represents the time it takes for half of the radioactive atoms in a sample to decay. Changes in the amount of a radioactive substance occur gradually, not all at once. Knowing the half-life, expressed in years, days, or even seconds, allows scientists to predict how much of a substance will remain over time.

Let's break it down with a simple example. Consider a sample of uranium-238 ( ^{92}U^{238} ), with a half-life of 4.51 billion years. If you start with a specific number of uranium-238 atoms, half will remain undecayed after 4.51 billion years. It's like cutting a cake in half repeatedly—you eat half, then half of half, and so on.

In our puzzle, we calculated how many half-lives fit into Earth's age of 10 billion years. This lets us know how often the uranium-238 atoms "halved" themselves. Using the formula:

• Number of half-lives = Total time elapsed / Half-life duration

With the Earth's age at 10^{10} years and the half-life at 4.51 billion years, we calculated around 2.21 half-lives. Each number representing halving gives us an idea of how many uranium-238 atoms remain.

Exponential decay formula

Radioactive decay is an exponential process, meaning it's based on the rate at which the number of remaining atoms decreases. The concept of exponential decay helps in predicting the behavior of a substance over time. It's a continuous and smooth process, unlike simple subtraction.

The exponential decay formula is crucial here: Remaining fraction = \( \left( \frac{1}{2} \right)^n \) where \(n\) is the number of half-lives. For our uranium-238 example, this formula tells us what's left after multiple half-lives. Plug in \(n = 2.21\) and you'll know the proportion of original uranium-238 atoms that remain.

The remaining fraction calculated, in our case, was approximately 0.203 after 2.21 half-lives. This means that only about 20.3% of the original uranium-238 atoms are still existing.

Using exponential decay mathematics allows scientists to understand and predict the future compositions of radioactive materials. It is an essential tool used extensively in fields like archaeology, geology, and even medicine.

Uranium-238 decay

Uranium-238 is one of the most well-known radioactive isotopes due to its long half-life of 4.51 billion years. This makes it particularly useful for dating geological formations and understanding the age of the Earth.

Over time, uranium-238 naturally decays into other elements through a series of decay steps, eventually becoming lead-206. This long process is a rich source of information for scientists tracking geological changes and the evolution of our planet.

In our original problem, uranium-238's decay is assessed over the Earth's assumed age of 10 billion years to determine how much uranium-238 remains. With the calculated result of 20.3% remaining, it reflects just how slow and steady this decay process truly is.

This slow decay rate also benefits various scientific fields; for instance, in studying minerals and even in fueling nuclear power plants. Understanding uranium-238's decay rhythm over billions of years opens windows into both the past and future of Earth.