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 \({ }_{92} \mathrm{U}^{218}\) is still in existence on earth \(\left(\mathrm{t}_{h}\right.\) of \({ }_{92} \mathrm{U}^{238}=4.51 \times 10^{9}\) years \()\) ? (a) \(10 \%\) (b) \(20 \%\) (c) \(30 \%\) (d) \(40 \%\)

Step-by-Step Solution

Answer

The remaining fraction of \(^{238}U\) is about 21.3%, so the answer is (b) 20%.

Step 1: Identify the Formula for Radioactive Decay

Radioactive decay follows an exponential function. The amount of a radioactive substance remaining after a certain amount of time can be modeled by the formula:\[N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_h}}\]where:- \(N(t)\) is the amount remaining after time \(t\),- \(N_0\) is the initial amount,- \(t\) is the total elapsed time, and- \(t_h\) is the half-life of the substance.

Step 2: Calculate the Number of Half-Lives

First, we need to determine how many half-lives have passed in the assumed 10,000,000,000 years of the Earth's existence. The half-life \(t_h\) of \(^{238}\text{U}\) is given as \(4.51 \times 10^9\) years.Number of half-lives = \(\frac{t}{t_h} = \frac{10^{10}}{4.51 \times 10^9} \approx 2.22\) half-lives.

Step 3: Calculate the Remaining Fraction of the Original Substance

Using the formula from Step 1:\[N(t) = N_0 \left(\frac{1}{2}\right)^{2.22}\]Calculate:\[N(t) = N_0 \times 0.213\]Thus, approximately 21.3% of the original \(^{238}U\) would remain after 10 billion years.

Key Concepts

Half-life

The concept of a half-life is crucial in understanding radioactive decay and isotopes. A half-life is the time it takes for half of the radioactive atoms in a sample to decay. This does not mean all atoms decay at once but half of them do, on average, over this period.

For example, if a substance has a half-life of 100 years, then in 100 years, only 50% of that substance will remain. After another 100 years, 25% will be left, and so on. The half-life provides a predictable way to calculate the amount of radioactive material remaining at various points in time.

Important properties of half-life include:

• It is a constant for a given substance.
• It is independent of the initial amount of substance.
• It is unaffected by environmental changes such as temperature and pressure.

Knowing the half-life allows scientists to estimate the ages of objects, like rocks, and understand how quickly a radioactive material will diminish.

Exponential Decay

Exponential decay is a mathematical model used to describe how a quantity decreases over time. In radioactive decay, a substance decreases at a rate proportional to its current value. This leads to a fast initial decrease that slows over time.

The formula for exponential decay in radioactive materials is similar to other types of exponential models: \[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_h}} \]This formula helps to calculate the remaining quantity of a substance at any time.

Key characteristics of exponential decay include:

• The decay process never completely reaches zero, it halves indefinitely.
• It follows a continuous and smooth curve that approaches zero.
• Exponential decay is common in natural processes like population decline or cooling of objects.

Understanding exponential decay helps scientists to interpret and predict the behavior of radioactive substances over time.

Uranium Isotopes

Uranium isotopes are variants of uranium atoms, differing in the number of neutrons. Uranium-238 is one of the most common isotopes, responsible for a lot of Earth's natural radioactivity.

Isotopes have the same atomic number but different mass numbers. In uranium-238, "238" represents the atomic mass number (92 protons and 146 neutrons).

Key points about uranium-238 and its role in radioactive decay:

• It has a very long half-life of about 4.5 billion years.
• It is widely used in understanding geological dating, helping scientists date the age of the Earth.
• It undergoes radioactive decay, transforming eventually into lead-206, which is stable.

Uranium-238 plays a critical role in fields such as geology, archaeology, and nuclear energy, providing insights into the age of objects and energy production.