Problem 76 - A Complete Resource Book in Chemistry for JEE Main

Question

Half-life of radioactive \({ }^{14} \mathrm{C}\) is 5760 years. In how many years, \(200 \mathrm{mg}\) of \({ }^{14} \mathrm{C}\) will be reduced to \(25 \mathrm{mg}\) ? (a) 5760 years (b) 11520 years (c) 17280 years (d) 23040 years

Step-by-Step Solution

Answer

It will take 17280 years for 200 mg of \(^{14}C\) to reduce to 25 mg.

Step 1: Understanding the Half-Life Formula

The half-life formula is essential for calculating the time required for a radioactive substance to decay to a certain amount. The formula is \( N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \), where \( N \) is the remaining quantity, \( N_0 \) is the initial quantity, \( T_{1/2} \) is the half-life period, and \( t \) is the time elapsed.

Step 2: Identifying Given and Required Values

We are given the initial quantity \( N_0 = 200 \) mg, the remaining quantity \( N = 25 \) mg, and the half-life \( T_{1/2} = 5760 \) years. We need to find the time \( t \) it takes for the substance to reduce to 25 mg.

Step 3: Setting Up the Equation

Using the half-life formula: \( 25 = 200 \left( \frac{1}{2} \right)^{\frac{t}{5760}} \). We need to solve for \( t \).

Step 4: Solving for the Power of 1/2

Rearranging the equation gives us: \( \left( \frac{1}{2} \right)^{\frac{t}{5760}} = \frac{25}{200} = \frac{1}{8} \). This implies\( \left( \frac{1}{2} \right)^3 = \frac{1}{8} \).

Step 5: Equating the Exponents

Since the powers of \( \frac{1}{2} \) are equal, we can equate the exponents: \( \frac{t}{5760} = 3 \).

Step 6: Calculating the Time

Solving for \( t \) gives \( t = 5760 \times 3 = 17280 \) years.

Step 7: Final Answer

The time it takes for the substance to decay from 200 mg to 25 mg is \( 17280 \) years.

Key Concepts

Half-Life Calculation

Calculating the half-life of a radioactive substance is crucial in understanding how long it takes for it to decay to a certain amount. Half-life is the time period required for half of the radioactive atoms in a sample to decay. To calculate the time for a quantity to decay to a specific amount, you need the initial amount, the remaining amount, and the half-life of the substance.
Here's how you approach it:

• Identify the initial amount (0) and the remaining amount (1).


• Use the half-life formula: \[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] where \(N\) is the remaining amount, \(N_0\) is the initial amount, \(T_{1/2}\) is the half-life, and \(t\) is the elapsed time.


• Rearrange the formula to solve for time \(t\) when \(N\) and \(N_0\) are known. This involves equating the exponents and solving for \(t\).

By taking these steps, you can determine how long it will take for the given substance to decay to a desired level.

Carbon-14 Decay

Carbon-14 is a naturally occurring radioactive isotope of carbon, which is used extensively in radiocarbon dating. This method is commonly used to date historical artifacts and archaeological findings.
Carbon-14 decays at a consistent rate, with a half-life of approximately 5760 years. This means every 5760 years, half of the carbon-14 in a sample will have decayed. The stable decay rate allows scientists to determine the age of objects by measuring the remaining carbon-14 content.
Some key points about carbon-14 decay include:

• It is produced in the atmosphere and is absorbed by living organisms.


• When the organism dies, it stops absorbing carbon-14, and the existing carbon-14 starts to decay.


• The decay follows the same exponential decay pattern as other radioactive isotopes.

By understanding the nature of carbon-14 decay, we can apply the half-life formula to determine how many years it will take for the isotope to decay from a given initial amount to a smaller, specified amount.

Half-Life Formula

The half-life formula is a powerful tool in the study of radioactive decay as it describes how quantities diminish over time.
This formula is given by:\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]Here, the formula components are:

• \(N\) = remaining quantity after time \(t\)


• \(N_0\) = initial quantity


• \(T_{1/2}\) = half-life period, the time taken for the quantity to reduce to half


• \(t\) = total time elapsed

To use the formula effectively, you need to solve for one variable if the others are known. For instance, if the remaining and initial quantities along with the half-life are given, you can calculate the time \(t\) it takes to decay using logarithms to solve the equation. This makes the half-life formula a fundamental equation for scientists and researchers working in fields like archaeology, geology, and nuclear physics.