Problem 93 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a radio-isotope is three hours. If the mass of the undecayed isotope at the end of 18 hours is \(3.125 \mathrm{~g}\), what was its mass initially? (a) \(300 \mathrm{~g}\) (b) \(200 \mathrm{~g}\) (c) \(180 \mathrm{~g}\) (d) \(400 \mathrm{~g}\)

Step-by-Step Solution

Answer

The initial mass was 200 g.

Step 1: Understanding Half-Life

The half-life of a radio-isotope is the time required for half of the isotope to decay. In this exercise, the half-life is given as 3 hours.

Step 2: Calculate Number of Half-Lives

We need to determine the number of half-lives that have passed in 18 hours. Since each half-life is 3 hours, divide 18 by 3: \[ \text{Number of half-lives} = \frac{18}{3} = 6 \]

Step 3: Use the Formula for Decay

The formula for radioactive decay is given by: \[ M = M_0 \left(\frac{1}{2}\right)^n \] where \( M \) is the remaining mass, \( M_0 \) is the initial mass, and \( n \) is the number of half-lives. We know \( M = 3.125 \) g and \( n = 6 \).

Step 4: Substitute Known Values and Solve for Initial Mass

Substituting in the known values, we have: \[ 3.125 = M_0 \left(\frac{1}{2}\right)^6 \] Calculate \( \left(\frac{1}{2}\right)^6 = \frac{1}{64} \). So, the equation becomes: \[ 3.125 = \frac{M_0}{64} \] Rearranging to solve for \( M_0 \): \[ M_0 = 3.125 \times 64 \]

Step 5: Calculate Initial Mass

Multiplying gives: \[ M_0 = 3.125 \times 64 = 200 \] So, the initial mass of the radio-isotope was 200 g.

Step 6: Validate Your Answer

Review the calculation: the remaining mass after 6 half-lives (18 hours) is \(3.125 \text{ g}\), and 6 half-lives from an original mass of \(200 \text{ g}\) checks out.

Key Concepts

Half-Life

Half-life is a fundamental concept in radioactive decay. It refers to the time it takes for half of a given amount of a radioactive isotope to decay. In simpler terms, if you start with a certain mass of a radioactive substance, after one half-life, only half of that mass will remain undecayed.

For example, if the half-life of a radio-isotope is 3 hours, and you start with 200g, after 3 hours, you would have 100g left. After 6 hours, you'd have 50g remaining, and so on. This exponential decay continues until the isotope is completely decayed.

• Important: Each element has a unique half-life. It must be measured separately for each radioisotope.

This rate of decay is consistent and measurable, which makes half-life a reliable method to measure the decay rate of isotopes.

Radio-Isotope

A radio-isotope, or radioactive isotope, is a variation of an element's atom that contains an unstable nucleus. This instability makes the isotope prone to radioactive decay, allowing it to emit radiation in the form of particles or electromagnetic waves.

Radio-isotopes can be naturally occurring or artificially produced in nuclear reactors or particle accelerators. They are used in a variety of applications, such as medical imaging (e.g., PET scans) and treatments, as well as in carbon dating and nuclear power generation.

• Tip: Not all isotopes of an element are radioactive; some have stable nuclei.

It's essential to understand that the behavior of radio-isotopes in terms of decay can be predicted using mathematical models like the half-life and decay formula mentioned in this exercise.

Initial Mass Calculation

Calculating the initial mass of a decayed radio-isotope involves working backwards from the remaining mass after a known number of half-lives. To determine how much of an isotope was originally present, you need to know:

• The remaining mass of the isotope.
• The number of half-lives that have passed.

In the provided problem, you already have the remaining mass (3.125g) and calculated the number of half-lives (6). Thus, you can use the decay formula to determine the original mass: \[ M = M_0 \left(\frac{1}{2}\right)^n \]where:\( M \) = final mass, \( M_0 \) = initial mass, and \( n \) = number of half-lives elapsed.Once you substitute the values and rearrange the equation, you solve for the initial mass \( M_0 \). This backward calculation is critical for understanding how much of the radio-isotope was present originally.

Decay Formula

The decay formula is a mathematical expression that describes how the mass of a radioactive isotope decreases over time. This formula ties together the half-life concept and the current mass of the isotope to find the initial mass or predict future masses.

• The formula is:
  \[ M = M_0 \left(\frac{1}{2}\right)^n \]

- \( M \) stands for the remaining mass of the isotope.
- \( M_0 \) represents the initial mass.
- \( n \) indicates the number of half-lives that have occurred.

Using this equation, you can find the mass of a radioactive isotope at any point in time, provided you know its half-life and either its initial or remaining mass. In the exercise example, by knowing the number of half-lives and the remaining mass, the initial mass could be deduced.