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 \(200 \mathrm{~g}\).

Step 1: Understanding Half-Life

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

Step 2: Total Time Elapsed

Since we need to determine how many complete half-lives have passed in 18 hours, we divide 18 by the 3-hour half-life: \[ \frac{18}{3} = 6 \] Therefore, there have been 6 half-lives.

Step 3: Calculating Remaining Mass after Each Half-Life

Each half-life reduces the mass of the isotope by half. Hence, if the mass after 6 half-lives is 3.125 g, the initial mass can be found by iteratively doubling 3.125 g six times: 1. Doubling once: \(3.125 \times 2 = 6.25 \) g.2. Doubling twice: \(6.25 \times 2 = 12.5 \) g.3. Doubling thrice: \(12.5 \times 2 = 25 \) g.4. By fourth doubling: \(25 \times 2 = 50 \) g.5. Fifth doubling: \(50 \times 2 = 100 \) g.6. Sixth doubling: \(100 \times 2 = 200 \) g.

Step 4: Finding the Initial Mass

Following the calculation, the initial mass of the radio-isotope before any decay, 6 doublings back, can be found as 200 g.

Key Concepts

Half-Life

Half-life is a fundamental concept in nuclear chemistry, particularly when studying radioactive decay. It refers to the time required for half of a radioactive substance to decay into another element or isotope. This means that for any given half-life period, the amount of the original substance present will decrease by 50%.

• If you start with 100 grams, after one half-life, you will have 50 grams remaining.
• After another half-life, 25 grams will remain, and so on.

In the context of the given exercise, the half-life of the radio-isotope is 3 hours. Thus, to determine how many half-lives have passed over 18 hours, you divide 18 by 3, resulting in 6 half-lives. This iterative halving of the amount is reversed to compute the initial mass when the final mass is known, like following a breadcrumb trail back to the original source.

Radio-Isotopes

Radio-isotopes are specific types of isotopes of an element that have unstable nuclei and emit radiation as they decay to a more stable form. These isotopes are integral to numerous applications, due to their radioactive properties.

• In medicine, they are used for imaging, treatment, and diagnostic procedures.
• In environmental science, radio-isotopes help track and date ancient materials.
• In industry, they are often used for inspecting welds and structural integrity.

Understanding the decay of radio-isotopes includes knowing their half-lives, making it crucial for managing their use safely and effectively. Each radio-isotope's decay is predictable when its half-life is known, allowing for precise calculations necessary in many scientific and industrial applications.

Nuclear Chemistry

Nuclear chemistry focuses on the reactions and processes involving atomic nuclei and radioactivity. It's a specialized field that addresses the behavior of nuclear reactions and their applications. This discipline includes the study of nuclear decay processes like those seen with radio-isotopes.

Nuclear reactions release or absorb much larger amounts of energy than typical chemical reactions, because they involve the forces holding the nucleus together. Understanding nuclear chemistry is essential for harnessing the power of nuclear energy and managing nuclear materials safely.

• Key areas of nuclear chemistry include nuclear decay, fission, fusion, and radiation.
• Applications range from generating power and developing new medical treatments to influencing modern technology and industry.

The study of half-life and the behavior of radio-isotopes falls under the umbrella of nuclear chemistry, providing essential insights into how we can safely and effectively use nuclear materials. By mastering these concepts, scientists and engineers ensure the responsible and innovative use of nuclear technology.