Problem 91 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life period of a radioactive element is 140 days. After 560 days, one gram of the element will reduce to (a) \(1 / 2 \mathrm{~g}\) (b) \(1 / 4 \mathrm{~g}\) (c) \(1 / 8 \mathrm{~g}\) (d) \(1 / 16 \mathrm{~g}\)

Step-by-Step Solution

Answer

After 560 days, one gram of the element will reduce to \(1/16 \mathrm{~g}\). So, the answer is (d).

Step 1: Understand the Half-Life Concept

The half-life of a radioactive element is the time taken for half of the radioactive nuclei in a sample to decay. For this problem, the half-life is given as 140 days. This means every 140 days, the amount of the radioactive element reduces to half its previous quantity.

Step 2: Calculate Number of Half-Life Periods

To find out how many half-life periods have passed after 560 days, divide the total time by the half-life period: \[ N = \frac{560}{140} = 4. \] This means that four half-life periods have passed in 560 days.

Step 3: Applying the Half-Life Formula

When a substance undergoes half-life decay, the remaining quantity can be calculated using: \[ A = A_0 \left( \frac{1}{2} \right)^N, \] where \( A_0 = 1 \) gram (the initial quantity) and \( N = 4 \). Substituting these values gives: \[ A = 1 \times \left( \frac{1}{2} \right)^4. \]

Step 4: Simplify the Expression

Calculate the expression: \[ \left( \frac{1}{2} \right)^4 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16}. \] So, \( A = 1 \times \frac{1}{16} = \frac{1}{16} \).

Step 5: Conclude the Result

After simplifying, we find that the remaining quantity of the element after 560 days is \( \frac{1}{16} \) grams. Therefore, the correct answer is option (d) \( 1/16 \mathrm{~g} \).

Key Concepts

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This process transforms the unstable parent nucleus into a more stable daughter nucleus. The rate at which a substance undergoes radioactive decay is characterized by its decay constant.

• The decay rate is spontaneous and happens at an atomic level.
• Common decay modes include alpha decay, beta decay, and gamma decay.
• The process is independent of external factors like temperature and pressure.

Understanding radioactive decay is fundamental in nuclear chemistry and has applications in fields such as archaeology for carbon dating and medicine for radiation therapies.

Half-Life Formula

The half-life formula is crucial in calculating the remaining quantity of a radioactive substance after a certain time period. Half-life is defined as the time required for half of the radioactive nuclei in a sample to undergo decay. It is a constant for any given isotope.

The half-life formula is given by:\[A = A_0 \left( \frac{1}{2} \right)^N,\]where:

• \( A \) is the remaining quantity of the substance,
• \( A_0 \) is the initial quantity,
• \( N \) is the number of half-life periods that have elapsed.

The half-life enables scientists to predict how long a radioactive substance will remain active and is integral in the fields of nuclear chemistry and radiometric dating.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. This occurs in many natural processes, including radioactive decay.

In the context of radioactivity, the exponential decay equation is represented as:\[A(t) = A_0 e^{-\lambda t},\]where:

• \( A(t) \) is the amount remaining at time \( t \),
• \( A_0 \) is the initial amount,
• \( \lambda \) is the decay constant,
• \( e \) is the base of the natural logarithm.

Exponential decay is characterized by a continuous, smooth decrease over time, crucial for modeling decay in nuclear chemistry and other fields.

Nuclear Chemistry

Nuclear chemistry focuses on the reactions and processes involving atomic nuclei. It covers a wide range of topics including radioactive decay, fission, and fusion.

• Radioactivity is a key concept, referring to the spontaneous emission of particles or energy from an unstable nucleus.
• Nuclear transmutation changes one element into another and is central to nuclear reactions.
• Applications of nuclear chemistry include energy production in nuclear reactors and the use of radioisotopes in medical diagnostics.

Nuclear chemistry provides powerful tools and insights into the atomic world, facilitating advancements in technology, medicine, and energy.