Problem 101 - A Complete Resource Book in Chemistry for JEE Main

Question

Half-life period of the radioactive element \(X\) is 10 hours. Amount of \(X\) left in the 11 th hour starting with one \(m o l X\) is (a) \((1 / 2)^{1 / 10}\) (b) \((1 / 2)^{11 / 10}\) (c) \((1 / 2)^{12 / 11}\) (d) \((1 / 2)^{1 / 11}\)

Step-by-Step Solution

Answer

The amount of \(X\) left in the 11th hour is \((1 / 2)^{11 / 10}\).

Step 1: Understanding the Half-Life Concept

The half-life of a radioactive element is the time it takes for half of the radioactive nuclei to decay. Here, the half-life of element \(X\) is 10 hours, meaning every 10 hours, half of the substance will decay.

Step 2: Determine the Formula to Use

We use the formula for decay: \(N = N_0 \left( \frac{1}{2} \right)^{t/T}\), where \(N\) is the remaining quantity, \(N_0\) is the initial quantity, \(t\) is the time elapsed, and \(T\) is the half-life period.

Step 3: Assign Values and Substitute

Given the problem, we have \(N_0 = 1\) mol, \(t = 11\) hours, and \(T = 10\) hours. Substitute these into the formula: \(N = 1 \cdot \left( \frac{1}{2} \right)^{11/10}\).

Step 4: Simplify the Expression

Simplifying the expression \(N = \left( \frac{1}{2} \right)^{11/10}\), we find \(N = \left( \frac{1}{2} \right)^{11/10}\).

Step 5: Select the Correct Answer Choice

Comparing our calculation \(\left( \frac{1}{2} \right)^{11/10}\) with the options given, the correct answer is option (b): \((1 / 2)^{11 / 10}\).

Key Concepts

Understanding Radioactive Decay

Radioactive decay is a natural process where an unstable atomic nucleus loses energy by emitting radiation. This process can transform an element into a different element or isotope over time.

Each radioactive isotope has a characteristic rate at which it decays. Key things to remember about radioactive decay include:

• Radioactive decay is spontaneous and occurs without external influence.
• It results in the production of particles such as alpha, beta particles, or gamma rays.
• This decay is predictable on a statistical level yet random in individual events.

The rate of decay is often expressed through the concept of half-life, which relates directly to how long it takes for half the radioactive atoms in a sample to decay.

Exponential Decay Formula

In radioactive decay, the amount of material decreases exponentially over time. The decay is described by an exponential decay formula. This formula helps us calculate the remaining quantity of a material after a given time.

The formula is:

• \[N = N_0 \left( \frac{1}{2} \right)^{t/T}\]

Where:

• \(N\) is the remaining amount of the substance.
• \(N_0\) is the initial amount of the substance.
• \(t\) represents the time that has passed.
• \(T\) is the half-life, the time it takes for half of the material to decay.

This formula allows calculations across different intervals, maintaining the relative decrease description provided by the half-life period. The exponential nature of the formula reflects the rate proportionality to the present amount of substance.

Half-Life of Radioactive Elements

The half-life of a radioactive element is an essential concept in understanding how quickly a sample decays. Calculating the time it takes for half of the initial sample to transform into a different substance, half-life offers a time frame for decay processes.

Characteristics of half-life include:

• It is specific to each radioactive element or isotope and remains constant.
• Half-life is an indirect measure of the stability of a nucleus; stable isotopes have long half-lives, while unstable ones have short half-lives.
• It serves as a basis for techniques like radiocarbon dating and nuclear medicine.

In calculations, knowing the half-life allows us to determine how much of a material will remain after any given time. In our exercise, we are given a half-life of 10 hours and used this to determine the remaining amount of substance after 11 hours, showcasing how half-life integrates into practical calculations.