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 1 1th hour starting with one \(\operatorname{mol} \mathrm{X}\) 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

(b) \((1 / 2)^{11 / 10}\)

Step 1: Understand the Half-Life Concept

The half-life of a radioactive element is the time required for half of the radioactive atoms in a sample to decay. In this case, the element X has a half-life of 10 hours, meaning that every 10 hours, half of the original amount of X remains.

Step 2: Apply the Half-Life Formula

The formula to calculate the remaining quantity of a radioactive element is \( N = N_0 \left( \frac{1}{2} \right)^{t/T} \), where \( N_0 \) is the initial quantity, \( t \) is the time elapsed, \( T \) is the half-life, and \( N \) is the remaining amount.

Step 3: Calculate the Remaining Amount after 11 Hours

Here, \( N_0 = 1 \) mol, \( t = 11 \) hours, and \( T = 10 \) hours. Substitute these values into the formula: \( N = 1 \left( \frac{1}{2} \right)^{11/10} \). This simplifies to \( N = \left( \frac{1}{2} \right)^{11/10} \).

Step 4: Determine the Correct Answer Option

Check the options given: (a) \((1 / 2)^{1 / 10}\)(b) \((1 / 2)^{11 / 10}\)(c) \((1 / 2)^{12 / 11}\)(d) \((1 / 2)^{1 / 11}\).The correct answer that matches our calculated expression is option (b).

Key Concepts

Radioactive Decay

Radioactive decay is a process by which an unstable atomic nucleus loses energy by emitting radiation. During this transformation, the nucleus of a radioactive element decays into a different element or a different isotope, releasing particles such as alpha particles, beta particles, or gamma rays. This release of particles leads to a change in the atomic structure of the radioactive substance.

A few key points about radioactive decay include:

• It is a random process at the level of single atoms, meaning it is impossible to predict when a particular atom will decay.
• However, with a large number of atoms, decay rates become predictable and are described by the half-life.
• Radioactive decay follows first-order kinetics, where the rate of decay is directly proportional to the number of undecayed nuclei present.

Understanding radioactive decay is crucial when studying elements with unstable nuclei, as it dictates how quickly a material will break down over time.

Half-Life Formula

The concept of a half-life is essential when discussing radioactive elements. It represents the time taken for half of a given quantity of a radioactive substance to decay. This period of time is consistent irrespective of the quantity present. The half-life provides a measure of the rate at which a radioactive element disintegrates.

For calculations involving half-life, scientists use the half-life formula:

• \[N = N_0 \left( \frac{1}{2} \right)^{t/T}\]
• Here, \(N\) is the remaining amount of the substance, \(N_0\) is the initial quantity, \(t\) is the time that has passed, and \(T\) is the half-life of the substance.

This formula allows us to compute how much of a radioactive material will remain after a certain amount of time has elapsed. The exponent \( t/T \) signifies how many half-lives have passed in the given time period. Employing the half-life formula can assist in predicting the behavior and transformation of substances over time.

Remaining Amount Calculation

Calculating the remaining amount of a radioactive substance after a certain period involves using the half-life formula. This is crucial for predicting how much of a substance will be left after time has passed. Let's break it down.

Suppose we start with an initial amount of 1 mol of element \(X\), as in the original exercise. If \(X\) has a half-life of 10 hours, we want to know how much of \(X\) is left after 11 hours.

• The formula given is:\[N = N_0 \left( \frac{1}{2} \right)^{t/T}\]
• Substitute \(N_0 = 1\) mol, \(t = 11\) hours, and \(T = 10\) hours:\[N = 1 \times \left( \frac{1}{2} \right)^{11/10}\]
• This calculation shows that the remaining amount of \(X\) is \[ \left( \frac{1}{2} \right)^{11/10} \] mol after 11 hours.

This approach allows us to answer the original exercise by recognizing which multiple-choice option corresponds to our calculated value, ensuring that we understand both the process of decay and the precise remaining quantity.