Problem 95 - A Complete Resource Book in Chemistry for JEE Main

Question

If \(5 \mathrm{~g}\) of a radioactive substance has \(\mathrm{t}_{1 / 2}=14 \mathrm{hr}, 10 \mathrm{~g}\) of the same substance will have a \(t_{1 / 2}\) equal to (a) 14 hours (b) 28 hours (c) 50 hours (d) 70 hours

Step-by-Step Solution

Answer

The correct answer is (a) 14 hours because the half-life is independent of the substance's mass.

Step 1: Understanding Half-life

The half-life (\(t_{1/2}\) ) of a substance is the time required for a quantity to fall to half its initial value due to decay. One important characteristic is that the half-life is independent of the initial amount of the substance.

Step 2: Applying Half-life Knowledge

Given that the half-life of the substance is 14 hours for 5 grams, we need to determine whether this changes when the mass is increased to 10 grams. Since half-life is an intrinsic property of the material and does not depend on the quantity, this value remains constant.

Step 3: Selecting the Correct Answer

Since the half-life remains constant irrespective of the initial mass, the half-life for 10 grams is the same as for 5 grams. Therefore, it is still 14 hours, which corresponds to option (a).

Key Concepts

Half-life

The half-life of a substance is a fascinating concept that helps us understand how certain materials degrade or lose their activity over time. Imagine a clock ticking on a piece of radioactive material, slowly losing its mass due to decay. Half-life is the period it takes for half of this material to disintegrate. It is a constant, unique property for each substance and is usually denoted by \( t_{1/2} \). Regardless of whether you start with 1 gram or 100 grams, the half-life remains the same.

**Example:** If you start with 10 grams of a substance with a half-life of 14 hours, after 14 hours, 5 grams will remain. After another 14 hours, 2.5 grams will remain, and so on.

• Half-life = Time for half of the substance to decay
• Constant over time
• Independent of initial quantity

Understanding half-life is crucial for fields like archaeology, medicine, and nuclear physics, where knowing how quickly a substance decays can have practical applications.

Radioactive Substance

A radioactive substance is a material that contains unstable atoms which release energy in the form of radiation as they decay into more stable forms. This decay process is natural and occurs spontaneously. What makes a substance "radioactive" is its ability to emit radiation over time as it loses its unstable atomic particles.

**Characteristics of Radioactive Substances:**

• Contain unstable atoms
• Emit radiation during decay
• Decay into more stable elements
• Have unique half-lives

These substances are common in nature, such as Uranium, Radon, and Carbon-14. They can be used in various applications such as dating archaeological findings, medicinal treatments such as cancer therapy, as well as in power generation within nuclear reactors. Understanding the nature of radioactive substances and their half-lives allows better management of risks and harnessing their practical benefits safely.

Decay Constant

The decay constant is a measure that helps to understand how quickly a radioactive substance undergoes decay. It is represented by the symbol \( \lambda \) and gives a sense of the proportionality of the decay rate of a substance. The larger the decay constant, the quicker the substance decays. Mathematically, it is related to the half-life and can be derived from the formula: \[ \lambda = \frac{\ln(2)}{t_{1/2}} \] Here, \( \ln(2) \approx 0.693 \) is the natural logarithm of 2, a constant value. The decay constant is incredibly useful because it provides insight on the time frame over which we can expect a radioactive substance to lose a significant amount of its mass.

**Key Points:**

• Relates to the rate of decay
• Inversely proportional to half-life
• Calculated using the natural logarithm of 2

This concept plays a significant role in fields dealing with nuclear sciences, where predicting the stability and longevity of material is essential for storage, disposal, or usage in various technologies.