Problem 77 - A Complete Resource Book in Chemistry for JEE Main

Question

One microgram of radioactive sodium \({ }_{11} \mathrm{Na}^{24}\) with a half- life of 15 hours was injected into a living system for a bio assay. How long will it take for the radioactivity to fall to \(25 \%\) of the initial value? (a) 60 hours (b) \(22.5\) hours (c) 375 hours (d) 30 hours

Step-by-Step Solution

Answer

It will take 30 hours for the radioactivity to fall to 25% of the initial value.

Step 1: Understanding the Half-Life

The half-life of a radioactive substance is the time it takes for half of the radioactive atoms to decay. In this problem, sodium 3
9erc{11}
99a^{24}
99
00 has a half-life of 15 hours.

Step 2: Calculating Decay Periods

To find the time it takes for the radioactivity to fall to 25% of the initial value, we first need to understand that this means the substance undergoes two half-lives, because 25% is half of half. Therefore, we calculate it using the formula: Number of half-lives = \( \frac{\log(\text{remaining percentage})}{\log(0.5)} \) = \( \frac{\log(0.25)}{\log(0.5)} = 2 \).

Step 3: Calculating Total Time

Since it takes two half-lives for the amount of radioactivity to fall to 25%, we multiply the half-life by 2. Hence, the total time is \( 2 \times 15 \text{ hours} = 30 \text{ hours} \).

Key Concepts

Half-Life Calculation

Understanding the concept of half-life is crucial for grasping the principles of radioactive decay. The half-life represents the time required for half of the radioactive nuclei in a sample to decay into a more stable form. Simply put, after one half-life, only 50% of the initial radioactive material remains. This decay follows an exponential process. To calculate the number of half-lives that pass for a substance to fall to a certain percentage of its original amount, the formula used is: \[ \text{Number of half-lives} = \frac{\log(\text{remaining percentage})}{\log(0.5)} \] Using this equation helps easily determine how many half-life periods have occurred for any given percentage decrease. Remember, however, that precise calculations also depend on knowing the exact half-life of the substance in question.

Sodium-24

Sodium-24 is a radioactive isotope of sodium, denoted as \(\mathrm{^{24}Na}\). It is often used in medical and scientific applications due to its relatively short half-life of 15 hours. This makes it suitable for experiments that require quick observations, such as bio assays in living systems. Here are a few important aspects of Sodium-24:

• Short Half-Life: With a half-life of only 15 hours, Sodium-24 decays quickly, making it ideal for short-term studies.
• Decay Product: It decays into stable magnesium isotope \(\mathrm{^{24}Mg}\), emitting beta radiation during the process.
• Uses in Research: It's commonly used to trace sodium ion movement in physiological studies, such as how ions flow through the bloodstream or various organs, due to its radioactive nature.

Whether for tracking metabolic pathways or conducting environmental studies, Sodium-24's characteristics make it a valuable tool in nuclear chemistry.

Nuclear Chemistry

Nuclear chemistry focuses on the reactions and properties of atomic nuclei, which are at the heart of processes like radioactive decay. It delves into how unstable isotopes transform into stable ones through the emission of particles, such as alpha, beta, or gamma radiation. Key aspects of nuclear chemistry include:

• Radioactive Decay: This natural process results in the transformation of an unstable nucleus into a more stable one. It involves the emission of radiation, which alters the composition of the nucleus.
• Half-Life: A critical concept that describes how quickly a radioactive substance decays. Half-life is used to determine the stability and duration of radioactive isotopes' activity.
• Applications: From generating power in nuclear reactors to medical imaging and treatments in isotopic labeling, nuclear chemistry finds its uses across various fields.

Overall, nuclear chemistry helps us understand the behavior of elements at a fundamental level, enabling advancements in technology, medicine, and environmental science.