Question

Half-life \(\left(t_{1 / 2}\right)\) for a radioactive decay is \(6930 \mathrm{sec}\). The time required to fall the rate of decay by \(\left(\frac{1}{100}\right)^{\text {th }}\) of it's initial value is : (a) \(69.3 \mathrm{sec}\) (b) \(20,000 \mathrm{sec}\) (c) 23030 sèc (d) none of these

Short answer

None of the options (a), (b), or (c) are correct. The correct time required to fall the rate of decay by \((1/100)^{th}\) of its initial value is approximately 46054 seconds.

Step-by-step solution

- Understand Half-life

The half-life of a substance is the time it takes for half of the radioactive nuclei to decay, or for the rate of decay to fall to half of its initial value. The given half-life for the radioactive decay is 6930 seconds.

- Determine the Fraction of Remaining Substance

The problem asks for the time it takes for the rate of decay to fall to \((1/100)^{th}\) of its initial value. To solve this, we must determine how many half-lives are required to reach \((1/100)^{th}\) of the initial rate of decay.

- Calculate the Number of Half-lives

Since each half-life reduces the quantity by half, we can use the formula \((1/2)^n = (1/100)\), where n is the number of half-lives. Taking the logarithm of both sides provides \(n \cdot \log(1/2) = \log(1/100)\).

- Solve for the Number of Half-lives

We know that \(\log(1/2) = -\log(2)\) and \(\log(1/100) = \log(1) - \log(100) = -2\). So the equation becomes \(-n \cdot \log(2) = -2\), which simplifies to \(n = 2/\log(2)\).

- Calculate the Time Required

To find the total time required to reach \((1/100)^{th}\) of the initial value, multiply the number of half-lives by the half-life duration: \(t = n \times t_{1/2}\).

- Compute the Final Answer

First, calculate \(n\) by solving the equation \(n = 2/\log(2)\). Since \(\log(2) \approx 0.3010\), we get \(n \approx 2 / 0.3010 \approx 6.6455\). Then, multiply this number by the half-life duration to obtain the time: \(t \approx 6.6455 \times 6930 \mathrm{sec} \approx 46053.9 \mathrm{sec}\).

Key concepts

Physical Chemistry Problems

Physical chemistry combines principles of physics and chemistry to understand the physical properties of molecules, the forces that act upon them, and their reactions. In solving physical chemistry problems, students must be familiar with various concepts like thermodynamics, kinetics, quantum mechanics, and spectroscopy.

Radioactive decay problems are a common topic in physical chemistry, serving as a primary example of first-order kinetics, where only the quantity of the radioactive substance affects the rate of its decay. It’s essential to approach these problems systematically: understanding the theory, identifying what is being asked, translating the problem into mathematical terms, and finally, applying appropriate formulas and constants to solve it.

For the Joint Entrance Examination (JEE) candidates, mastering these steps takes practice and a solid understanding of the underlying principles. Consistent practice with problems of increasing complexity helps in preparing for questions on radioactive decay, among other key topics in JEE chemistry.

Half-life Calculation

Understanding the calculation of half-life is crucial for students studying physical chemistry. The half-life, typically denoted as \( t_{1/2} \), is the time required for a quantity to reduce to half its initial value. In the context of radioactive decay, it indicates how quickly unstable atoms undergo decay.

The formula to calculate the time it takes for a substance to decay to any fraction of its original quantity is given by \( t = n \times t_{1/2} \) where \( n \) represents the number of half-lives. To find \( n \) when the final amount and the half-life are known, logarithmic equations are employed. The application of half-life calculations is not confined to education; it extends to fields like nuclear medicine, radiocarbon dating, and environmental science, showcasing the real-world importance of understanding this concept.

Logarithmic Equations

Logarithmic equations are indispensable tools for solving problems related to exponential decay and growth, such as radioactive decay. These equations translate multiplicative relationships into additive ones, which are often easier to manipulate algebraically.

To solve for the number of half-lives in a decay process, as in the given problem, logarithms help us find the exponent that a number (in this case, 1/2 for half-life) must be raised to produce another number (like 1/100).

Application of Logarithms

By representing the problem with a logarithmic equation, \( n \cdot \log(1/2) = \log(1/100) \), and subsequently solving for \( n \) using known logarithm properties, we identify the number of half-lives involved. This approach is not only valuable for half-life calculation but also for a wide range of problems in physics and chemistry where exponential change is observed.

JEE Chemistry Preparation

Preparation for JEE, or the Joint Entrance Examination, an integral part of the engineering college admission process in India, requires a deep understanding of concepts across physics, chemistry, and mathematics. Specifically for chemistry, students must grasp the fundamentals of organic, inorganic, and physical chemistry.

Focusing on physical chemistry, problems like radioactive decay feature prominently in JEE chemistry preparation because they test a student’s ability to integrate concepts of kinetics and mathematics. To excel, students should practice a wide range of problems, adopt a problem-solving mindset, and learn to swiftly apply mathematical tools like logarithmic equations.

Strategies for Success

Successful preparation entails consistently practicing problems, understanding the reasoning behind each step, and being able to apply concepts to unfamiliar situations. This holistic approach to understanding, rather than memorization, is key for tackling JEE chemistry questions effectively.