Problem 78 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a radioactive nuclide is \(0.693\) minutes. The time (in minutes) required for the disintegration of this nuclide from 10 grams to one gram is ........ (a) 1 (b) \(0.693\) (c) \(6.93\) (d) \(2.303\)

Step-by-Step Solution

Answer

The time required is option (d) 2.303 minutes.

Step 1: Understanding Half-Life

The half-life of a radioactive substance is the time it takes for half of the substance to decay. In this problem, the half-life is given as 0.693 minutes. This means every 0.693 minutes, the amount of the substance is reduced by half.

Step 2: Using the Half-Life Formula

The formula for the remaining quantity of a radioactive substance after a certain time is given by: \[ N(t) = N_0 imes 0.5^{t/t_{1/2}} \] where \( N(t) \) is the remaining amount, \( N_0 \) is the initial amount, \( t \) is the elapsed time, and \( t_{1/2} \) is the half-life.

Step 3: Applying the Formula

In this exercise, \( N_0 = 10 \) grams, \( N(t) = 1 \) gram, and \( t_{1/2} = 0.693 \) minutes. Substitute these values into the formula to find \( t \): \[ 1 = 10 imes 0.5^{t/0.693} \]

Step 4: Solving the Exponential Equation

Rearrange the equation: \[ 0.1 = 0.5^{t/0.693} \] Take the logarithm on both sides to solve for \( t \): \[ \log_{10}(0.1) = \frac{t}{0.693} \cdot \log_{10}(0.5) \]

Step 5: Calculating the Logarithms

Calculate \( \log_{10}(0.1) = -1 \) and \( \log_{10}(0.5) \approx -0.3010 \). Substitute these values into the equation:\[ -1 = \frac{t}{0.693} imes -0.3010 \]

Step 6: Finding Elapsed Time \( t \)

Solve for \( t \):\[ -1\times0.693 = t \times -0.3010 \] which gives \[ t = \frac{0.693}{0.3010} \] Thus, \( t \approx 2.303 \) minutes.

Key Concepts

Understanding Radioactive Decay

Radioactive decay is a natural process whereby unstable atomic nuclei lose their energy by emitting radiation. This process leads to the transmutation of one element into another over time. During radioactive decay, an atom will release particles and energy until it becomes stable. The rate of this decay is defined by its half-life, which is a significant property of the radioactive material.

It's important to know that radioactive decay is random, but statistically predictable. If you have a large enough sample of a radioactive substance, you can predict how much will remain after a given period. In a simpler sense, you can think of it as a "clock" that ticks down as the substance decreases in quantity. With each tick, part of the substance is transformed into something else.

Characteristics of radioactive decay include:

• It is a spontaneous process.
• The rate of decay is constant over time.
• It is not influenced by external physical or chemical conditions.

Understanding these properties is crucial for calculating how long it will take for a specific amount of a substance to decay to a different amount, as seen in this exercise.

Explaining Exponential Decay

Exponential decay is a vital concept used to understand how quantities diminish over time, particularly in scenarios like radioactive decay. Unlike linear decay, where a quantity decreases at a constant rate, exponential decay means that the quantity decreases proportionally to its current value.

This can be expressed with the formula:\[ N(t) = N_0 \times e^{-\lambda t} \]where:

• \(N(t)\) is the quantity at time \(t\)
• \(N_0\) is the initial quantity
• \(\lambda\) is the decay constant
• \(t\) is the elapsed time

In this context, the decay constant \(\lambda\) is related to the half-life by the equation:\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]Exponential decay exhibits a rapid reduction initially, which slows over time. The computational method used in the solution, involving logarithms, allows us to solve exponential equations accurately to find out for how long a substance needs to decay to reach a particular state.

The Half-Life Formula and Calculation

The half-life formula is a critical tool for calculating radioactive decay. It defines the time required for a quantity of a substance to reduce to half its initial amount. This is not only useful in chemistry and physics but also in fields like archaeology and medicine.

In simple terms, if you start with a certain amount of a radioactive material, the half-life is the period it takes for half of it to "disappear" due to decay. The formula frequently used is:\[ N(t) = N_0 \times 0.5^{t/t_{1/2}} \]where each term is as defined before.

This formula is used to determine the time needed for a particular material to decay from an initial amount to a specified amount. For example, in our exercise, we start with 10 grams and want to find out when it will be down to just 1 gram. By using logarithms, you can solve for the time \(t\) when the condition is met. This serves as a typical example of applying math to solve real-world scientific problems with practical applications in many areas of research and industry.