Problem 137 - A Complete Resource Book in Chemistry for JEE Main

Question

The disintegration rate of a certain radioactive sample at any instant is 5400 dpm. After 5 min the rate becomes 2700 dpm. The half life of the sample in min is approximately

Step-by-Step Solution

Answer

The half-life of the sample is approximately 5 minutes.

Step 1: Understand the Problem

We are given the initial disintegration rate of a radioactive sample as 5400 dpm (disintegrations per minute) and the rate after 5 minutes as 2700 dpm. We need to find the half-life of the sample. The half-life is the time it takes for the disintegration rate to reduce to half of its original value.

Step 2: Use the Exponential Decay Formula

The decay of radioactive material follows the exponential decay formula: \[N(t) = N_0 e^{-kt}\]where \(N(t)\) is the disintegration rate at time \(t\), \(N_0\) is the initial rate, \(k\) is the decay constant, and \(t\) is the time elapsed.

Step 3: Calculate the Decay Constant \(k\)

Using the given rates, set up the equation for 5 minutes: \[ 2700 = 5400 \times e^{-k \times 5} \]This simplifies to: \[ e^{-5k} = \frac{1}{2} \]Take the natural logarithm of both sides to solve for \(k\): \[ -5k = \ln\left(\frac{1}{2}\right) \]Therefore, \(k = -\frac{\ln(\frac{1}{2})}{5}\).

Step 4: Calculate the Half-Life

The half-life \(t_{1/2}\) satisfies the equation: \[ N(t_{1/2}) = \frac{N_0}{2} \]Using the decay formula, \[ \frac{N_0}{2} = N_0 \times e^{-kt_{1/2}} \]which simplifies to \[ e^{-kt_{1/2}} = \frac{1}{2} \]Solving for \(t_{1/2}\), we have \[ t_{1/2} = \frac{\ln(2)}{k} \].

Step 5: Substitute \(k\) and Simplify

Substitute the expression for \(k\) from Step 3 into the expression for \(t_{1/2}\): \[ t_{1/2} = \frac{\ln(2)}{-\frac{\ln(\frac{1}{2})}{5}} \]Further simplify: \[ t_{1/2} = \frac{5 \ln(2)}{\ln(2)} = 5 \text{ minutes} \].

Key Concepts

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. This is a common phenomenon in radioactive decay, where a radioactive substance breaks down over time. The rate at which this decay happens is initially rapid but gradually slows, creating an exponential curve if you were to graph it.
In mathematical terms, exponential decay is represented by the equation \(N(t) = N_0 e^{-kt}\). Here, \(N(t)\) represents the amount of substance remaining at time \(t\), \(N_0\) is the initial quantity, \(k\) is the decay constant, and \(e\) is the base of natural logarithms, approximately equal to 2.718.

• **Initial Rate**: This is the starting rate of decay, like the 5400 disintegrations per minute (dpm) of the exercise.
• **Exponential Function**: The negative exponent \(-kt\) indicates a decrease over time, typical of processes like radioactive decay.

If you are trying to understand how fast a material decays, you watch how the quantity \(N(t)\) changes compared to \(N_0\). The faster \(N(t)\) decreases, the larger the decay constant \(k\). But this connection also highlights why half-life is so important—it gives a measurable way to predict decay over time.

Half-Life

Half-life refers to the time it takes for a radioactive substance to decay to half of its initial amount. In other words, it is the time period over which the disintegration rate of a sample is reduced by 50%. This concept is pivotal in fields like nuclear physics and pharmacology because it provides a consistent measure to assess long-term decay.
The half-life, \(t_{1/2}\), can be determined using the expression:

• \[ t_{1/2} = \frac{\ln(2)}{k} \]
• **\(\ln(2)\):** This constant arises because when a substance reduces to half, the proportionality derived involves the natural logarithm of 2.
• **Decay Constant \(k\):** This is specific to the material under study and influences how rapidly the half-life is reached.

Understanding half-life helps in determining how quickly or slowly a radioactive material will reduce in activity, which is essential for safety and handling in practical applications. Calculating the half-life in the example given, we see that the sample's half-life is 5 minutes, meaning every 5 minutes the activity is halved.

Decay Constant

The decay constant, denoted as \(k\), is a critical factor in understanding how quickly a radioactive substance decays over time. It is unique to each type of substance and represents the probability per unit time that a nucleus will decay. The larger the decay constant, the faster the decay process.
You can think of \(k\) as setting the pace of the decay process.
This constant allows us to use the formula \(N(t) = N_0 e^{-kt}\) to gain insights into how a substance breaks down over time. For example, from our exercise, using the initial and 5-minute rates (5400 dpm and 2700 dpm respectively), we calculated the decay constant \(k\) as:

• \[ k = -\frac{\ln(\frac{1}{2})}{5} \]

This shows how the decay constant can help us relate the rate of decay, half-life, and the time it takes for a substance to reach a certain level of decay.
In summary, knowing \(k\) and understanding its relationship with half-life and the exponential decay equation enables scientists to predict future states of radioactive material, ensuring accurate and efficient usage and containment.