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

Question

The disintegration rate of a certain radioactive sample at any instant is \(5400 \mathrm{dpm}\). After 5 min the rate becomes \(2700 \mathrm{dpm}\). 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 Decay Formula

Radioactive decay can be represented by the formula \( R = R_0 e^{-kt} \), where \( R \) is the rate of disintegration at time \( t \), \( R_0 \) is the initial rate of disintegration, \( k \) is the decay constant, and \( t \) is time. We will use this formula to find the decay constant \( k \).

Step 2: Substitute Known Values

We know the initial rate \( R_0 = 5400 \text{ dpm} \) and the rate after 5 minutes \( R = 2700 \text{ dpm} \). Substitute these into the formula: \( 2700 = 5400 e^{-5k} \).

Step 3: Solve for the Decay Constant \( k \)

Divide both sides by 5400 to get \( \frac{1}{2} = e^{-5k} \). Taking the natural logarithm on both sides gives us \( \ln \left( \frac{1}{2} \right) = -5k \). Thus, \( k = -\frac{\ln(0.5)}{5} \).

Step 4: Calculate the Half-life

The half-life \( T_{1/2} \) is given by \( T_{1/2} = \frac{\ln(2)}{k} \). Substitute \( k \) from the previous step: \( T_{1/2} = \frac{\ln(2)}{-\frac{\ln(0.5)}{5}} \). Simplifying, this yields \( T_{1/2} \approx 5 \text{ minutes} \), because \( \ln(0.5) = -\ln(2) \).

Key Concepts

Decay Constant

The decay constant, often represented by the symbol \( k \), is a crucial factor in understanding radioactive decay. It quantifies the rate at which a radioactive substance decays. This constant essentially dictates how fast or slow the decay process occurs. In the formula for radioactive decay \( R = R_0 e^{-kt} \), the decay constant \( k \) determines the rate at which the initial disintegration rate \( R_0 \) decreases over time. A larger \( k \) would mean a more rapid decay, leading to a faster drop in the disintegration rate. Conversely, a smaller \( k \) suggests a slower decay. It is calculated based on the natural logarithm of half and the time it takes for the rate to drop by half, using the relationship \( k = -\frac{\ln(0.5)}{t} \). Understanding and calculating \( k \) helps predict how a radioactive sample will behave over time, which is vital in scientific and medical applications.

Half-Life

The concept of half-life is fundamental in the study of radioactive materials. It is the time required for the quantity of a radioactive material to decrease to half of its initial value. In mathematical terms, it is represented as \( T_{1/2} \). This period is unique to each radioactive isotope and is a direct measure of its stability and rate of decay. Half-life helps in predicting how long a substance will remain active or hazardous. For our example, we find the half-life using the calculation \( T_{1/2} = \frac{\ln(2)}{k} \). This formula shows that the half-life is inversely proportional to the decay constant. Thus, knowing the half-life can inform you not only about the time it takes for a substance to lose half its potency but also gives insight into the decay constant, and vice versa.

Natural Logarithm

Natural logarithms are an essential mathematical tool in understanding exponential decay processes like radioactive decay. The natural logarithm, denoted as \( \ln \), simplifies the equations involved in these processes. In radioactive decay, it is used to solve for the decay constant and half-life. When we write exponential decay formulas like \( R = R_0 e^{-kt} \), we often need to isolate terms involving the exponent. Using the natural logarithm helps us remove the exponential part and solve for variables like the decay constant \( k \). For instance, if we have \( e^{-5k} = \frac{1}{2} \), taking the natural logarithm on both sides results in \( -5k = \ln(\frac{1}{2}) \). This simplification is a key step in deriving values that describe the decay process. It also provides insight into changes, as the natural logarithm relates exponential growth or decay back to linear rates.

Disintegration Rate

The disintegration rate is the speed at which a radioactive substance decays, usually measured in disintegrations per minute (dpm). It reflects how many atoms of a material are actively decaying at any given time. Initially, a sample might have a high disintegration rate, which gradually decreases over time as the sample decays. In our exercise, the initial disintegration rate was \( 5400 \mathrm{dpm} \), which then reduced to \( 2700 \mathrm{dpm} \) after 5 minutes. This decline is predicted using the decay formula. The disintegration rate is crucial in understanding how long a radioactive substance will remain dangerous or useful. It is directly influenced by the decay constant and provides practical data on the activity of a sample. By tracking changes in the disintegration rate over time, one can confirm the calculated half-life and ensure predictions align with actual observations.