Problem 75 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a radioactive element is 40 days. Calculate the average life. (a) \(5.76\) days (b) \(57.6\) days (c) 646 days (d) \(4.56\) days

Step-by-Step Solution

Answer

The average life is approximately 57.6 days (Option b).

Step 1: Understanding Half-Life

The half-life of a radioactive element is defined as the time taken for half of the radioactive nuclei in a sample to decay. It is denoted by the symbol \( T_{1/2} \). In this exercise, \( T_{1/2} = 40 \) days.

Step 2: Formula for Average Life

The average life (also known as the mean life) \( \tau \) of a radioactive element is related to its half-life by the formula: \[ \tau = \frac{T_{1/2}}{\ln(2)} \] where \( \ln(2) \approx 0.693 \).

Step 3: Substituting the Half-Life

Substitute the given half-life value into the formula: \[ \tau = \frac{40}{0.693} \]

Step 4: Calculating Average Life

Perform the division to find \( \tau \): \[ \tau \approx \frac{40}{0.693} \approx 57.68 \] days.

Step 5: Choosing the Closest Option

Compare the calculated average life \( 57.68 \) days with the given options and select the nearest value, which is \( 57.6 \) days.

Key Concepts

Half-Life Calculation

Half-life is a fascinating concept in the field of radioactive decay. It is essentially the time it takes for half of a sample of radioactive nuclei to decay into other elements or isotopes. This means that if you start with a certain amount of a radioactive substance, after one half-life period, only half of that substance will remain undecayed.

• The half-life is a constant for any given material and does not change over time.
• This property makes it particularly useful in fields such as archaeology, where it's used in methods like carbon dating.

To calculate half-life, you first need to know this constant value for the substance you are dealing with. In our exercise, it was given as 40 days. We denote the half-life with the symbol \( T_{1/2} \). Understanding this foundational concept allows one to predict how quickly or slowly a radioactive decay process will occur. It serves as a stepping stone for more complex calculations such as determining the average life.

Average Life

Once the half-life is determined, we can calculate the average life, also known as the mean life, of a radioactive substance. Average life gives a snapshot of how long, on average, a nucleus remains intact before decaying.

The formula connecting half-life with average life is:\[ \tau = \frac{T_{1/2}}{\ln(2)} \]where \( \tau \) represents the average life and \( \ln(2) \) is the natural logarithm of 2, approximately 0.693.

Calculating average life involves dividing the half-life by 0.693. For example, if the half-life is 40 days, the calculation is:\[ \tau = \frac{40}{0.693} \approx 57.68 \] days.

The average life is always longer than the half-life since it accounts for the entire decay time of all nuclei rather than just half. It's crucial for understanding the lifespan and behavior of radioactive substances.

Exponential Decay

Exponential decay is a principal feature of radioactive decay processes. It describes how the quantity of a substance decreases at a rate proportional to its current value. This means the decay process speeds up or slows down as the quantity changes over time.

During exponential decay:

• The number of decays in a particular period is proportional to the amount of substance present initially.
• This results in a curve that starts steep but flattens over time as less radioactive material remains available to decay.

The mathematical expression for exponential decay is described by the equation:\[ N(t) = N_0 e^{-\frac{t}{\tau}} \]Here, \( N(t) \) is the quantity of the substance at time \( t \), \( N_0 \) is the initial quantity, \( e \) is the base of the natural logarithm, and \( \tau \) is the average life.

Comprehending exponential decay is vital for predicting how quickly a radioactive substance will lose its radioactivity over time. It’s used in many scientific and engineering fields where decay or reduction over time needs to be quantified.