Question

The half-life of a radioactive nuclide is 10 months. The fraction of the substance left behind after 40 months is (a) \(1 / 2\) (b) \(1 / 4\) (c) \(1 / 8\) (d) \(1 / 16\)

Short answer

The fraction of the substance left after 40 months is \( \frac{1}{16} \). Hence, option (d) is correct.

Step-by-step solution

Understanding Half-Life

The half-life of a substance is the time it takes for half of the substance to decay. For this problem, the half-life is given as 10 months.

Determine Number of Half-Lives

We need to determine how many half-lives have passed in 40 months. This is calculated by dividing the total time by the half-life. Thus, the number of half-lives is \( \frac{40}{10} = 4 \).

Calculating the Remaining Fraction

After each half-life, half of the remaining substance decays. Therefore, after 4 half-lives, the fraction of substance left is \( \left( \frac{1}{2} \right)^4 \).

Simplifying the Expression

Calculate \( \left( \frac{1}{2} \right)^4 = \frac{1}{2^4} = \frac{1}{16} \). Thus, the fraction of the substance left after 40 months is \( \frac{1}{16} \).

Key concepts

Half-Life

The concept of half-life is central to understanding radioactive decay. It represents the time required for half of a given amount of a radioactive substance to disintegrate. The idea is simple: after the half-life period, only 50% of the original material remains unchanged. This periodic halving continues until the substance becomes negligible.
In this exercise, the half-life of the radioactive nuclide is 10 months. Purpose of half-life is to provide a predictable measure for the decay of radioactive substances, crucial for fields like archeology (radiocarbon dating), medicine, and nuclear energy management. By grasping the concept of half-life, you unlock the key to predicting how substances diminish over specific time frames.

Number of Half-Lives

To determine the number of half-lives that have passed, simply divide the total time by the half-life duration. In the given problem, this principle allows us to transform the question of decay over 40 months into a more manageable number of half-life periods.
For instance, if the half-life is 10 months, and the total time observed is 40 months, you would calculate the number of half-lives as follows:

• Number of half-lives = Total time / Half-life = 40 months / 10 months = 4 half-lives

This calculation is critical as it converts temporal duration into units of decay cycles, allowing us to apply exponential decay formulas effectively.

Fraction Remaining

After determining the number of half-lives, we can calculate the remaining fraction of the substance. Each half-life reduces the remaining substance by half. It follows a consistent division, where each cycle multiplies the remaining fraction by 1/2.
In the example:

• After 1 half-life, 50% or 1/2 of the original amount remains.
• After 2 half-lives: (1/2) x (1/2) = 1/4 remains.
• After 3 half-lives: (1/4) x (1/2) = 1/8 remains.
• After 4 half-lives: (1/8) x (1/2) = 1/16 remains.

Thus, for our specific problem, after 40 months or 4 half-lives, the fraction remaining is 1/16. This outcome is calculated using the formula:\[\left( \frac{1}{2} \right)^n\]where \(n\) is the number of half-lives.

Exponential Decay

Exponential decay describes how a quantity decreases increasingly rapidly in correspondence to its size. It's not a simple linear reduction but rather a proportional decrease that accelerates the smaller it gets.
This pattern is characterized by the formula:\[N(t) = N_0 \left( \frac{1}{2} \right)^{t/T_{1/2}}\]where \(N(t)\) is the quantity of substance remaining at time \(t\), \(N_0\) is the initial quantity, and \(T_{1/2}\) is the half-life.
Exponential decay is a crucial concept in understanding how radioactive substances diminish. It implies that even a small segment remaining from the initial substance follows the same pattern, diminishing predictably over successive half-lives. This powerful, fundamental concept helps us manage and predict the behavior of radioactive materials in a variety of applications.