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 substance left is \( \frac{1}{16} \). Answer: (d).

Step-by-step solution

Understanding Half-Life

The half-life of a substance is the time taken for half of the radioactive nuclide to decay. Here, the half-life is given as 10 months.

Determining Number of Half-Lives

Calculate how many half-lives have passed in 40 months by dividing 40 months by the half-life duration. \[ \text{Number of half-lives} = \frac{40 \, \text{months}}{10 \, \text{months/half-life}} = 4 \]

Calculating Remaining Fraction

For each half-life that passes, the substance is reduced by half. After 4 half-lives, the remaining fraction is calculated by \[ \left( \frac{1}{2} \right)^4 = \frac{1}{16} \]

Matching the Answer

Compare the calculated remaining fraction with the given options. The matching choice is (d) \( \frac{1}{16} \).

Key concepts

Understanding Half-Life

Half-life is a critical concept in the study of radioactive decay. It refers to the time required for half of the radioactive nuclides in a sample to decay or transform into other substances. In the given exercise, we are dealing with a nuclide that has a half-life of 10 months. This means that every 10 months, half of the remaining radioactive nuclide will have decayed. Thus, if you start with a certain amount of substance, only half will remain after one half-life period. This predictable rate of decay allows scientists to determine the age of materials and to forecast how long a radioactive substance will be active.

Defining a Nuclide

A nuclide is a term that refers to a specific type of atom characterized by the composition of its nucleus. This includes the number of protons, neutrons, and its energy state. In radioactive materials, different nuclides have unique properties and decay rates. The nuclide in the given problem undergoes a process known as radioactive decay, where it transforms into a different element or a different isotope of the same element. The concept of a nuclide is central in understanding the behavior and properties of radioactive materials because it helps in identifying and calculating aspects like half-life and atomic mass.

Calculating Fraction Remaining

When dealing with radioactive decay, calculating the fraction of a substance remaining after a certain period is a common task. The fraction remaining refers to the proportion of the original radioactive material still present after a series of half-lives.

To find this, you need to know the number of half-lives that have passed. For each half-life, the remaining substance is halved. This can be expressed mathematically as

• After 1 half-life: \( \left( \frac{1}{2} \right)^1 = \frac{1}{2} \)
• After 2 half-lives: \( \left( \frac{1}{2} \right)^2 = \frac{1}{4} \)
• After 3 half-lives: \( \left( \frac{1}{2} \right)^3 = \frac{1}{8} \)
• After 4 half-lives: \( \left( \frac{1}{2} \right)^4 = \frac{1}{16} \)

Using this pattern, you can accurately determine the fraction remaining after any given time period.

Radioactive Decay Calculation

Radioactive decay calculations involve understanding how a substance diminishes over time due to the transformation of unstable nuclides. To solve problems like the one in the exercise, you need a systematic approach:

1. **Determine the half-life**: Start by identifying the half-life of the nuclide, which is the time it takes for half of the substance to decay. In the exercise, it's 10 months.2. **Calculate the number of half-lives**: Divide the total time period by the half-life to find out how many half-lives have passed. For example, 40 months divided by 10 months per half-life equals 4 half-lives.3. **Compute the remaining fraction**: Use each half-life to reduce the substance by half. Thus, after four half-lives, the fraction of the radioactive material still remaining would be \( \left( \frac{1}{2} \right)^4 = \frac{1}{16} \).By following these steps, you can effectively perform radioactive decay calculations and understand the nature of radioactive materials over time.