Question

The half lives of two radioactive nuclides \(\mathrm{A}\) and \(\mathrm{B}\) are 1 and 2 min respectively. Equal weights of \(A\) and \(B\) are taken separately and allowed to disintegrate for \(4 \mathrm{~min}\). What will be the ratio of weights of \(\mathrm{A}\) and \(\mathrm{B}\) disintegrated? (a) \(1: 2\) (b) \(1: 1\) (c) \(1: 3\) (d) \(5: 4\)

Short answer

The ratio of disintegrated weights is \(5:4\), option (d).

Step-by-step solution

Understand the given data

The half-lives of nuclides \(A\) and \(B\) are given as 1 minute and 2 minutes, respectively. We want to calculate the disintegrated weight of each after 4 minutes to find the ratio.

Identify formula to use

We will use the formula for decay: \( N_t = N_0 \times \left(\frac{1}{2}\right)^{t/T_{1/2}} \), where \(N_t\) is the amount remaining, \(N_0\) is the initial amount, \(t\) is time passed, and \(T_{1/2}\) is the half-life.

Calculate disintegrated amount for nuclide A

Using the formula for decay for nuclide \(A\), where \(T_{1/2} = 1\) minute: \[ N_4 = N_0 \times \left(\frac{1}{2}\right)^{4/1} = N_0 \times \left(\frac{1}{2}\right)^4 = \frac{N_0}{16} \] Thus, the disintegrated amount of \(A\) is: \[ \Delta N_A = N_0 - \frac{N_0}{16} = \frac{15N_0}{16} \]

Calculate disintegrated amount for nuclide B

For nuclide \(B\), where \(T_{1/2} = 2\) minutes: \[ N_4 = N_0 \times \left(\frac{1}{2}\right)^{4/2} = N_0 \times \left(\frac{1}{2}\right)^2 = \frac{N_0}{4} \]Thus, the disintegrated amount of \(B\) is:\[ \Delta N_B = N_0 - \frac{N_0}{4} = \frac{3N_0}{4} \]

Calculate ratio of disintegrated weights

The ratio of disintegrated weights of \(A\) to \(B\) is:\[ \text{Ratio} = \frac{\Delta N_A}{\Delta N_B} = \frac{\frac{15N_0}{16}}{\frac{3N_0}{4}} = \frac{15}{12} = \frac{5}{4} \]

Choose the correct answer

The ratio of weights disintegrated is \(5:4\), corresponding to option (d).

Key concepts

Half-life

The concept of half-life is central to understanding radioactive decay and how quickly a substance disintegrates. Half-life is the time required for half of the radioactive nuclides in a sample to decay. This decay process is exponential, meaning that after each half-life, the amount of radioactive material remaining is halved.
For example, if you have 100 grams of a substance with a half-life of 1 minute, after 1 minute only 50 grams will remain radioactive. After 2 minutes, this amount will again halve, leaving you with 25 grams of the original radioactive substance. After 3 minutes, it will reduce to 12.5 grams, and so on.

• Half-life is specific to each radioactive nuclide, meaning different isotopes have different half-lives.
• Knowing the half-life helps scientists and students predict how much of a radioactive substance remains after a given time, which is crucial in fields like nuclear chemistry and medical radiology.
• The half-life not only helps in academic exercises but is vital in practical applications such as dating archaeological finds and managing nuclear waste.

Disintegration Equation

The disintegration equation provides a way to calculate the remaining amount of a radioactive substance after a certain period. This formula is particularly important because it models the exponential nature of decay, which is key to making accurate predictions.
The formula used for radioactive decay is: \[ N_t = N_0 \times \left(\frac{1}{2}\right)^{t/T_{1/2}} \]Here:

• \(N_t\) is the remaining amount of the substance after time \(t\).
• \(N_0\) is the initial amount of the substance.
• \(t\) is the time elapsed.
• \(T_{1/2}\) is the half-life of the substance.

This equation helps us understand how much of the initial substance has disintegrated over time, by calculating the amount remaining and subtracting it from the initial amount. In exercises, this tool is used to quantify disintegration, such as in calculating the amount of nuclides disintegrated in the provided problem.
Using this equation allows students to visualize the decay process and reinforces the idea that radioactive decay is a predictable, mathematical process, not subject to random chance.

Nuclide Comparison

Nuclide comparison involves analyzing the decay behaviors of different radioactive substances, particularly looking at how different half-lives affect their disintegration over time. This concept is crucial for understanding that not all radioactive substances decay at the same pace.
For example, given two nuclides, like A with a half-life of 1 minute and B with a half-life of 2 minutes, their rates of decay will differ significantly over the same time span.

• Nuclide A, with a shorter half-life, will lose more of its mass faster compared to nuclide B.
• In practical terms, this means that after a certain period, like the 4 minutes specified in the exercise, the proportion of material remaining will be less for nuclide A than for nuclide B.
• This is why the exercise calculates the disintegrated weight for each nuclide and then compares them to find a ratio, helping assess their relative rates of decay.

Understanding nuclide comparison is essential in scenarios ranging from nuclear reactors to dating geological formations. It provides insight into the behavior of radioactive materials and helps in making informed decisions in scientific and industrial applications.