Problem 84 - General Chemistry: Principles and Modern Applications

Question

Given a radioactive nuclide with \(t_{1 / 2}=1.00 \mathrm{h}\) and a current disintegration rate of 1000 atoms \(s^{-1}\), three hours from now the disintegration rate will be (a) 1000 atoms \(s^{-1} ;\) (b) 333 atoms \(s^{-1} ;\) (c) 250 atoms \(s^{-1}\); (d) 125 atoms \(s^{-1}\)

Step-by-Step Solution

Answer

The disintegration rate for the given radioactive nuclide three hours from now will be 125 atoms \(s^{-1}\), hence option (d) is the correct answer.

Step 1: Understand the half-life concept

Half-life (\(t_{1 / 2}\)) is the time required for a quantity to reduce to half of its initial value. This is a specific case of a more general concept called exponential decay. The half-life of a substance undergoing decay is constant, not dependent on its amount. In this problem, it's given that the half-life (\(t_{1 / 2}\)) is 1 hour. This implies that the quantity of the radioactive nuclide will halve every hour.

Step 2: Apply the half-life principle to the time specified

As per the half-life principle, after 1 hour (1 half-life), the disintegration rate will reduce to half of its initial rate, i.e., from 1000 atoms \(s^{-1}\) to 500 atoms \(s^{-1}\). Similarly, after 2 hours (2 half-lives), the rate will further reduce to its half, i.e., from 500 atoms \(s^{-1}\) to 250 atoms \(s^{-1}\). Finally, after 3 hours (3 half-lives), the rate will again reduce to its half, i.e., from 250 atoms \(s^{-1}\) to 125 atoms \(s^{-1}\).

Step 3: Compare the result with the given options

Comparing the final disintegration rate (125 atoms \(s^{-1}\)) calculated in step 2 with the given options, it is observed that it matches with option (d).

Key Concepts

half-life

The concept of half-life is central to understanding radioactive decay. It measures the time it takes for half of a given quantity of a radioactive substance to decay. This property is unique in that it is constant for each radioactive element, irrespective of the initial amount of the substance.

Consider a radioactive nuclide with a half-life of 1 hour. If you begin with a disintegration rate, or rate of radioactive decay, of 1000 atoms per second, the half-life tells us that in one hour, this rate will reduce to half, which is 500 atoms per second.

The calculation of half-life is a repetitive division by two, showing its reliance on time rather than the quantity of substance. Each hour reduces the disintegration rate by half, indicating a predictable exponential decay pattern.

exponential decay

Exponential decay is a process in which the quantity of a substance decreases at a rate proportional to its current value. This kind of decay is common among radioactive substances, including the nuclides studied in physics and chemistry.

The exponential nature of radioactive decay implies that over equal time intervals, the substance reduces by a consistent fraction of its existing amount, generally half. For instance, if the decay follows the exponential pattern with a half-life of 1 hour, it will consistently halve every hour.

Mathematically, this can be represented by the formula: \[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/t_{1/2}} \]where \( N(t) \) is the quantity remaining after time \( t \), \( N_0 \) is the initial quantity, and \( t_{1/2} \) is the half-life. This equation best showcases how time governs decay, not the magnitude of the initial amount.

disintegration rate

The disintegration rate is a measure of how quickly atoms in a radioactive sample are decaying. It indicates the number of atoms that disintegrate, or decay, per unit time.

In practical terms, measuring the disintegration rate allows scientists to determine the activity of a radioactive substance. For a nuclide with a half-life of 1 hour and an initial disintegration rate of 1000 atoms per second, the activity decreases as time progresses.

With each passing hour, or each half-life, that rate halves because the remaining quantity of radioactive material continues to decrease, significantly influencing the overall activity. Over three hours, the disintegration rate changes from 1000 atoms per second to 125 atoms per second, following a consistent pattern of exponential decay.