Question

The half-life of a sample of \(10^{11}\) atoms that decay by alpha emission is \(10 \mathrm{~min} .\) How many alpha particles are emitted between the time interval 100 min and 200 min?

Short answer

Answer: Approximately \(8.95 \times 10^9\) alpha particles are emitted between the time interval 100 and 200 minutes.

Step-by-step solution

Write down the radioactive decay formula

The radioactive decay formula is the following: \(N(t) = N_0 e^{-\lambda t}\) Here, \(N(t)\) is the number of remaining atoms at time t, \(N_0\) is the initial number of atoms, \(\lambda\) is the decay constant, and \(t\) is the time.

Find the decay constant using the half-life

The decay constant \(\lambda\) can be found using the following half-life formula: \(\lambda = \frac{\ln(2)}{T_{1/2}}\) Given that the half-life (\(T_{1/2}\)) is 10 minutes, we can calculate the decay constant: \(\lambda = \frac{\ln(2)}{10} = 0.0693\)

Find the number of remaining atoms at t = 100 minutes

Now we will use the radioactive decay formula with the initial number of atoms (\(N_0 = 10^{11}\) atoms), the decay constant (\(\lambda = 0.0693\)), and the time interval t = 100 minutes: \(N(100) = 10^{11}e^{-(0.0693)(100)} = 10^{11}e^{-6.93} \approx 9.93 \times 10^9\)

Find the number of remaining atoms at t = 200 minutes

Similarly, we will find the number of remaining atoms at t = 200 minutes using the radioactive decay formula: \(N(200) = 10^{11}e^{-(0.0693)(200)} = 10^{11}e^{-13.86} \approx 9.86 \times 10^8\)

Find the number of alpha particles emitted between t = 100 and t = 200 minutes

To find the number of alpha particles emitted during the time interval, we will find the difference between the remaining atoms at the beginning and end of the interval: Number of alpha particles emitted = \(N(100) - N(200) = 9.93 \times 10^9 - 9.86 \times 10^8 = (9.93 - 0.986) \times 10^9 \approx 8.95 \times 10^9\) Therefore, approximately \(8.95 \times 10^9\) alpha particles are emitted between the time interval 100 and 200 minutes.

Key concepts

Half-life

The concept of half-life is fundamental to understanding radioactive decay. It refers to the amount of time it takes for half of a sample of radioactive material to decay. In the given exercise, the half-life is 10 minutes, meaning every 10 minutes, half of the remaining atoms will decay.
Characteristics of half-life include:

• The half-life is independent of the initial amount of substance present.
• It describes the exponential rate at which radioactive atoms decay over time.
• It is used to calculate the time required for a given percentage of material to decay.

This allows us to predict how much of the radioactive sample will remain after a certain period.

Alpha Emission

Alpha emission is a type of radioactive decay in which an unstable atom releases an alpha particle. An alpha particle consists of 2 protons and 2 neutrons, equivalent to a helium nucleus. During alpha emission, the parent nucleus loses these particles, resulting in a daughter nucleus with a mass number decreased by 4 and an atomic number decreased by 2.
Key features of alpha emission include:

• It reduces the mass of the parent atom, leading to changes in both element and isotope.
• It usually occurs in heavier elements, like uranium and plutonium, leading them toward more stable configurations.
• Alpha particles can cause significant ionization, which is why they are often used in smoke detectors, despite having low penetration power (they can be blocked by a sheet of paper).

In the exercise, as atoms decay by alpha emission, they transform, releasing energy and transforming the sample's composition over time.

Decay Constant

The decay constant, denoted as \(\lambda\), is a vital parameter that characterizes the rate at which a radioactive substance decays. It is directly related to the half-life through the equation: \[ \lambda = \frac{\ln(2)}{T_{1/2}} \] where \(T_{1/2}\) is the half-life of the substance. This relationship allows us to calculate the decay constant from the known half-life.
Some important points about the decay constant:

• It is the probability per unit time that a single atom will decay.
• The larger the decay constant, the faster the rate of decay.
• In the exercise, the decay constant was calculated to be approximately 0.0693, using the given half-life of 10 minutes.

The decay constant is crucial for using the exponential decay formula to predict how much of the substance remains after a given period.

Exponential Decay Formula

The exponential decay formula is a mathematical model used to describe the process of radioactive decay. It calculates the number of atoms remaining in a sample after a certain time has elapsed. The formula is given by: \[ N(t) = N_0 e^{-\lambda t} \] where \(N(t)\) is the number of atoms at time \(t\), \(N_0\) is the initial number of atoms, and \(\lambda\) is the decay constant.
The formula is essential because:

• It provides a way to quantify how quickly radioactive materials decay over time.
• It helps in calculating how many atoms remain, or have decayed, over specified time intervals, as showcased in the exercise.
• It reflects the decay process as a continuous, exponential reducibility, not a linear one, which is characteristic of radioactive decay.

Using this formula, the exercise determined the number of atoms remaining at 100 and 200 minutes to derive the number of alpha particles emitted during this period.