Question

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

Short answer

Answer: Approximately 9.77 x 10^7 alpha particles are emitted in the time interval from 100 to 200 minutes.

Step-by-step solution

Define the decay formula and given values

The radioactive decay formula is: \(N_t = N_0 * (1/2)^{t/t_{1/2}}\) where: - \(N_t\) is the number of atoms remaining at time t - \(N_0\) is the initial number of atoms - \(t\) is the time (in minutes) - \(t_{1/2}\) is the half-life of the sample Given values: - \(N_0 = 10^{11}\) atoms - \(t_{1/2} = 10\) minutes

Calculate the remaining atoms at 100 minutes

To find the remaining atoms after 100 minutes, plug in the given values into the decay formula: \(N_{100} = 10^{11} * (1/2)^{100/10}\) \(N_{100} = 10^{11} * (1/2)^{10}\) \(N_{100} = 10^{11} * (1/1024)\) \(N_{100} = \frac{10^{11}}{1024}\)

Calculate the remaining atoms at 200 minutes

To find the remaining atoms after 200 minutes, plug in the given values into the decay formula: \(N_{200} = 10^{11} * (1/2)^{200/10}\) \(N_{200} = 10^{11} * (1/2)^{20}\) \(N_{200} = 10^{11} * (1/1048576)\) \(N_{200} = \frac{10^{11}}{1048576}\)

Calculate the number of alpha particles emitted

To find the number of alpha particles emitted between 100 and 200 minutes, subtract the remaining number of atoms after 200 minutes from the remaining number of atoms after 100 minutes: \(\Delta N = N_{100} - N_{200} = \frac{10^{11}}{1024} - \frac{10^{11}}{1048576}\) To make it easier to calculate the difference, we can find a common denominator in the fractions: \(\Delta N = \frac{10^{11}(1024 - 1)}{1048576}\) \(\Delta N = \frac{10^{11}(1023)}{1048576}\) Now we can calculate the value of \(\Delta N\): \(\Delta N = 9.77 \times 10^7\) The number of alpha particles emitted in the time interval from 100 to 200 minutes is approximately \(9.77 \times 10^7\).

Key concepts

Understanding Half-Life in Radioactive Decay

In nuclear physics, 'half-life' is a term that describes the time required for half of the unstable nuclei in a radioactive sample to decay. The concept of half-life is essential in predicting how a radioactive sample will behave over time. For instance, if the half-life of a sample is 10 minutes, then after 10 minutes, only half of the original number of radioactive nuclei will remain.

The half-life helps us understand long-term radioactive processes, such as carbon dating and medical radiotherapy. It's a key feature of nuclear chemistry that influences fields ranging from archaeology to space travel. In problem-solving, knowing the half-life enables us to predict the remaining quantity of a substance at any given time using the formula:
N_t = N_0 * (1/2)^{t/t_{1/2}}

Breaking down the formula, N_t represents the number of atoms remaining after time t, N_0 is the initial number of atoms, and t_{1/2} is the half-life. Understanding this concept is crucial for tasks like determining the safety of storing nuclear waste or calculating dosages for radiotherapy treatments.

Alpha Particle Emission in Radioactive Decay

Alpha particle emission is a type of radioactive decay where an unstable atom ejects an alpha particle, which consists of two protons and two neutrons (essentially a helium-4 nucleus). This process results in a decrease in the mass number by four and an atomic number by two, leading to the formation of a new element.

Alpha particles are relatively large and carry a +2 charge, making them highly ionizing but with low penetration ability. They can be stopped by a sheet of paper or the outer layer of human skin. However, if alpha emitters are ingested or inhaled, they can cause significant damage to living tissues, making understanding alpha decay important for health and safety precautions.

In the provided exercise, each alpha particle emitted represents the disintegration of one atomic nucleus. Therefore, tracking the number of emitted alpha particles gives insight into the extent of decay and the reduction of the radioactive sample.

Exponential Decay in Radioactive Substances

Radioactive decay follows a pattern known as 'exponential decay', where the decay rate is proportional to the number of atoms present. Unlike linear decay, where a fixed amount decays over each time period, exponential decay means that the number of decaying atoms decreases over time.

By nature, exponential processes can be deceiving; at first, the quantity changes slowly, but the rate of change becomes more drastic over time. Mathematically, this is represented by the exponential decay formula we've discussed. To visualize exponential decay, one can graph the number of remaining atoms against time, showing a distinct, gradually steepening curve.

Exponential decay is how we describe processes that quickly become less intense—the dimming of stars, the cooling of hot objects, and, of course, the disintegration of radioactive materials.

Physics Problem-Solving: Radioactive Decay

Problem-solving in physics, particularly in scenarios involving radioactive decay, requires a methodical approach: Understand the concept, identify known values, apply the formulae appropriately, and solve for the unknown quantities. In the exercise presented, we follow these steps by defining our variables, applying the decay formula, and calculating atom counts at different time intervals.

Improving Exercise Understanding

To improve understanding, one should approach problems with visual aids, like decay curves or conceptual diagrams. Another key element is practicing the rearrangement of decay equations to solve for different variables, such as solving for half-life instead of remaining atoms. Real-world examples, such as dating fossils or predicting the behavior of nuclear reactors, can also provide context that enriches the learning experience.
In summary, combining methodical problem-solving with an understanding of real-world applications builds competence in tackling physics problems involving radioactive decay.