Question

Explain the meaning of half-life.

Short answer

Answer: Half-life is the time required for half of the original quantity of a substance to decay or break down. It is typically used in the context of radioactive elements, which undergo radioactive decay. Radioactive decay is a spontaneous process where an unstable atomic nucleus loses energy by emitting radiation, transforming the radioactive element into another element over time. Half-life is a convenient way to describe the decay process because it is independent of the initial amount of the substance. The amount of remaining substance after time t is usually calculated through exponential decay, which involves a constant decay rate.

Step-by-step solution

Definition of Half-Life

Half-Life (denoted as \(T_{1/2}\)) is the time required for half of the original quantity of a substance to decay or break down. It is typically used in the context of radioactive elements, which decay into other elements over time. It can also be applied in other contexts like the decay of chemicals or the breakdown of certain drugs in the body. The amount of remaining substance after time t is usually calculated through exponential decay, which involves a constant decay rate.

Radioactive Decay and Half-Life

Radioactive decay is a spontaneous process where an unstable atomic nucleus loses energy by emitting radiation. This causes the radioactive element to transform into another element over time. The rate of radioactive decay depends on the nature of the element and is usually characterized by its half-life. Half-life is a convenient way to describe the decay process because it is independent of the initial amount of the substance.

Exponential Decay

The process of radioactive decay can be described by the exponential decay equation, which relates the remaining amount of a substance to its initial amount using the decay constant, k. The equation for exponential decay is: \(N(t) = N(0) * e^{-kt}\) Here, \(N(0)\) is the initial amount of the substance, \(N(t)\) is the remaining amount after time t, and k is the decay constant. The half-life (\(T_{1/2}\)) is related to the decay constant by the following equation: \(T_{1/2} = \frac{\ln(2)}{k}\)

Half-Life Example

Let's consider an example to illustrate the concept of half-life. Suppose we have a sample of 1000 atoms of a radioactive substance with a half-life of 4 hours. We want to know how many atoms of the substance remain after 12 hours. First, we can calculate the decay constant using the half-life formula: \(k = \frac{\ln(2)}{T_{1/2}} = \frac{\ln(2)}{4}\) Next, we can use the exponential decay equation to calculate the remaining amount after 12 hours: \(N(12) = N(0) * e^{-kt} = 1000 * e^{-\frac{\ln(2)}{4} * 12}\) \(N(12) \approx 28.17\) atoms So after 12 hours, there will be approximately 28 atoms remaining in the sample.

Key concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atom loses energy by emitting radiation, such as alpha particles, beta particles, or gamma rays. This emission of radiation results in the transmutation of the atom into a different element or a different isotope of the same element.

The process is random for individual atoms, meaning we cannot predict exactly when a single atom will decay. However, for a large number of atoms, we can statistically estimate the decay rate. It's important to understand that radioactive decay is a natural process that occurs in all radioactive materials.

To demonstrate the concept, let's take an example: Radium-226 is a radioactive element that decays into Radon-222. Measuring the rate of decay helps scientists understand the longevity and stability of different elements, which is vital in fields such as archaeology (carbon dating), medical diagnostics (imaging), and nuclear energy (reactor stability).

Exponential Decay

Exponential decay is a mathematical model that describes how the quantity of a substance decreases over time at a rate proportional to its current value. This model is essential in understanding not just radioactive decay, but also in areas ranging from finance (compound interest) to biology (population dynamics).

In the context of radioactive decay, the rate at which the substance diminishes follows an exponential curve, implying that it decreases more rapidly at first, and the rate of decay slows down as time passes. The model is characterized by the decay constant and the initial amount of the material. The exponential nature ensures that the same fraction of the substance decays over equal time intervals - for example, half of the substance will decay by the end of each half-life period, regardless of how much material was present initially.

To visualize the principle, think of it this way: If you start with 1000 unstable atoms, after one half-life period you'll have 500; after two half-lives, you'll have 250, and so forth, always reducing to half the amount from the previous period.

Decay Constant

The decay constant, denoted by the symbol 'k', is a pivotal parameter in the exponential decay equation. It is pivotal because it directly influences the rate at which a radioactive substance decays. The decay constant is unique for each radioactive substance and is inversely related to the half-life – substances with a high decay constant decay more quickly, and therefore have a shorter half-life.

Mathematically, the decay constant provides the proportionality between the disappearance rate of the substance and the current amount. In essence, it's a measure of probability: the decay constant represents the likelihood of a single atom decaying in a given time period, allowing calculation of material loss over time.

To illustrate, in a substance with a higher decay constant, a larger number of atoms will transform into a different element in a shorter amount of time compared to a substance with a low decay constant. This directly affects how quickly the material ceases to be radioactive, which is crucial for safety in radioactive waste management and estimating the duration of treatments in radiotherapy.