A radioactive sample contains 2.45 g of an isotope with a half-life of 3.8 days. How much of the isotope in grams remains after 11.4 days?

Radioactive decay is a fundamental concept in physics and chemistry, describing the process through which an unstable atomic nucleus loses energy by emitting radiation. As a natural phenomenon, it occurs in all radioactive substances, including isotopes, which are variants of chemical elements with different numbers of neutrons.

When discussing radioactive decay, it's crucial to understand that it occurs randomly at a level of single atoms and is measured statistically across a large number of atoms. However, it is described by a consistent rate characteristic to each radioactive isotope, which can be quantified in terms of 'half-life.' A vital aspect of radioactive decay is its spontaneity; decay will proceed regardless of the physical state or chemical combination of the element.

To improve comprehension of this complex material, it's important to visualize radioactive decay as an ongoing process. Imagine if you had a pile of coins and with each flip, half of them always turned up tails and were removed from the pile. Similarly, in radioactive decay, atoms 'flip' and transform into different atoms or isotopes over time by emitting particles, such as alpha particles, beta particles, or gamma rays. The rate of this decay presents a consistent pattern that is modeled by exponential decay.

The half-life formula is a mathematical expression used to describe the rate at which a radioactive substance decays over time. A half-life is defined as the time it takes for half of the radioactive isotope in a sample to decay. This concept is imperative when trying to understand how much of a substance remains after a given period.

The formula for calculating the remaining amount of a substance after a certain number of half-lives is expressed as:\[\begin{equation}Remaining\,mass = Initial\,mass \times \left(\frac{1}{2}\right)^{Number\,of\,half-lives}\end{equation}\]In this equation, the initial mass refers to the mass of the radioactive sample before decay has begun, and the number of half-lives is calculated by dividing the elapsed time by the substance's half-life.

As an example, applying the half-life formula can help us understand exercises involving the decay of isotopes. For your quick reference, the calculation for the number of half-lives is an integer division of the time elapsed by the individual half-life of the substance. Following this formula, students can chart the exponential decay of an isotope and the corresponding reduction in mass over time.

At the core of the half-life formula lies the principle of exponential decay, which describes how quantities decrease rapidly and then level off over time. In the context of radioactive decay, this principle illustrates that the quantity of a radioactive isotope decreases by half over each half-life period. What is notable about exponential decay is that it occurs at a rate proportional to the remaining amount, thus generating a predictable decay pattern.

Understanding Exponential Curves

The decay over time can be plotted on a graph, with the curve starting steep and tapering off as time progresses, which visually represents the concept of a quantity being halved repeatedly over equal intervals. The steepness of the initial curve shows the rapid loss of material at the beginning of the decay process.

Mathematical Representation

Mathematically, exponential decay is expressed by the formula:\[\begin{equation}N(t) = N_0 e^{-\lambda t}\end{equation}\]where:

• N(t) is the quantity at time t,
• N_0 is the initial quantity,
• e is the base of the natural logarithm,
• λ is the decay constant (which is related to the half-life).

One can observe that this expression exponentiates to the negative power, portraying the essential characteristic of decay—reduction over time. Understanding exponential decay helps in comprehending a wide range of phenomena, from the half-life of radioactive substances to the depreciation of asset values or the spread of certain diseases.