Question

Calculate how much of a 10.0-g sample of americium-241 remains after four half-lives. Americium-241 is a radioisotope commonly used in smoke detectors and has a half-life of 430 y.

Short answer

After four half-lives, 0.625 grams of americium-241 remain in the 10.0-g sample.

Step-by-step solution

Write down the given information

We are given: Initial amount of americium-241 (N0) = 10.0 g Half-life (t1/2) = 430 years Number of half-lives (n) = 4

Use the half-life formula

The amount of americium-241 remaining after a certain number of half-lives can be found by using the following formula: \(N = N_0 \cdot (\frac{1}{2})^n\) where: N = remaining amount of americium-241 N0 = initial amount of americium-241 n = number of half-lives

Plug in the given values and solve for N

Now, we'll plug in the values for N0 and n, and solve for N: \(N = 10.0 \cdot (\frac{1}{2})^4\) \(N = 10.0 \cdot \frac{1}{16}\) N = 0.625 g After four half-lives, 0.625 grams of americium-241 remain in the 10.0-g sample.

Key concepts

Half-Life

The concept of half-life is key to understanding radioactive decay. It defines the timeframe in which half of a radioactive substance decays into a more stable material. This 'half-life' is constant and unique for each radioisotope. For example, americium-241, which is utilized in household smoke detectors, has a half-life of 430 years.

What does this mean in practice? Let's take a sample of 10 grams of americium-241. After 430 years—just one half-life—only 5 grams of the original substance will remain. The other 5 grams will have transformed into other elements or energy forms. After yet another 430 years, which adds up to two half-lives, only 2.5 grams of the original americium-241 would remain, and the pattern continues.

Understanding half-life allows us to calculate how much of a radioactive substance will be left after a certain time has elapsed, just like in the provided exercise above.

Radioisotope

Radioisotopes are atoms that have unstable nuclei and release energy in order to achieve a more stable state. This energy is emitted in the form of radiation, which can be alpha, beta, or gamma rays. The type of radiation depends on the nature of the radioisotope. In our case, americium-241 is a radioisotope that is commonly used in smoke detectors because it emits alpha particles, which are helpful in detecting smoke particles.

Each radioisotope has practical applications based on its radioactive properties, rate of decay, and the type of radiation it emits. For instance, apart from americium-241, other radioisotopes like carbon-14 are used in archaeological dating, and iodine-131 is used in medical treatments.

Their unstable nature and resulting decay make radioisotopes both useful and sometimes hazardous, which is why understanding their behavior and half-lives is crucial for their safe handling and application.

Exponential Decay

Radioactive decay follows a pattern known as exponential decay, which describes a rapid decrease at first, that slows down over time. This rate of decay is not linear; it does not decrease by the same amount each time period. Instead, the reduction is a percentage of the remaining amount. This is why we use an exponential function to model this behavior.

The mathematical expression for this model is typically written as:
\[N = N_0 \cdot (\frac{1}{2})^n\]
where \(N\) is the remaining amount of substance, \(N_0\) is the initial amount, and \(n\) is the number of half-lives elapsed.

With each passing half-life, the quantity of the substance is divided by two, illustrating why the graph of the function is a curve that approaches zero without ever reaching it. This concept is crucial in fields ranging from environmental science to nuclear engineering, as it helps predict how substances decrease over time.