Question

Radioactive Decay What percent of a present amount of radioactive cesium \(\left({ }^{137} \mathrm{Cs}\right)\) will remain after 100 years? Use the fact that radioactive cesium has a half-life of 30 years.

Short answer

The remaining percent of Cesium-137 after 100 years can be found using the decay function derived using the decay constant and the given half-life. The exact value can be calculated by substituting the time (100 years) into this decay function.

Step-by-step solution

Find the Decay Constant

The decay constant (\(λ\)) can be found using the following formula: \(λ = \frac{\ln{2}}{\text{half-life}}\). Here, the half-life of Cesium-137 is given as 30 years. So, \(λ = \frac{\ln{2}}{30}\). Calculate it to obtain the value of \(λ\).

Determine the Decay Function

The decay function which models the percent remaining of a substance after a certain time is given by: \(N(t) = N_{0}e^{-λt}\), where \(N_{0}\) is the initial amount of the substance, \(λ\) is the decay constant we have just found, \(t\) is the time, and \(N(t)\) is the remaining amount after time \(t\). Since we are asked to find the percentage remaining, let \(N_{0}\) = 100%. Then, our function becomes: \(N(t) = 100e^{-λt}\).

Apply the Decay Function for 100 years

We are to find the remaining amount after 100 years. So, replace \(t\) with 100 in the decay function calculated in step 2 to get \(N(100) = 100e^{-λ*100}\). Calculate it to get the percentage remaining.

Key concepts

Decay Constant Formula

The decay constant formula is pivotal in the realm of nuclear physics, especially when it deals with radioactive decay. To understand how quickly a radioactive substance diminishes over time, we use this formula. Mathematically, it's represented by \( \lambda = \frac{\ln{2}}{\text{half-life}} \).

For cesium-137 \( {}^{137}Cs \), which has a half-life of 30 years, finding the decay constant \( \lambda \) allows us to predict its rate of decay. We simply plug in the half-life to get \( \lambda = \frac{\ln{2}}{30} \). The natural logarithm of 2, \( \ln{2} \), is a constant stemming from the fact that half-life is the time taken for half of the radioactive substance to decay. This decay constant provides the basis for creating an exponential decay model, which we can apply to determine how much cesium-137 remains after any given time frame, such as 100 years.

Half-Life Calculation

The concept of half-life is crucial for scientists and engineers who work with materials that undergo radioactive decay. It represents the period required for half of a radioactive substance to decay. In calculations involving half-life, we use the concept to both estimate the decay rate and predict how much of the substance will remain at a future point.

To calculate the half-life of a substance, you could rearrange the decay constant formula: from \( \lambda = \frac{\ln{2}}{\text{half-life}} \) to \( \text{half-life} = \frac{\ln{2}}{\lambda} \). However, since we often know the half-life, as is the case with cesium-137, we use it to find the decay constant. Moreover, understanding half-life enables us to set up the exponential decay function and directly apply it to real-world problems, such as determining the percent of a radioactive substance remaining after a specific number of years.

Exponential Decay Function

An exponential decay function is ideal for modeling the decline of a radioactive substance over time. This mathematical representation is expressed as \( N(t) = N_0e^{-\lambda t} \), where \( N_0 \) is the initial quantity, \( \lambda \) is our decay constant, \( t \) is time, and \( N(t) \) signifies the remaining quantity after time \( t \).

For our cesium-137 example, with an initial amount of 100%, we rewrite the formula as \( N(t) = 100e^{-\lambda t} \). When we want to find out what percentage of cesium-137 remains after a century, we plug \( t = 100 \) years and the previously calculated decay constant into our exponential decay function. We get \( N(100) = 100e^{-\lambda \times 100} \). This result will give us the percentage of cesium-137 that has not decayed after 100 years, helping us to understand the longevity and potential risks of the substance in the environment or for its use in various applications.