Question

Radioactive plutonium- \(239\left(t_{\frac{1}{2}}=2.44 \times 10^{5} \mathrm{yr}\right)\) is used in nuclear reactors and atomic bombs. If there are \(5.0 \times 10^{2} \mathrm{~g}\) of the isotope in a small atomic bomb, how long will it take for the substance to decay to \(1.0 \times 10^{2} \mathrm{~g},\) too small an amount for an effective bomb? (Hint: Radioactive decays follow first-order kinetics.)

Short answer

The time it will take for the substance to decay to \(1.0 \times 10^{2} \mathrm{~g}\) is found by evaluating the expression given in step 3. You can perform the calculations using a calculator, making sure to use natural logarithms (ln) rather than base-10 logarithms.

Step-by-step solution

Recognize the problem's variables

In this problem, the initial amount of the substance \(N_{0}\) is \(5.0 \times 10^{2} \mathrm{~g}\), the final amount of the substance \(N\) is \(1.0 \times 10^{2} \mathrm{~g}\), and the half-life of the substance \(t_{\frac{1}{2}}\) is \(2.44 \times 10^{5} \mathrm{~yr}\). Our task is to find the time it will take for the substance to decay, t.

Express the decay constant in terms of the half-life

The decay constant k can be expressed in terms of the half-life using the equation \(k = ln(2) / t_{\frac{1}{2}}\). Substituting the given half-life into this equation gives \(k = ln(2) / (2.44 \times 10^{5})\).

Use the first-order decay equation to find the time

We rearrange the formula \(N = N_{0}e^{-kt}\) to solve for t, yielding \(t = -ln(N / N_{0}) / k\). Substituting the values for N, \(N_{0}\) and k gives \(t = -ln((1.0 \times 10^{2}) / (5.0 \times 10^{2})) / (ln(2) / (2.44 \times 10^{5}))\).

Key concepts

Nuclear Reactors

Nuclear reactors are devices that control nuclear fission, a process where atomic nuclei split, releasing energy. People often use these reactors in power plants to generate electricity. The key to a reactor's function is the fuel, often a radioactive isotope like plutonium-239. When plutonium undergoes fission, it releases heat and neutrons.

• Heat produced during fission boils water, generating steam.
• Steam moves turbines connected to generators, creating electricity.
• Neutrons continue the reaction, as they collide with other atomic nuclei.

Reactor designs are carefully crafted to manage the fission reaction safely and efficiently. A key element in this process is the control of the reaction rate. Depending on the isotope's properties, like its half-life, the reaction can vary significantly. Understanding the decay characteristics of a fuel, such as first-order kinetics, helps in predicting how long the fuel will last, which is crucial for safety and efficiency in design.

First-order Kinetics

Radioactive decay is commonly modeled as a first-order kinetic process. This model assumes that the rate of decay of a substance is directly proportional to the amount present. In other words, the more substance there is, the faster it decays.

• The general formula for first-order kinetics is given by: \[N = N_{0}e^{-kt}\]
• Here, \(N\) is the amount remaining after time \(t\), \(N_{0}\) is the initial amount, and \(k\) is the decay constant.
• First-order kinetics is used for calculating the radioactive decay over time.

Understanding this model allows one to determine how quickly a substance will decay and predict how much time will pass before it becomes non-hazardous. This is particularly important in contexts like nuclear reactors and atomic bombs, where precise knowledge of decay rates can help control energy output or ensure safety.

Half-life

The concept of half-life is central to understanding radioactive decay. It is defined as the time required for half of a given sample of a radioactive substance to decay. This remains constant, regardless of the initial amount of substance present.

• For example, plutonium-239 has a half-life of \(2.44 \times 10^{5}\) years.
• This constancy means that in every half-life period, the substance's amount reduces by half.
• Knowing the half-life helps calculate how long it will take for the substance to reach a safe level.

The half-life not only helps in determining safety and efficiency in nuclear reactors but also in strategic military applications. Using the half-life formula \[k = \frac{ln(2)}{t_{1/2}}\], where \(t_{1/2}\) is the half-life, scientists can derive the decay constant \(k\), which is critical in solving decay equations. Thus, mastering these understandings is essential when dealing with radioactive materials.