The Galileo space probe was launched on its long journey past Venus and Earth in \(1989,\) with an ultimate goal of Jupiter. Its power source is \(11.0 \mathrm{kg}\) of \(^{238} \mathrm{Pu}\) a by-product of nuclear weapons plutonium production. Electrical energy is generated thermoelectrically from the heat produced when the \(5.59-\mathrm{MeV} \quad \alpha\) particles emitted in each decay crash to a halt inside the plutonium and its shielding. The half-life of \(^{238} \mathrm{Pu}\) is 87.7 years. (a) What was the original activity of the \(^{238} \mathrm{Pu}\) in becquerels? (b) What power was emitted in kilowatts? (c) What power was emitted 12.0 y after launch? You may neglect any extra energy from daughter nuclides and any losses from escaping \(\gamma\) rays.

Understanding radioactive decay is crucial for fields like nuclear physics and radiology. It's the process by which an unstable atomic nucleus loses energy by emitting radiation, such as alpha particles, beta particles, or gamma rays. Each of these emissions has different properties and penetrative abilities.

Alpha decay, for example, involves the release of alpha particles, which are helium-4 nuclei. These particles consist of two protons and two neutrons bound together. Although they don't penetrate very far into materials and can be stopped by a sheet of paper, when emitted within a material—like the plutonium used on the Galileo space probe—they release significant amounts of energy. This energy release is what enables thermoelectric power generation on spacecraft.

Radioactive decay happens randomly but at a rate characteristic to each isotope, which is measured by the term 'activity.' The unit of activity is the becquerel (Bq), with one Bq corresponding to one decay per second. Knowing the activity level of a substance is essential in calculating the amount of energy it can produce, which directly affects the design and functionality of devices like space probes.

The concept of half-life is pivotal when studying nuclear physics and predicting the behavior of radioactive substances. A half-life is the time required for half of the radioactive nuclei in a sample to undergo decay. This decay process follows an exponential decay model, meaning the quantity of undecayed nuclei decreases exponentially over time.

In calculations, especially when dealing with power generation from radioactive sources, understanding the half-life of the material in question is critical. Knowing that plutonium-238 has a half-life of 87.7 years allows scientists to calculate the diminishing power output over time, essential when planning long-term space missions such as the Galileo probe's journey. Even though the activity reduces over time, due to the significant half-life of plutonium-238, the power output only decreases marginally over the first few years, ensuring a reliable power source for the spacecraft's instruments.

Thermoelectric power generation is an innovative technology that converts heat directly into electricity using thermoelectric materials. These materials generate power through the Seebeck effect, where an electric potential is produced between two different conductive materials at different temperatures.

In the case of the Galileo space probe, the heat source is the radioactive decay of plutonium-238. The energy from this decay is transformed into power without any moving parts, which is ideal for spacecraft because it limits the number of potential mechanical failures. However, the efficiency of thermoelectric generators is generally lower than other power conversion methods, which is why their usage is primarily in situations where long-term, maintenance-free operation is more critical than efficiency.

Through precise calculations involving the emitted energy per decay and the current activity of the radioactive material, as in the exercise provided, scientists can determine the power output that a thermoelectric generator can provide at any given time. As part of the calculation, understanding the reduction in activity due to half-life must be taken into account to accurately predict the future power availability.