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Chapter 26: Problem 53

At a typical nuclear power plant, refueling occurs about every 18 months. Assuming that a plant has operated continuously since the last refueling and produces \(1.2 \mathrm{GW}\) of electric power at an efficiency of \(33 \%\), how much less massive are the fuel rods at the end of the 18 months than at the start? (Assume 30 -day months.)

Short Answer

Expert verified
The fuel rods are approximately 0.1746 kg less massive at the end of 18 months.

Step by step solution

01

Calculate Total Energy Produced

First, we determine the total energy produced by the nuclear power plant over 18 months. Given that the plant operates at a power output of \(1.2 \text{GW}\), we calculate the energy output in kilojoules. One month has 30 days, so 18 months have \(18 \times 30 \times 24 \times 3600\) seconds. The total energy produced is:\[E = 1.2 \times 10^9 \text{W} \times 18 \times 30 \times 24 \times 3600 = 5.184 \times 10^{15} \text{J}\]
02

Account for Efficiency

Since the nuclear power plant operates at an efficiency of 33%, not all nuclear energy is converted to electrical energy. We need to calculate the total nuclear energy actually produced. The energy extracted from nuclear reactions is:\[ E_{\text{nuclear}} = \frac{E}{0.33} = \frac{5.184 \times 10^{15}}{0.33} = 1.5715 \times 10^{16} \text{J}\]
03

Calculate Mass Loss Using E=mc^2

According to Einstein's equation \(E = mc^2\), where \(E\) is the energy, \(m\) is the mass change, and \(c\) is the speed of light \((3 \times 10^8 \text{m/s})\). We can rearrange the equation to solve for the mass change:\[ m = \frac{E_{\text{nuclear}}}{c^2} = \frac{1.5715 \times 10^{16}}{(3 \times 10^8)^2} \]Calculate:\[ m = \frac{1.5715 \times 10^{16}}{9 \times 10^{16}} = 0.1746 \text{kg}\]
04

Conclusion

Thus, the mass lost by the fuel rods as they produce energy over 18 months is approximately \(0.1746\) kg.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy Conversion in Nuclear Power Plants
A nuclear power plant generates electricity by converting nuclear energy into electrical energy. This process primarily involves the fission of uranium atoms. When a uranium atom splits, it releases a considerable amount of energy, which heats water to produce steam.
The steam then drives turbines that activate electricity generators. This is the basic cycle of energy conversion in a nuclear plant.
  • The energy originates from nuclear reactions within the reactor core.
  • Heat is transferred from the reactor to water, converting it to steam.
  • Steam is used to spin turbines connected to electrical generators.
Understanding this chain of conversion helps us calculate how much energy is produced and the efficiency of the process. Not all energy is converted to electricity due to losses in the system, which leads us directly into the next concept.
Mass-Energy Equivalence in Nuclear Reactions
The principle of mass-energy equivalence, encapsulated by Einstein鈥檚 famous equation \(E=mc^2\), plays a critical role in nuclear energy. This equation implies that mass can be converted into energy and vice versa.
Within nuclear power plants, even a small loss in mass results in the production of a considerable amount of energy.When we apply \(E=mc^2\):
  • \(E\) is the energy liberated in the reaction.
  • \(m\) is the mass that is converted into energy.
  • \(c\) is the speed of light, \(3 \times 10^8\) meters per second, a large number demonstrating how a tiny mass converts into high energy output.
In the given problem, this principle explains how the diminishing mass of the fuel rods correlates with the energy produced in the nuclear plant.
Efficiency Calculation of Nuclear Power Plants
Efficiency in nuclear power plants is about how well they convert nuclear energy to electrical energy. The stated efficiency is 33%, meaning just a third of the nuclear energy actually becomes electrical energy.
Calculating efficiency involves understanding the proportion of energy effectively used for power generation.Here's how we account for it:
  • The formula used is \( \text{Efficiency} = \frac{\text{Electrical Energy Output}}{\text{Total Nuclear Energy Produced}} \).
  • With an efficiency of 33%, the energy output after energy losses aligns with the equation result.
  • This implies out of all the energy produced, only a part is transformed into electric power due to systemic losses and inherent inefficiencies.
Efficiency calculations are essential as they reflect the plant's performance in terms of energy utilization and guide improvements to minimize energy losses.

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Most popular questions from this chapter

Sirius is about 9.0 light-years from Earth. (a) To reach the star by spaceship in 12 years (ship time), how fast must you travel? (b) How long would the trip take according to an Earth-based observer? (c) How far is the trip according to you?

One of a pair of 25 -year-old twins takes a round trip through space while the other twin remains on Earth. The traveling twin moves at a speed of \(0.95 c\) for a total of 39 years, according to Earth time. Assuming that special relativity applies for the entire trip (that is, neglect accelerations at the start, end, and turnaround), (a) what are the twins' ages when the traveling twin returns to Earth? (b) On the round trip, how far did the traveling twin go according to the traveling twin? (c) On the round trip, how far did the traveling twin go according to the Earth twin? Which of your answers to parts (b) and (c) are proper lengths, if any?

You fly your 15.0 -m-long spaceship at a speed of \(c / 3\) relative to your friend. Your velocity is parallel to the ship's length. (a) How long is your spaceship, as observed by your friend? (b) What is the speed of your friend relative to you?

The distance to Planet X from Earth is 1.00 light-year. (a) How long does it take a spaceship to reach \(X\), according to the pilot of the spaceship, if the speed of the ship is \(0.700 c\) relative to \(X ?\) (b) How long does it take the ship to make the trip according to an astronaut already stationed on Planet \(X ?\) (c) Determine the distance between Earth and Planet \(X\) according to the pilot and according to the X-based astronaut and explain why the two answers are different.

A small airplane has an airspeed (speed with respect to air) of \(200 \mathrm{~km} / \mathrm{h}\). Find the time for the airplane to travel \(1000 \mathrm{~km}\) if there is (a) no wind, (b) a headwind of \(35 \mathrm{~km} / \mathrm{h},\) and \((\mathrm{c})\) a tailwind of \(35 \mathrm{~km} / \mathrm{h}\).
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