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Chapter 31: Problem 54

Consider a system of nuclear power plants that produce 2100MW. (\(a\)) What total mass of \(^{235}_{92}\)U fuel would be required to operate these plants for 1 yr, assuming that 200 MeV is released per fission? (\(b\)) Typically 6% of the \(^{235}_{92}\)U nuclei that fission produce strontium \(-\)90, \(^{90}_{38}\)Sr, a \(\beta^-\) emitter with a half-life of 29 yr. What is the total radioactivity of the \(^{90}_{38}\)Sr, in curies, produced in 1 yr? (Neglect the fact that some of it decays during the 1-yr period.)

Short Answer

Expert verified
(a) Requires about 80,690 kg of 235U; (b) Produces approximately 4.93 million curies of 90Sr.

Step by step solution

01

Convert Power to Energy

First, note that the power output is 2100 MW. Over the course of one year (365 days), we need to convert this power to energy in joules; knowing that 1 year is equal to 31,536,000 seconds. Therefore, we calculate the total energy produced in one year as:\[ E = 2100 \times 10^6 \text{ W} \times 31,536,000 \text{ s} \]This results in:\[ E = 66.2256 \times 10^{15} \text{ J} \]
02

Convert Energy Per Fission Event

We know from the problem that 200 MeV is released per fission. We need to convert this to joules where 1 MeV = \(1.60218 \times 10^{-13}\) J:\[ 200 \text{ MeV} = 200 \times 1.60218 \times 10^{-13} \text{ J} = 3.20436 \times 10^{-11} \text{ J/fission} \]
03

Calculate Number of Fissions Required

Using the total energy produced over the year and the amount of energy per fission, the number of fissions can be calculated with:\[ N = \frac{66.2256 \times 10^{15}}{3.20436 \times 10^{-11}} \]Calculating the above expression gives:\[ N = 2.0679 \times 10^{26} \text{ fissions} \]
04

Calculate Mass of Uranium Required

Each fission event involves one uranium nucleus. To find the mass of uranium, we use Avogadro's number, \(6.022 \times 10^{23}\), and the molar mass of \(^{235}_{92}U\) which is 235 g/mol:\[ \text{Mass} = \frac{2.0679 \times 10^{26}}{6.022 \times 10^{23}} \times 235 \text{ g} \]This results in an approximate mass of:\[ 8.069 \times 10^4 \text{ kg} \]
05

Calculate Amount of Strontium-90 Produced

6% of the fission reactions produce Strontium-90. Therefore, the number of \(^{90}_{38}Sr\) nuclei produced is:\[ N_{Sr} = 0.06 \times 2.0679 \times 10^{26} \]This results in:\[ N_{Sr} = 1.2407 \times 10^{25} \]
06

Convert to Curies

Each curie (Ci) equals \(3.7 \times 10^{10}\) decays per second. To find the total curies, calculate the decay rate using the number of nuclei and the half-life, 29 years. The decay constant \(\lambda\) is given by:\[ \lambda = \frac{\ln(2)}{T_{1/2}} = \frac{0.693}{29 \times 365 \times 86400} \text{ s}^{-1} \]The activity \(A\) in becquerels (1/s) is:\[ A = \lambda \times N_{Sr} \]Convert this to curies:\[ \text{Activity in Curies} = \frac{A}{3.7 \times 10^{10}} \]Calculate these expressions based on provided constants to determine the activity in curies.

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Key Concepts

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

Nuclear Power Plants
Nuclear power plants are facilities that generate electricity through the process of nuclear fission. This is when the nucleus of an atom splits into two or smaller nuclei. Most commonly, uranium-235 is used as fuel in these plants. The fission of uranium-235 releases a significant amount of energy, which is then converted into electricity.
  • Nuclear power plants are designed to manage the nuclear fission process safely and efficiently.
  • They produce electricity without emitting greenhouse gases, unlike fossil fuel plants.
  • Water plays a crucial role in nuclear power plants as both a coolant and in generating steam to drive turbines.
Each plant is equipped with several layers of safety systems to prevent any accidents or to mitigate them should they occur. Understanding how nuclear power plants operate is vital to appreciate their role in meeting the world's energy demands.
Uranium-235
Uranium-235 (\(^{235}_{92}\text{U}\)) is a naturally occurring isotope of uranium and a crucial element in nuclear fission. It is one of the few materials capable of sustaining a nuclear chain reaction.
  • During fission, uranium-235 absorbs a neutron and becomes an unstable isotope, leading to its split into smaller elements and the release of a substantial amount of energy (around 200 MeV per fission event).
  • Uranium-235 has a fission probability that makes it especially useful in both nuclear power plants and nuclear weapons.
  • This isotope has a relatively long half-life of about 703.8 million years, which contributes to its suitability as a fuel in nuclear reactors.
Producing energy from uranium-235 is an efficient and powerful way to generate electricity, but it requires careful management to avoid environmental contamination and ensure safety.
Strontium-90 Decay
Strontium-90 (\(^{90}_{38}\text{Sr}\)) is a byproduct of uranium-235 fission. It is a radioactive isotope that emits \(\beta^-\) particles during its decay process.
  • Strontium-90 has a half-life of 29 years, meaning it remains radioactive and poses potential health risks for an extended period.
  • Once released into the environment, Strontium-90 can be absorbed into bones, as it mimics calcium.
  • This can lead to increased risks of bone cancer and leukemia.
Understanding the decay process of strontium-90 is important for managing nuclear waste and assessing the environmental impacts of nuclear power plant operations.
Radioactivity Measurement
Measuring radioactivity is crucial to ensuring safety and monitoring nuclear processes. The radioactivity of a substance is typically measured in terms of its decay rate.
  • The decay constant (\(\lambda\)) is a vital parameter representing the probability of decay per unit time for a radioactive isotope.
  • Activity (\(A\)) quantifies the number of decays per second, often expressed in becquerels or curies (1 curie equals \(3.7 \times 10^{10}\) decays per second).
To calculate the activity of strontium-90 produced in a year, we consider:
  • The decay constant, determined using the half-life formula: \(\lambda = \frac{\ln(2)}{T_{1/2}}\).
  • The number of strontium-90 particles produced during fission.
Accurate measurement and interpretation of radioactivity help in the assessment and mitigation of potential risks associated with nuclear power and the environment.

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Most popular questions from this chapter

(II) If \(^6_3\)Li is struck by a slow neutron, it can form \(^4_2\)He and another nucleus. (\(a\)) What is the second nucleus? (This is a method of generating this isotope.) (\(b\)) How much energy is released in the process?

(I) How many fissions take place per second in a 240-MW reactor? Assume 200 MeV is released per fission.

(III) Assume a liter of milk typically has an activity of 2000 pCi due to \(^{40}_{19}\)K. If a person drinks two glasses (0.5 L) per day, estimate the total effective dose (in Sv and in rem) received in a year. As a crude model, assume the milk stays in the stomach 12 hr and is then released. Assume also that roughly 10\(\%\) of the 1.5 MeV released per decay is absorbed by the body. Compare your result to the normal allowed dose of 100 mrem per year. Make your estimate for (\(a\)) a 60-kg adult, and (\(b\)) a 6-kg baby.

(I) The energy produced by a fission reactor is about 200 MeV per fission. What fraction of the mass of a \(^{235}_{92}\)U nucleus is this?

(II) How much energy is deposited in the body of a 65-kg adult exposed to a 2.5-Gy dose?
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