/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 36 In 2010 , the United States used... [FREE SOLUTION] | 91影视

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In 2010 , the United States used approximately \(1.4 \times 10^{19} \mathrm{~J}\) of elec- trical energy. If all this energy came from the fission of \({ }^{235} \mathrm{U},\) which releases \(200 \mathrm{MeV}\) per fission event, (a) how many kilograms of \({ }^{235} \mathrm{U}\) would have been used during the year? (b) How many kilograms of uranium would have to be mined to provide that much \({ }^{235} \mathrm{U} ?\) (Recall that only \(0.70 \%\) of naturally occurring uranium is \({ }^{235} \mathrm{U} .\) )

Short Answer

Expert verified
(a) About 1.71脳10鈦 kg of 虏鲁鈦礥 is used. (b) About 2.44脳10鈦 kg of uranium needs to be mined.

Step by step solution

01

Convert Energy Released per Fission Event to Joules

The energy released per fission event is given as 200 MeV. We need to convert this energy into joules using the conversion factor: \[ 1 \, \text{MeV} = 1.602 \times 10^{-13} \, \text{J} \]Thus, for 200 MeV, the energy in joules is: \[ 200 \, \text{MeV} \times 1.602 \times 10^{-13} \, \text{J/MeV} = 3.204 \times 10^{-11} \, \text{J} \]
02

Find Number of Fission Events

We need to find out how many fission events are required to generate a total energy of \(1.4 \times 10^{19} \, \text{J}\): \[ \text{Number of events} = \frac{1.4 \times 10^{19} \, \text{J}}{3.204 \times 10^{-11} \, \text{J/event}} \approx 4.37 \times 10^{29} \text{ events} \]
03

Calculate Mass of Used \(^{235}U\)

Find the mass of \(^{235}U\) needed. The atomic mass of \(^{235}U\) is approximately 235 grams per mole, and Avogadro's number is \(6.022 \times 10^{23} \text{ atoms/mole}\):First, calculate the number of moles:\[ \text{Number of moles} = \frac{4.37 \times 10^{29} \text{ atoms}}{6.022 \times 10^{23} \text{ atoms/mole}} \approx 7.26 \times 10^{5} \text{ moles} \]Next, find the mass:\[ \text{Mass} = 7.26 \times 10^{5} \text{ moles} \times 235 \text{ g/mole} \times \frac{1 \text{ kg}}{1000 \text{ g}} \approx 1.71 \times 10^{5} \text{ kg} \]
04

Calculate Mass of Mined Uranium

Since only 0.70% of naturally occurring uranium is \(^{235}U\), the mass of the mined uranium needs to be:\[ \text{Mass of mined uranium} = \frac{1.71 \times 10^{5}}{0.007} \approx 2.44 \times 10^{7} \text{ kg} \]

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Key Concepts

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

Nuclear Physics
Nuclear physics is a fascinating field that deals with the nucleus of atoms, studying their structure, components, and interactions. One of the most intriguing aspects of nuclear physics is the fission process, which is the splitting of a heavy atomic nucleus into two lighter nuclei, releasing a significant amount of energy in the process. This energy release is due to the mass-to-energy conversion principle described by Einstein's famous equation \(E=mc^2\). In the nuclear fission process, a small amount of mass is converted into a massive amount of energy.
This principle is particularly useful for producing energy because it can be harnessed in nuclear reactors to generate electricity. Understanding nuclear reactions provides insights into not just electricity generation, but also the principles of nuclear weapons, the origins of chemical elements, and the behavior of astronomical bodies.
Energy Conversion
Energy conversion in the context of uranium fission involves transforming the nuclear energy released during fission into usable electrical energy. In a nuclear power plant, the heat generated from fission reactions is used to produce steam, which then drives turbines connected to generators, converting kinetic energy into electrical energy.
This conversion process highlights the efficiency challenges because not all the energy released from fission can be transformed into electricity. Various losses occur due to heat dissipation and system inefficiencies, affecting the overall energy yield. Nonetheless, nuclear fission remains a potent source of energy, providing a significant share of electricity in many countries, emphasizing its role in energy sustainability and reduction of greenhouse gas emissions.
Uranium Isotopes
Uranium isotopes are different forms of the uranium element, distinguished by the number of neutrons in the nucleus. The most notable isotope is \(^{235}U\), which is the primary fuel used in nuclear reactors because it is fissile, meaning it can sustain a chain reaction. Another isotope is \(^{238}U\), more abundant but not directly usable for fission in most types of reactors.
Only about 0.7% of naturally occurring uranium is \(^{235}U\), which makes it rare and valuable for nuclear power generation. This scarcity requires mining large amounts of uranium ore to extract enough \(^{235}U\) for energy production. Consequently, refining and enriching uranium is crucial to preparing it for use as reactor fuel, involving complex processes to increase the proportion of \(^{235}U\) to a levels sufficient for efficient fission reactions.
Fission Reactions
Fission reactions occur when a heavy nucleus, such as \(^{235}U\), absorbs a neutron and becomes unstable, splitting into two smaller nuclei along with several neutrons and a large amount of energy. This chain reaction occurs because the emitted neutrons can initiate further fission events with nearby nuclei.
Each fission event of \(^{235}U\) releases around 200 MeV of energy, a substantial amount for a single atomic event, and emphasizes why fission is so effective for power generation. By maintaining a controlled chain reaction, nuclear reactors can harness these reactions steadily, providing a continuous power supply. Ensuring safety and maintaining control over the nuclear reactions are essential, highlighting the need for advanced technology and robust systems in nuclear power plants.

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Most popular questions from this chapter

In 1952, spectral lines of the element technetium-99 ( \(\left.{ }^{99} \mathrm{Tc}\right)\) were discovered in a red-giant star. Red giants are very old stars, often around 10 billion years old, and near the end of their lives. Technetium has no stable isotopes, and the half-life of \({ }^{99} \mathrm{Tc}\) is 200,000 years. (a) For how many half-lives has the \({ }^{99} \mathrm{Tc}\) been in the red-giant star if its age is 10 billion years? (b) What fraction of the original \({ }^{99} \mathrm{Tc}\) would be left at the end of that time?

It has become popular for some people to have yearly whole-body scans (CT scans, formerly called CAT scans), using X-rays, just to see if they detect anything suspicious. A number of medical people have recently questioned the advisability of such scans, due in part to the radiation they impart. Typically, one such scan gives a dose of \(12 \mathrm{mSv},\) applied to the whole body. By contrast, a chest X-ray typically administers \(0.20 \mathrm{mSv}\) to only \(5.0 \mathrm{~kg}\) of tissue. How many chest X-rays would deliver the same total amount of energy to the body of a \(75-\mathrm{kg}\) person as one whole-body scan?

The common isotope of uranium, \({ }^{238} \mathrm{U},\) has a half-life of \(4.47 \times 10^{9}\) years, decaying to \({ }^{234} \mathrm{Th}\) by alpha emission. (a) \(\mathrm{What}\) is the decay constant? (b) What mass of uranium is required for an activity of 1.00 curie? (c) How many alpha particles are emitted per second by \(10.0 \mathrm{~g}\) of uranium?

Assuming that \(200 \mathrm{MeV}\) is released per fission, how many fissions per second take place in a \(100 \mathrm{MW}\) reactor?

(a) If a chest X-ray delivers \(0.25 \mathrm{mSv}\) to \(5.0 \mathrm{~kg}\) of tissue, how many total joules of energy does this tissue receive? (b) Natural radiation and cosmic rays deliver about \(0.10 \mathrm{mSv}\) per year at sea level. Assuming an \(\mathrm{RBE}\) of \(1,\) how many rem and rads is this dose, and how many joules of energy does a 75-kg person receive in a year? (c) How many chest X-rays like the one in part (a) would it take to deliver the same total amount of energy to a \(75-\mathrm{kg}\) person as she receives from natural radiation in a year at sea level, as described in part (b)?

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