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Chapter 19: Problem 80

Estimate the temperature needed to achieve the fusion of deuterium to make an \(\alpha\) particle. The energy required can be estimated from Coulomb's law [use the form \(E=9.0 \times 10^{9}\) \(\left(Q_{1} Q_{2} / r\right)\), using \(Q=1.6 \times 10^{-19} \mathrm{C}\) for a proton, and \(r=2 \times\) \(10^{-15} \mathrm{~m}\) for the helium nucleus; the unit for the proportionality constant in Coloumb's law is \(\left.\mathrm{J} \cdot \mathrm{m} / \mathrm{C}^{2}\right]\).

Short Answer

Expert verified
Using Coulomb's Law formula, given values for the charges and the helium nucleus's distance, we can calculate the energy required for the fusion of deuterium to make an 伪 particle. After obtaining the energy (E) in Joules, we can estimate the necessary temperature for the fusion using the kinetic energy formula with the Boltzmann constant. The temperature (T) in Kelvin can be calculated with the equation \(T = \frac{2E}{3k}\), where k is the Boltzmann constant \(1.38 \times 10^{-23} \mathrm{J/K}\). Plug in the energy value and the Boltzmann constant to compute the required temperature for the fusion process.

Step by step solution

01

Using the Coulomb's Law formula: \(E = 9.0 \times 10^{9} \left(\frac{Q_1 Q_2}{r}\right)\), we have to find the energy required for the fusion of deuterium to make an 伪 particle. Given, \(Q_1\) and \(Q_2\) are the charges of the helium nucleus, which can be expressed as the charge of a proton \(1.6 \times 10^{-19} \mathrm{C}\), and the distance \(r = 2 \times 10^{-15} \mathrm{m}\). We'll plug these values into the formula to find the energy (E) required. #Step 2: Substitute the given values and calculate the energy#

Now, we will substitute the given values into the formula: \(E = 9.0 \times 10^{9} \left(\frac{(1.6 \times 10^{-19})(1.6 \times 10^{-19})}{2 \times 10^{-15}}\right)\) Now calculate the energy (E) by performing the required operations. #Step 3: Convert the energy to temperature#
02

Once we have calculated the energy (E) in Joules required for the fusion process, we can estimate the temperature needed for the fusion using the kinetic energy formula and the energy calculated: \(E = \dfrac{3}{2}kT\), where k is the Boltzmann constant \(k = 1.38 \times 10^{-23} \mathrm{J/K}\), and \(T\) is the required temperature in Kelvin. Now solve for the temperature (T). #Step 4: Estimate the temperature needed for the fusion of deuterium to 伪 particles#

With the calculated energy (E) and the Boltzmann constant (k) in hand, we can finally estimate the temperature (T) needed for the fusion of deuterium to form 伪 particles by rearranging the equation: \[T = \frac{2E}{3k}\] Plug in the energy value and the Boltzmann constant, and compute the temperature.

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Most popular questions from this chapter

Natural uranium is mostly nonfissionable \({ }^{238} \mathrm{U} ;\) it contains only about \(0.7 \%\) of fissionable \({ }^{235} \mathrm{U}\). For uranium to be useful as a nuclear fuel, the relative amount of \({ }^{235} \mathrm{U}\) must be increased to about \(3 \%\). This is accomplished through a gas diffusion process. In the diffusion process, natural uranium reacts with fluorine to form a mixture of \({ }^{238} \mathrm{UF}_{6}(g)\) and \({ }^{235} \mathrm{UF}_{6}(g) .\) The fluoride mixture is then enriched through a multistage diffusion process to produce a \(3 \%\) \({ }^{235} \mathrm{U}\) nuclear fuel. The diffusion process utilizes Graham's law of effusion (see Chapter 5, Section 5.7). Explain how Graham's law of effusion allows natural uranium to be enriched by the gaseous diffusion process.

The most significant source of natural radiation is radon- \(222 .\) \({ }^{222} \mathrm{Rn}\), a decay product of \({ }^{238} \mathrm{U}\), is continuously generated in the earth's crust, allowing gaseous Rn to seep into the basements of buildings. Because \({ }^{222} \mathrm{Rn}\) is an \(\alpha\) -particle producer with a relatively short half-life of \(3.82\) days, it can cause biological damage when inhaled. a. How many \(\alpha\) particles and \(\beta\) particles are produced when \({ }^{238} \mathrm{U}\) decays to \({ }^{222} \mathrm{Rn}\) ? What nuclei are produced when \({ }^{222} \mathrm{Rn}\) decays? b. Radon is a noble gas so one would expect it to pass through the body quickly. Why is there a concern over inhaling \({ }^{222} \mathrm{Rn}\) ? c. Another problem associated with \({ }^{222} \mathrm{Rn}\) is that the decay of \({ }^{222} \mathrm{Rn}\) produces a more potent \(\alpha\) -particle producer \(\left(t_{1 / 2}=3.11\right.\) min) that is a solid. What is the identity of the solid? Give the balanced equation of this species decaying by \(\alpha\) -particle production. Why is the solid a more potent \(\alpha\) -particle producer? d. The U.S. Environmental Protection Agency (EPA) recommends that \({ }^{222} \mathrm{Rn}\) levels not exceed \(4 \mathrm{pCi}\) per liter of air \((1 \mathrm{Ci}=\) 1 curie \(=3.7 \times 10^{10}\) decay events per second; \(1 \mathrm{pCi}=1 \times\) \(10^{-12} \mathrm{Ci}\). Convert \(4.0 \mathrm{pCi}\) per liter of air into concentrations units of \(^{222} \mathrm{Rn}\) atoms per liter of air and moles of \({ }^{222} \mathrm{Rn}\) per liter of air.

Using the kinetic molecular theory (Section 5.6), calculate the root mean square velocity and the average kinetic energy of \({ }_{1}^{2} \mathrm{H}\) nuclei at a temperature of \(4 \times 10^{7} \mathrm{~K}\). (See Exercise 46 for the appropriate mass values.)

Consider the following reaction to produce methyl acetate: When this reaction is carried out with \(\mathrm{CH}_{3} \mathrm{OH}\) containing oxygen18, the water produced does not contain oxygen-18. Explain.

Which type of radioactive decay has the net effect of changing a neutron into a proton? Which type of decay has the net effect of turning a proton into a neutron?
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