Typical house driven by nuclear fusion
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The following quantities appear in the problem:
Zeit \(t\) / Masse \(m\) / Arbeit \(W\) / Energie \(E\) / Leistung \(P, \Phi\) / molare Masse \(M\) / Stoffmenge \(n\) / Anzahl \(N\) / Wirkungsgrad \(\eta\) /
The following formulas must be used to solve the exercise:
\(P = \dfrac{E}{t} = \dfrac{W}{t} = \dfrac{Q}{t} \quad \) \(E = mc^2 \quad \) \(\Delta m = \sum m_{\rm Edukte} - \sum m_{\rm Produkte} \quad \) \(m = nM \quad \) \(E = N \cdot E_1 \quad \) \(\eta = \dfrac{W_{\rm nutz}}{E_{\rm zu}} = \dfrac{P_{\rm nutz}}{P_{\rm zu}} \quad \)
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Exercise:
If a typical house requires PO of electric power on average what minimum of deuterium fuel would have to be used in a year to supply these electrical needs? Ase the fusion reaction isotopeD+isotopeD rightarrow isotopeHe+mboxn. The mass of deuterium is mDO the mass of Helium- mHeO and the neutron has a mass of ncmnu.
Solution:
If we take the mass difference of the considered reaction we get Delta m sscmD - sscmHe - sscmn mD - mHe - ncmnu dmu dmk. According to Einstein's energy-mass-equivalence this corresponds to E_ Delta m c^ dmk qtyncc^ Ep EpM of energy which comes out of the reaction as kinetic energy i.e. velocity of the neutron. A typical house requires E Pt P t Eh EhM per year. The fusion reaction above hence needs to run N' fracEE_ fracEhEp N times. Each reaction requires two isotopeH atoms. Therefore N N' N Nz n nQ isotopeH atoms are required. They have a mass of m n M n M m mgQ.
If a typical house requires PO of electric power on average what minimum of deuterium fuel would have to be used in a year to supply these electrical needs? Ase the fusion reaction isotopeD+isotopeD rightarrow isotopeHe+mboxn. The mass of deuterium is mDO the mass of Helium- mHeO and the neutron has a mass of ncmnu.
Solution:
If we take the mass difference of the considered reaction we get Delta m sscmD - sscmHe - sscmn mD - mHe - ncmnu dmu dmk. According to Einstein's energy-mass-equivalence this corresponds to E_ Delta m c^ dmk qtyncc^ Ep EpM of energy which comes out of the reaction as kinetic energy i.e. velocity of the neutron. A typical house requires E Pt P t Eh EhM per year. The fusion reaction above hence needs to run N' fracEE_ fracEhEp N times. Each reaction requires two isotopeH atoms. Therefore N N' N Nz n nQ isotopeH atoms are required. They have a mass of m n M n M m mgQ.