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Problem 62 Hard Difficulty

Approximately $1$ of every $3300$ water molecules contains one deuterium atom. (a) If all the deuterium nuclei in $1 \mathrm{L}$ of water are fused in pairs according to the $\mathrm{D}-\mathrm{D}$ fusion reaction $^{2} \mathrm{H}+^{2} \mathrm{H} \rightarrow^{3} \mathrm{He}+\mathrm{n}+3.27 \mathrm{MeV},$ how much energy in joules is liberated? (b) What If? Burning gasoline produces approximately $3.40 \times 10^{7} \mathrm{J} / \mathrm{L}$. State how the energy obtainable from the fusion of the deuterium in $1\mathrm{L}$ of water compares with the energy liberated from the burning of $1 \mathrm{L}$ of gasoline.

Answer

$2.65 \times 10^6 \ \text{J}$

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Top Physics 103 Educators
Elyse G.

Cornell University

Andy C.

University of Michigan - Ann Arbor

LB
Liev B.

Numerade Educator

Jared E.

University of Winnipeg

Video Transcript

question 62 states that approximately one out of 3300 water molecules contains the deuterium. Adam put a states based on the reaction that I wrote out here. If all the deuterium nuclei in one liter water refused to create killing three, how much energy would be produced. So this ratio of deteriorate toe water and this relationship So from this we can see that the amount of energy released perfusion is 3.27 a m et. So her reaction there's this much energy that's produced. So in order to determine makes me values I use Ian. The scenario for energy, not that it really matters, is one is your consistent. So in order to find the total amount of energy released in this reaction of all the water, all the deuterium atoms in the one leader amount of water were to be released. Then we simply need to find the energy per reaction, multiply by the number of reactions. So so I mean, I'll say this way. So this is the energy her, um, number of reactions. What? We need most bias by the number of deuterium atoms because that's what we're looking for here. so, actually, I already on the side. So to solve for the number of deuterium atoms. First thing he recognizes that in a sample of water. Um, basically, she before it's the number of water molecules over upside over 3300. So, what you solve for number of water molecules? All polls 133 1/3. 3300 outfront. Number of water molecules. This based on what? The mass of water over the molar mass of water. I don't mind. I have a god rose number. All right, so we're not giving massive water in this scenario, but we are given volume of water volume of one leader so we can use density 10 volume of water to solve. Okay, so, looking up, we're gonna look a bowl of our values now. 3300 such that the density of water is a approximately one gram per leader. Most everyone leader over the molar mass. So to hydrogen, one oxygen about 18.2 grams per mole on my point by avocados. Number 6.2 times 10 to the 23 Adams. Provolone don't units. We do see leaders cancel grams. Cancel one of her moles Cancel me are just left with tandems as unit. And so we find that the number at deuterium atoms present is see the world quick. Yes, it would be about 1.2 My point, sir. 012 1.12 times 10 to the native Adams again of deuterium so nd so they actually go ahead and plug in the civilians here now. So the energy of these per reaction is 3.27 Actually, get myself a little more room here. So is 3.27 and maybe per reaction. However, I'm gonna assume gonna want jewels as my energy. So, actually, cause we will want to look at that after it's a 1.6 temps in the to get 13 Jules, pertinent TV is me years jewels it mostly by the number of deuterium atoms of 1.12 times 10 to the 19 Adams number 10 Adams. So actually, it may be to see me to say and D, but note for one reaction, we need to deuterium to divide this. Now you bite to get the number of reactions and so let me double check here, we'll try this out. We find that the amount of energy release is 2.67 Well, im you see hero Quick way double check. Feel free to check on your own as well. Yes, it's about 2.65 story 2.65 times 10 to the six Jules Barbie This questions asks, How is this compared to the amount of energy released based on gasoline? So it does say that for gasoline, this energy is 3.4 old times, 10 to the seven jewels for leader. So again, this initial values based on one leader from part A. So you see it is 10 times less. Um, but it is certainly very similar. However, um, it is easier to obtain hunt gasoline and, uh, do term is very is not very abundant in a sample of water. So if you were to harvest, be able to hurt us more deuterium, it would be much more useful as a gasoline as a fuel string out of the guest

Carleton University
Top Physics 103 Educators
Elyse G.

Cornell University

Andy C.

University of Michigan - Ann Arbor

LB
Liev B.

Numerade Educator

Jared E.

University of Winnipeg