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Problem 45 Hard Difficulty

(a) Show that about $1.0 \times 10^{10} \mathrm{J}$ would be released by the fusion of the deuterons in 1.0 gal of water. Note that 1 of every 6500 hydrogen atoms is a deuteron. (b) The average energy consumption rate of a person living in the United States is about $1.0 \times 10^{4} \mathrm{J} / \mathrm{s}(\text { an average power of } 10 . \mathrm{kW}) .$ At this rate, how long would the energy needs of one person be supplied by the fusion of the deuterons in 1.0 gal of water? Assume the energy released per deuteron is 1.64 $\mathrm{MeV}$

Answer

a. There total energy released is
$Q_{\text {tot }}=N Q$
$=\left(3.898 \times 10^{22} \text { atoms }\right)(1.64 \mathrm{MeV})$
$=6.393 \times 10^{22} \mathrm{MeV}$
$=6.393 \times 10^{22} \mathrm{MeV}$
$=10.393 \times 10^{22} \times 1.6 \times 10^{-13} \mathrm{J}$
$=1.023 \times 10^{10} \mathrm{J}$
Thus about $1.0 \times 10^{10} \mathrm{J}$ of energy in released by the fusion of deuterons in 1 $\mathrm{gal}$ of water.
b. (b) Rate of energy consumption is $P=1.0 \times 10^{4} \mathrm{J} / \mathrm{s}$
Then $P=\frac{Q_{\text {tot }}}{t}$
Then time taken to consume $t=\frac{1.023 \times 10^{10} \mathrm{J}}{1.0 \times 10^{4} \mathrm{J} / \mathrm{s}}$
$=1.023 \times 10^{6} \mathrm{s}$
$=1.023 \times 10^{6} \mathrm{s}\left(\frac{1 \text { day }}{86400 \mathrm{s}}\right)$
$=11.84$ days
$\approx 12$ days

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Video Transcript

so to start this question, we know that energy off fusion, the energy released by the fusion of deuterium. He's given a street point to seven MTV, and we want to find out what is the energy released by one gallon off water. We are forgiven debt out off 6500 hydrogen atoms. One would be a duty. And so from this one, get, Although we knew, find how many deuterium there are, and so we can then find out how many fusion reactions chemical. And from there we multiply this by the energy released perfusion to get our energy. All right, So first off, modesty amount off water the number off most off water quite. And we take the mess off the water divided by t more la mess. The mess of the water in one gallon is given as the density multiply by the volume. The density of water is one Gramps Centimeter cube. The volume we are given in terms off one gallon. So we had to convert that gallon into leaders. So this Dimon off details in one gallon, very from a convert from leaders to Centimeter Cube, right, since our density is in terms off centimeter que Do you fight that by De mola Miss which in our case, if the emoluments off water 18 grams for more and we get 2.1 time simple to force it's from this we can find a number of hydrogen atoms. The more hydrogen atoms is two times off the number of water molecules. So two times number water molecules, which is the number most multiplied by avocados constant you ever got was constant, is given s six point 6.2 time stands post in a tree. So we get 2.5. Understand politically. Six. You see number off hydrogen atoms, but we're giving it one or 6500 is a duty. Even so, a number of to TV in just to be ghost 2.5% Part 26 right by 6500. It's when we get your point. Eat times tended polity to finally we can get what is the energy that's available energy that can be released in one gallon of water. That is the number off deuterium divide by two since each fusion reaction worked in to duty rooms. So this will give us the number off fusion reactions Monty Bury Dead by a 3.27 MTV Convert this MTV into juice by multiplying by 1.6 stand part minister. Teen Jews per MTV. We should get our value one times £10.10 juice. So and this it's given in the question until this were proven the same amount of energy if released, not in expert. We're looking at how long this amount off energy can supply. Persons need were given that an average person we're taking a power off tend ApoE four Jews per second. So the time that which we can supply this person will be take a et Fatah party power, right. This will give us in terms off seconds, but ah, because this is a very large number We converted into some off days can committed to a number of days by dividing, but a number of seconds in a day that is it. 6400 seconds in a d. We get softies. So this morning energy can supply a person's needs for softies

National University of Singapore
Top Physics 103 Educators
Marshall S.

University of Washington

Farnaz M.

Other Schools

Aspen F.

University of Sheffield

Meghan M.

McMaster University