Approximately 1 of every 3300 water molecules contains one...

Approximately 1 of every 3300 water molecules contains one deuterium atom. (a) If all the deuterium nuclei in 1 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 L of water compares with the energy liberated from the burning of 1 L of gasoline.

Key Concepts

Energy Density Comparison

Energy density refers to the amount of energy stored or released per unit volume or mass. Comparing the energy density of nuclear fusion reactions to that of chemical processes, such as the burning of fossil fuels, helps illustrate the vast differences in efficiency and potential energy output between these two types of energy sources.

Energy Unit Conversion

Energy unit conversion is essential for comparing energies expressed in different units. In nuclear physics, energy is often given in mega electronvolts (MeV), which must be converted to joules (J) to enable meaningful comparisons with energies from chemical reactions and other macroscopic energy scales.

Nuclear Fusion

Nuclear fusion is the process where two light atomic nuclei combine to form a heavier nucleus, releasing energy as a result of the mass lost in the reaction. This process, governed by principles such as Einstein's E=mc², is the fundamental mechanism powering stars and is explored as a potential energy source on Earth.

Isotopic Abundance

Isotopic abundance describes the relative proportions of isotopes of an element found in nature. This concept is crucial for determining the number of reactive nuclei available in a substance, which impacts the overall energy yield in reactions such as nuclear fusion.

Mole Concept and Avogadro's Number

The mole is a basic unit used to quantify the amount of a substance, where one mole contains approximately 6.022×10²³ particles (Avogadro's number). This concept allows for the conversion between mass, number of particles, and the volume of a substance, facilitating stoichiometric calculations in both chemical and nuclear reactions.

Transcript

00:01 Question 62 states that approximately 1 out of 3 ,300 water molecules contains a deuterium atom.

00:09 Part a states, based on the reaction that i wrote out here, if all the deuterium nuclei in one liter of water were fused to create helium 3, how much energy would be produced.

00:24 So there's this ratio of deuterion to water and this relationship.

00:28 So from this, we can see that the amount of energy released per fusion is 3.

00:34 3 .27 m .ev.

00:36 So per reaction, there's this much energy that's produced.

00:42 So in order to determine, i guess maybe i'll use e in the scenario for energy.

00:47 Not that it really matters as long as you're consistent.

00:50 So in order to find the total amount of energy released in this reaction of all the water, all the deteriorate atoms in the one literate amount of water were to be released.

01:02 Then we simply just need to find the energy per reaction multiplied by the number of reactions.

01:17 So maybe i'll say this way.

01:19 So this is the energy per number of reactions, but we need to multiply this by the number of deuterium atoms, because that's what we're looking for here.

01:30 So actually, i'll write it on the side.

01:33 So to solve for the number of deuterium atoms, first thing we recognize is that in a sample of water, based on the ratio before, it's the number of water molecules over 3 ,000.

02:02 So in order to solve for number of water molecules, all plus plus 133, 1 over 3 ,300 out front.

02:10 Number of water molecules is based on the mass of water over the molar mass of water multiplied by avagadro's number.

02:26 So we're not given mass of water in this scenario, but we are given a volume of water, a volume of 1 liter, so we can use density times volume of water to solve.

02:49 So looking up, we can look up all of our values now, 3 ,300, such as the density, of water is approximately one gram per liter multiplied by one liter over the molar mass so two hydrogens one oxygen is about 18 .02 grams per mole and multiplied by avagadro's number 6 .02 times 10 to the 23 atoms per mole and double -shank units we do see liters cancel grams cancel one over moles cancel and the are just left with atoms as a unit and so we find that the number at deuterium atoms present is, let me see, check real quick.

03:46 Yes, it would be about 1 .02, sorry, 0 .1, 1 .012 times 10 to the 19 atoms, again, of deuterium.

04:06 So that's nd.

04:13 So i think actually we can go ahead and plug in some values here now.

04:17 So the energy release per reaction is 3 .27...

Please give Ace some feedback