Question

A water sample is found to have \(0.016 \%\) deuterium content (that is, $0.016 \%\( of the hydrogen nuclei in the water are \)^{2} \mathrm{H}$ ). If the fusion reaction \(\left(^{2} \mathrm{H}+^{2} \mathrm{H}\right)\) yields 3.65 MeV of energy on average, how much energy could you get from \(1.00 \mathrm{L}\) of the water? (There are two reactions with approximately equal probabilities; one yields \(4.03 \mathrm{MeV}\) and the other \(3.27 \mathrm{MeV} .\) ) Assume that you are able to extract and fuse \(87.0 \%\) of the deuterium in the water. Give your answer in kilowatt hours.

Short answer

Express your answer in kilowatt hours (kWh). Answer: To find the total energy produced, follow these steps: 1. Calculate the number of moles of water in 1.00 L: Number of moles of water = (1000 g) / (18.015 g/mol) 2. Calculate the total number of hydrogen atoms in the water: Number of hydrogen atoms = (Number of moles of water) x (Avogadro's number) x 2 3. Calculate the number of deuterium atoms in the water: Number of deuterium atoms = (Number of hydrogen atoms) x (0.016%) 4. Calculate the number of deuterium atoms to be fused: Number of deuterium atoms to be fused = (Number of deuterium atoms) x (87.0%) 5. Calculate the total energy produced: Total energy produced = (Number of deuterium atoms to be fused) x (3.65 MeV) 6. Convert the energy produced from MeV to kWh: Total energy produced (kWh) = (Total energy produced in MeV) x (1.602 x 10^-13 J/MeV) x (2.7778 x 10^-7 kWh/J)

Step-by-step solution

Calculating the number of moles of water in 1.00 L

The first step is to calculate the number of moles of water in 1.00 L. Assuming that 1.00 L of water weighs 1000 g, we can calculate the number of moles using the molar mass of water (18.015 g/mol): Number of moles of water = (mass of water) / (molar mass of water) Number of moles of water = (1000 g) / (18.015 g/mol)

Calculating the number of hydrogen atoms in the water

Now, we can calculate the total number of hydrogen atoms in the water. Since there are two hydrogen atoms in every water (H2O) molecule, we can multiply the number of moles by Avogadro's number (6.022 x 10^23) and then by 2: Number of hydrogen atoms = (Number of moles of water) x (Avogadro's number) x 2

Calculating the number of deuterium atoms in the water

Knowing the number of hydrogen atoms in the water, we can determine the number of deuterium atoms by multiplying the total number of hydrogen atoms by the deuterium content (0.016%): Number of deuterium atoms = (Number of hydrogen atoms) x (0.016%)

Calculating the number of deuterium atoms to be fused

Now, we will calculate the number of deuterium atoms that will be fused by multiplying the number of deuterium atoms by 87.0%: Number of deuterium atoms to be fused = (Number of deuterium atoms) x (87.0%)

Calculating the total energy produced

We can now calculate the total energy produced from the fusion reaction by multiplying the number of deuterium atoms to be fused by the average energy yield per reaction (3.65 MeV): Total energy produced = (Number of deuterium atoms to be fused) x (3.65 MeV)

Converting energy to kilowatt hours

Finally, we need to convert the energy produced from MeV to kilowatt hours. To do this, we first convert MeV to joules using the conversion factor 1.602 x 10^-13 J/MeV and then convert joules to kilowatt hours using the conversion factor 2.7778 x 10^-7 kWh/J: Total energy produced (kWh) = (Total energy produced in MeV) x (1.602 x 10^-13 J/MeV) x (2.7778 x 10^-7 kWh/J)