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Chapter 19: Problem 77

Calculate the energy released (in joules) from the following fusion reaction: $${ }_{1}^{2} \mathrm{H}+{ }_{1}^{3} \mathrm{H} \longrightarrow{ }_{2}^{4} \mathrm{He}+{ }_{0}^{1} \mathrm{n}$$ The atomic masses are \({ }_{1}^{2} \mathrm{H}=2.0140 \mathrm{amu},{ }_{1}^{3} \mathrm{H}=3.01603\) \(\mathrm{amu},{ }_{2}^{4} \mathrm{He}=4.00260 \mathrm{amu},{ }_{0}^{1} \mathrm{n}=1.008665 \mathrm{amu}\).

Short Answer

Expert verified
The energy released in the fusion reaction is \(2.52 \times 10^{-12}\) Joules.

Step by step solution

01

Calculate Mass of Reactants

The first step is to calculate the total mass of the reactants which includes one atom of hydrogen-2 and one atom of hydrogen-3. The total mass of the reactants is: \(2.0140 \, \text{amu} + 3.01603 \, \text{amu} = 5.03003 \, \text{amu}\)
02

Calculate Mass of Products

The second step is to calculate the total mass of the products which includes one atom of helium-4 and one neutron. The total mass of the products is: \(4.00260 \, \text{amu} + 1.008665 \, \text{amu} = 5.011265 \, \text{amu}\)
03

Calculate Change in Mass

The third step is to calculate the change in mass which is the difference between the total mass of the reactants and the total mass of the products: \(5.03003 \, \text{amu} - 5.011265 \, \text{amu} = 0.018765 \, \text{amu}\)
04

Calculate Energy Released

Finally, convert the change in mass from atomic mass units (amu) to kilograms (kg) by multiplying with \(1.66053906660 \times 10^{-27} \, \text{kg/amu}\), and then use the formula for calculating energy from mass, according to Einstein's mass-energy equivalence principle: \(E = mc^2\), where \(m\) is the mass in kilograms and \(c\) is the speed of light in a vacuum (\(2.99792458 \times 10^8 \, \text{m/s}\)). So the energy released is: \(E = 0.018765 \, \text{kg} \times (2.99792458 \times 10^8 \, \text{m/s})^2 = 2.52 \times 10^{-12} \, \text{J}\)

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Most popular questions from this chapter

How does Einstein's equation, \(E=m c^{2},\) enable us to calculate nuclear binding energy?

Define nuclear fission, nuclear chain reaction, and critical mass.

Determine the symbol \({ }_{Z}^{A} \mathrm{X}\) for the parent nucleus whose \(\alpha\) decay produces the same daughter as the \({ }_{-1}^{0} \beta\) decay of \({ }^{220} \mathrm{At} .\)

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Complete the following nuclear equations and identify \(\mathrm{X}\) in each case: (a) \({ }_{53}^{135} \mathrm{I} \longrightarrow{ }_{54}^{135} \mathrm{Xe}+\mathrm{X}\) (b) \({ }_{19}^{40} \mathrm{~K} \longrightarrow{ }_{-1}^{0} \beta+\mathrm{X}\) (c) \({ }_{27}^{59} \mathrm{Co}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{25}^{56} \mathrm{Mn}+\mathrm{X}\) (d) \({ }_{92}^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{40}^{99} \mathrm{Zr}+{ }_{52}^{135} \mathrm{Te}+2 \mathrm{X}\).
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