Calculating Energy Released in Nuclear Reactions

The binding energy is equal to the amount of energy released in forming the nucleus, and can be calculated using:

E = (Δm)c²

Where:
- E = Binding energy released (J)
- Δm = mass defect (kg)
- c = speed of light (m s^-1)

The daughter nuclei produced as a result of both fission and fusion have a higher binding energy per nucleon than the parent nuclei
Therefore, energy is released as a result of the mass difference between the parent nuclei and the daughter nuclei

Worked example

When uranium-235 nuclei undergo fission by absorbing slow-moving neutrons, two reactions are possible:

$Reaction 1 : {}_{92}^{235}U + {}_{0}^{1}n \to {}_{54}^{139}{Xe} + {}_{38}^{95}{Sr} + 2 {}_{0}^{1}n + energy$

$Reaction 2 : {}_{92}^{235}U + {}_{0}^{1}n \to 2 {}_{46}^{116}{Pd} + x c + energy$

(a)

For reaction 2, identify the particle c, and state the number, x, of such particles produced in the reaction.

(b)

The binding energy per nucleon, E, for a number of nuclides is given by the table below. Use the table to show that the energy produced in reaction 1 is about 210 MeV.

(c)

The energy produced in reaction 2 is 163 MeV. Suggest, with supporting reason, which one of the two reactions is more likely to happen.

nuclide	E / MeV
${}_{38}^{95}{Sr}$	8.74
${}_{54}^{139}{Xe}$	8.39
${}_{92}^{235}U$	7.60

Answer:

Part (a)

Step 1: Balance the number of protons on each side (bottom number)

92 = (2 × 46) + xn_p (where n_p is the number of protons in c)

xn_p = 92 – 92 = 0

Therefore, c must be a neutron

Step 2: Balance the number of nucleons on each side

235 + 1 = (2 × 116) + x

x = 235 + 1 – 232 = 4

Therefore, 4 neutrons are generated in the reaction

Part (b)

Step 1: Find the binding energy of each nucleus

Total binding energy of each nucleus = Binding energy per nucleon × Mass number

Binding energy of ⁹⁵Sr = 8.74 × 95 = 830.3 MeV

Binding energy of ¹³⁹Xe = 8.39 × 139 = 1166.21 MeV

Binding energy of ²³⁵U = 7.60 × 235 = 1786 MeV

Step 2: Calculate the difference in energy between the products and reactants

Energy released in reaction 1 = E_Sr + E_Xe – E_U

Energy released in reaction 1 = 830.3 + 1166.21 – 1786

Energy released in reaction 1 = 210.5 MeV

Part (c)

Since reaction 1 releases more energy than reaction 2, its end products will have a higher binding energy per nucleon
- Hence they will be more stable

This is because the more energy is released, the further it moves up the graph of binding energy per nucleon against nucleon number (A)
- Since at high values of A, the binding energy per nucleon gradually decreases with A

Nuclear reactions will tend to favour the more stable route, therefore, reaction 1 is more likely to happen

Calculating Energy Released in Nuclear Reactions (CIE A Level Physics)

Revision Note

Calculating energy released in nuclear reactions

Worked example

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Author: Leander