Nuclear Fission Examples: 44 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Nuclear Fission.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Nuclear fission is the splitting of a heavy nucleus into smaller nuclei, releasing energy and often additional neutrons.

A large unstable nucleus can split apart and release a huge amount of energy.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Nuclear Fission asks whether the system is nuclear, quantum, or relativistic before using an everyday model.

Common stuck point: Students often know a formula related to nuclear fission but skip the recognition step: Does the situation involve particles, nuclei, photons, or relativistic speeds where everyday mechanics is not enough? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Does the situation involve particles, nuclei, photons, or relativistic speeds where everyday mechanics is not enough?

Worked Examples

Example 1

medium

A mass defect of

0.20\text{ u}

(

1\text{ u} = 1.66\times 10^{-27}\text{ kg}

) is converted to energy in fission. Find

E

in joules. (

c = 3\times 10^8

Answer

E \approx 2.99 \times 10^{- 11} J

First step

m = 0.20\times 1.66\times 10^{-27} = 3.32\times 10^{-28}\text{ kg}

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Example 2

medium

Fission of

1\text{ kg}

of U-235 releases roughly

8\times 10^{13}\text{ J}

. Burning

1\text{ kg}

of coal releases about

3\times 10^7\text{ J}

. By what factor is U-235 more energy-dense per kg?

Example 3

medium

A fission fuel pellet of mass

5\text{ g}

4\%

U-235 by mass. If all U-235 fissions (releasing

200\text{ MeV}/

atom;

M_{U-235}=235\text{ g/mol}

N_A = 6\times 10^{23}

), find total energy.

Example 4

hard

A subcritical assembly has

k = 0.95

and starts with

10^{10}

fissions in the first generation. Roughly how many total fissions occur before the chain dies out? (Use

\sum_{n=0}^{\infty} k^n = 1/(1-k)

Example 5

hard

A reactor with

k = 1.01

doubles its fission rate every how many generations? (

\log_2(1.01)\approx 0.01435

Example 6

challenge

A reactor's prompt-neutron generation time is

\tau \approx 10^{-4}\text{ s}

and

k = 1.001

. Estimate how long until the fission rate grows by a factor

e

. (

T = \tau/(k-1)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Nuclear fission is the splitting of what kind of nucleus?

Example 2

easy

Fission of a heavy nucleus typically releases what particles that can trigger further fission?

Example 3

easy

Is the energy released in fission far larger or far smaller than in a chemical reaction?

Example 4

easy

A chain reaction in fission is sustained by what released particles?

Example 5

easy

Using

E = mc^2

with

c = 3\times10^8 \text{ m/s}

, find the energy from a mass defect of

1\times10^{-27} \text{ kg}

Example 6

easy

In fission, is the total mass of products slightly more or slightly less than the original nucleus?

Example 7

easy

Which heavy element is most commonly used as fission fuel in reactors?

Example 8

easy

Does fission occur in light nuclei like hydrogen or in heavy nuclei like uranium?

Example 9

medium

A single U-235 fission releases about

3.2\times10^{-11} \text{ J}

. Find the energy from

1\times10^{20}

fissions.

Example 10

medium

A fission reaction has mass defect

3\times10^{-28} \text{ kg}

. Find the energy released (

c = 3\times10^8

Example 11

medium

If each fission yields

2.5

neutrons on average and all cause new fissions, how many fissions occur in the 3rd generation starting from

1

Example 12

medium

A reactor produces

9\times10^8 \text{ J}

per second. Using

E = mc^2

(

c = 3\times10^8

), find the mass converted per second.

Example 13

medium

Compare the energy per reaction: fission releases about

200 \text{ MeV}

, a chemical bond about

4 \text{ eV}

. Find the ratio.

Example 14

medium

A fission fuel pellet converts

2\times10^{-9} \text{ kg}

of mass to energy. Find the energy (

c = 3\times10^8

Example 15

medium

In a controlled reactor, the neutron multiplication factor is kept at exactly

1

. What does this mean for the reaction rate?

Example 16

medium

Each fission releases

3.2 imes10^{-11} ext{ J}

. How many fissions are needed to release

6.4 imes10^{8} ext{ J}

Example 17

medium

A fission converts

4 imes10^{-28} ext{ kg}

of mass. Find the energy released (

c = 3 imes10^8

Example 18

challenge

A reactor runs at

600 \text{ MW}

for

1

day. Using

E = mc^2

(

c = 3\times10^8

), find the total mass converted. (

1 \text{ day} = 86400 \text{ s}

)

Example 19

challenge

Fission of U-235 yields about

200 \text{ MeV}

per atom. Find the energy from

1 \text{ mol}

(

6\times10^{23}

atoms) in joules. (

1 \text{ MeV} = 1.6\times10^{-13} \text{ J}

)

Example 20

challenge

A critical mass is needed for a chain reaction because below it too many neutrons escape. If a sphere's surface-to-volume ratio falls as radius grows, why does a larger sphere reach criticality?

Example 21

easy

Using

E = mc^2

with

c = 3\times 10^8\text{ m/s}

, find the energy released when

2\times 10^{-27}\text{ kg}

of mass disappears.

Example 22

easy

Each U-235 fission releases about

200\text{ MeV}

. How many MeV are released by

5

fissions?

Example 23

easy

Convert

200\text{ MeV}

to joules. (

1\text{ MeV} = 1.6\times 10^{-13}\text{ J}

Example 24

easy

A reactor releases

4\times 10^{19}

fissions per second. If each releases

3.2\times 10^{-11}\text{ J}

, find the power output.

Example 25

medium

A nuclear power plant outputs

1.0\text{ GW}

thermal for

1

hour. Using

E = mc^2

with

c=3\times 10^8

, find the mass converted.

Example 26

medium

If a fission generates

2

neutrons that each cause new fission (with no losses), how many fissions occur in the

5

th generation, starting from

1

in the

1

st generation?

Example 27

medium

A reactor needs to release

1.6\times 10^{14}\text{ J}

. If each fission gives

3.2\times 10^{-11}\text{ J}

, how many fissions are required?

Example 28

medium

U-235 absorbs a neutron, becomes U-236, and splits into Ba-141 and Kr-92 plus some neutrons. How many free neutrons are emitted? (Conserve nucleon number;

235+1 = 141 + 92 + n

Example 29

medium

A reactor with multiplication factor

k = 1.001

starts with

10^{16}

fissions in one generation. After

100

generations, how many fissions per generation? (Use

10^{16}(1.001)^{100}

Example 30

medium

A reactor uses

1

g/day of U-235 (mass converted, not burned mass). Find the average power. (

c=3\times 10^8

1\text{ day} = 86400\text{ s}

Example 31

medium

Fissioning all atoms in

1\text{ mol}

of U-235 releases

\sim 1.92\times 10^{13}\text{ J}

. A typical household uses

30\text{ kWh/day} = 1.08\times 10^8\text{ J/day}

. For how many days could one mole of U-235 power one household?

Example 32

hard

A nuclear plant runs

900\text{ MW}

thermal continuously. Over

30

days it converts mass

m

to energy. Find

m

. (

c = 3\times 10^8

1\text{ day} = 86400\text{ s}

Example 33

hard

30\%

of a reactor's released energy actually reaches the grid as electricity, how much electrical energy comes from

1\text{ kg}

of U-235 fissioned, given

1\text{ kg}

releases

8\times 10^{13}\text{ J}

thermal?

Example 34

hard

Each U-235 fission releases roughly

0.21\text{ u}

of mass-equivalent energy (

1\text{ u}\approx 931\text{ MeV}

). Find the energy per fission in MeV.

Example 35

hard

A pile of fissile material has

k = 1

. Doubling the radius (with density fixed) makes surface area grow as

r^2

but volume as

r^3

. How does the surface-to-volume ratio change?

Example 36

hard

A reactor consumes

3.0\text{ g}

of U-235 to release

2.4\times 10^{11}\text{ J}

over a run. Find the energy released per gram of U-235.

Example 37

challenge

A reactor at

1.2\text{ GW}

thermal runs continuously for

1

year (

3.15\times 10^7\text{ s}

). At a yield of

3.2\times 10^{-11}\text{ J/fission}

, find the total U-235 mass fissioned (atomic mass

235\text{ u}

1\text{ u} = 1.66\times 10^{-27}\text{ kg}

Example 38

challenge

A neutron with kinetic energy

2\text{ MeV}

(

3.2\times 10^{-13}\text{ J}

) and mass

1.67\times 10^{-27}\text{ kg}

has what speed? (Nonrelativistic OK since

v\ll c

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Related Concepts

Radioactive Decay Speed of Light

Background Knowledge

These ideas may be useful before you work through the harder examples.

radioactive decayspeed of light