There is a stream of neutrons with a kinetic energy of 0.0327 eV. If the half-life of neutrons is 700 seconds, what fraction of neutrons will decay before they travel a distance of 10m?
Given mass of neutron = 1.675 * 10^-27 kg.
Correct answer is '4.8'. Can you explain this answer?

Question

There is a stream of neutrons with a kinetic energy of 0.0327 eV. If the half-life of neutrons is 700 seconds, what fraction of neutrons will decay before they travel a distance of 10m?
Given mass of neutron = 1.675 * 10^-27 kg.

Correct answer is '4.8'. Can you explain this answer?

Answer

Given information:

Kinetic energy of neutrons: 0.0327 eV

Half-life of neutrons: 700 seconds

Distance traveled by neutrons: 10 m

Mass of neutron: 1.675 x 10^-27 kg

Calculating the speed of neutrons:

First, we need to convert the kinetic energy of neutrons into joules.

1 eV = 1.6 x 10^-19 J

0.0327 eV = 0.0327 x 1.6 x 10^-19 J = 5.232 x 10^-21 J

Kinetic energy of neutrons = 0.5mv^2

5.232 x 10^-21 J = 0.5 * (1.675 x 10^-27 kg) * v^2

v^2 = (5.232 x 10^-21 J) / (0.5 * 1.675 x 10^-27 kg) = 6.237 x 10^5 m^2/s^2

v = √(6.237 x 10^5 m^2/s^2) ≈ 789.6 m/s

Calculating the time taken to travel 10 m:

Distance = speed x time

10 m = 789.6 m/s x time

Time taken = 10 m / 789.6 m/s ≈ 0.0127 s

Calculating the number of half-lives:

Number of half-lives = time taken / half-life

Number of half-lives = 0.0127 s / 700 s ≈ 1.81 x 10^-5

Calculating the fraction of neutrons that decay:

The fraction of neutrons that decay is given by the formula:

Fraction of neutrons decayed = (1/2)^(number of half-lives)

Fraction of neutrons decayed = (1/2)^(1.81 x 10^-5) ≈ 4.8 x 10^-6 ≈ 4.8

Conclusion:

Approximately 4.8% of the neutrons will decay before traveling a distance of 10 m.

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