Question:

A nuclear reactor is designed to operate with a critical mass of uranium-235. The reactor is currently producing 3.6 x 10^18 fissions per second. Each fission releases an average of 200 MeV of energy.

a) Calculate the total power output of the reactor in watts.

b) The reactor is shut down and left undisturbed for 12 hours. During this time, its power output decreases to half its initial value. Determine the number of fissions that occurred during this period.

c) Given that the mass of a uranium-235 atom is 3.9 x 10^-25 kg, calculate the total mass of uranium-235 that was consumed during the 12-hour period mentioned in part b).

Assume the conversion factor 1 eV = 1.60 x 10^-19 J.

Answer:

a) To calculate the total power output of the reactor, we need to multiply the number of fissions per second by the energy released per fission. First, we need to convert the energy released from MeV to joules.

Given that 1 MeV = 1.60 x 10^-13 J, the energy released per fission is: E_fission = 200 MeV = 200 x 1.60 x 10^-13 J = 3.20 x 10^-11 J

The total power output is given by: Power = Number of fissions per second x Energy released per fission

Given the number of fissions per second is 3.6 x 10^18, we have: Power = (3.6 x 10^18 fissions/s) x (3.20 x 10^-11 J/fission) = 1.152 x 10^8 J/s

Therefore, the total power output of the reactor is 1.152 x 10^8 Watts.

b) If the power output decreases to half its initial value during the 12 hours, the number of fissions that occurred can be calculated by finding the ratio of the power output after 12 hours to the initial power output.

Given that the power output is halved, we have: Power_after = (1/2) x Power_initial = (1/2) x 1.152 x 10^8 Watts = 5.76 x 10^7 Watts

Since power is directly proportional to the number of fissions per second, we can write the following proportion:

(Number of fissions after 12 hours) / (Number of fissions per second after 12 hours) = (Number of fissions initially) / (Number of fissions per second initially)

Plugging in known values, we have: (Number of fissions after 12 hours) / [(5.76 x 10^7 fissions/s)] = (3.6 x 10^18 fissions) / [(3.6 x 10^18 fissions/s)]

Simplifying and solving for the number of fissions after 12 hours, we get: Number of fissions after 12 hours = (5.76 x 10^7 fissions/s) x (3.6 x 10^18 fissions) / [(3.6 x 10^18 fissions/s)] = 5.76 x 10^7 fissions

Therefore, the number of fissions that occurred during the 12-hour period is 5.76 x 10^7 fissions.

c) To calculate the mass of uranium-235 consumed during the 12-hour period, we can use the equation:

Mass = (Number of fissions) x (Mass of each fissioned uranium-235 atom)

Given that the mass of a uranium-235 atom is 3.9 x 10^-25 kg, and the number of fissions during the 12-hour period is 5.76 x 10^7, we can calculate the mass consumed as follows:

Mass = (5.76 x 10^7 fissions) x (3.9 x 10^-25 kg/fission) = 2.2464 x 10^-17 kg

Therefore, the total mass of uranium-235 consumed during the 12-hour period is 2.2464 x 10^-17 kg.