A nuclear reactor for producing neutron beams is refuelled with about 10 kg

of uranium, which contains initially 90% of the fissile isotope 235U.
The reactor operates continuously at a constant thermal power of 57 MW until the quantity of 235U is reduced by
30%, at which point the fuel is too dilute in 235U for the chain reaction to continue. At 57MW, how many neutrons are released per second, and what is the longest possible period of
continuous running?
You may assume that the fission of one 235U nucleus releases on average
2.4 neutrons and 185 MeV of energy in the form of heat.

To find the number of neutrons released per second and the longest possible period of continuous running, we need to calculate the number of fissions that occur and the time it takes for the quantity of 235U to reduce by 30%.

First, let's calculate the initial quantity of 235U in the reactor:
Initial quantity of 235U = 10 kg * 90% = 9 kg

We know that the fission of one 235U nucleus releases on average 2.4 neutrons. So, the number of neutrons released per fission = 2.4.

To calculate the number of fissions:
Number of fissions = (Initial quantity of 235U) / (mass of one 235U nucleus)
= (9 kg) / (235 g/mol * Avogadro's number)

Here, Avogadro's number is approximately 6.022 x 10^23 mol^-1.

Next, we need to calculate the energy released per second:
Energy released per fission = 185 MeV
Power output of the reactor = 57 MW

Power output in J/s = 57 MW * (10^6) W/MW = 57 * 10^6 J/s
Energy released per second = (Power output in J/s) / (Energy released per fission)

Now, we can calculate the number of neutrons released per second:
Number of neutrons released per second = (Number of fissions) * (Number of neutrons released per fission)

Finally, to find the longest possible period of continuous running, we need to calculate the time taken for the quantity of 235U to reduce by 30%.

Remaining quantity of 235U = 9 kg - 30% = 9 kg * (1 - 0.30)

The longest possible period of continuous running can be found using the formula:
Time taken = (Remaining quantity of 235U) / (Number of fissions per second)

Now that we have the necessary equations and values, let's calculate the answer.