please show me all works thank you so much In one nuclear reactor reaction, a 92

Question

please show me all works thank you so much

In one nuclear reactor reaction, a 92 235U is bombarded with a neutron, temporarily forming a highly unstable 92 236U, which promptly decays via fission into two smaller daughter nuclei 92 37 Rb and 55 140 Cs. This reaction also liberates several neutrons. (Note: this is also typical of a reaction that occurs in a fission bomb.) Calculate the energy released in this reaction. Compare this with the energy released in a typical chemical reaction (for example, the combustion of a hydrogen molecule.) Estimate how many such fission reactions are necessary in order to bring a cup of room temperature water to boiling. Calculate how much mass of uranium (in kg) this amounts to.

Explanation / Answer

The nuclear reaction is

(235,92)U = (92,37)Rb +(140,55)Cs +3*(1,0)n + energy

atomic masses are

M(235U) =235.0439 amu

(http://en.wikipedia.org/wiki/Uranium-235)

M(92Rb) =91.9197 amu         

(http://en.wikipedia.org/wiki/Isotopes_of_rubidium)

M(140Cs) =139.9173 amu

(http://en.wikipedia.org/wiki/Isotopes_of_caesium)

M(1n) =1.00866 amu

mass defect is

Delta(M)= 235.0439 - 91.9197 -139.9173 -3*1.00866 =0.18092 amu

(E = m*c^2)

1 amu = 1.6605*10^-27 kg = 931.5 MeV =1.4904*10^-10 J

Energy released is by a single (235,92)U atom

E = 0.18092*931.5 =168.527 MeV = 2.69*10^-11 J

For hydrogen the energy of combustion is

E(H) = 286 KJ/mol =286*10^3/6.022*10^23 =4.75*10^-19 J

(http://en.wikipedia.org/wiki/Heat_of_combustion#Heat_of_combustion_tables)

The energy released in the fission reaction is 8 orders of magnitude higher!

A cup of water mas mass m =200 g.

Caloric capacity of water is c=4.18 J/g/K

Heat needed to increse temperature of water from 300 K to 373 K is

Q =200*4.18*(373-300) =61028 J

Number of fissions is

N = Q/E =61028/2.69*10^-11 =2.27*10^15 atoms

One mole of 235U has mass M =235 g

The mass of U is

m = M*N/Na =235*2.27*10^15/6.023*10^23 =8.85*10^-7 grams! = 8.85*10^-10 kg of U

Na is Avogadro number.

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