The element curium `._96^248 Cm` has a mean life of `10^13s`. Its primary decay modes are spontaneous fission and `alpha`-decay, the former with a probability of `8%` and the later with a probability of `92%`, each fission releases `200 MeV` of energy. The masses involved in decay are as follows
`._96^248 Cm=248.072220 u`,
`._94^244 P_u=244.064100 u` and `._2^4 He=4.002603u`. Calculate the power output from a sample of `10^20` Cm atoms. (`1u=931 MeV//c^2`)

Explanation

Energy generated in fission event is given to us but we are required to calculate energy released in a-decay process. We shall calculate m ass defect and then corresponding equivalent am ount of energy. We can w rite equation of a-decay as follows:
`""_(96)Cm^(248) to ""_(94)Pu^(244) + ""_(2)He^(4)`
Mass defect can be written as follows:
`Deltam = m(""_(96)Cm^(248)) -m(""_(94)Pu^(244)) -m(""_(2)He)^(4)`
`rArr Deltam=248.072220 - 244.064100 - 4.002603`
`rArr Deltam = 0.005517` amu
Hence energy released in `alpha`-decay process can be written as follows:
`DeltaE = 0.005517 xx 931 MeV = 5.136` MeV
lb calculate power output, we need num ber of disintegrations per second. We have to find power output corresponding to `102^(@)` curium atoms. Rate of disintegration can be written as follows:
`-(dN)/(dt) = lambdaN`
`-(dN)/(dt) =1/("averge -life") xx N`
`rArr -(dN)/(dt) =1/10^(13) xx 10^(20) = 10^(7)s^(-1)`
Probability of rate of fission is given to be 8% and hence ra te of fission can be w ritten as follows:
Rate of fission = `0.08 xx 10^(7)s^(-1)`
Probability of rate of a-decay is given to be 92% and hence rate of a-decay can be w ritten as follows:
Rate of `alpha`-decay =`0.92 xx 10^(7)s^(-1)`
One fission event generates 200 MeV and we have th at one a-decay that generates energy 5.136 MeV, hence energy released per second can be w ritten as follows:
Power (energy released per second)
`=[200 xx (0.08 xx 10^(7)) + 5.136 xx (0.92 xx 10^(7))]MeV//s`
`rArr` Power =20.73 `xx 10^(7) MeV//s`
We know that 1 MeV `=1.6 xx 10^(-13)` Joules, hence power can be written as follows:
`rArr` Power `=20.73 xx 10^(7) xx 1.6 xx 10^(-13)` J/s
`rArr` Power `=3.31 xx 10^(-5)` watt,