Uranium-238 is radioactive. A uranium nucleus (mass 238 u, charge 92 e) at rest
ID: 3161948 • Letter: U
Question
Uranium-238 is radioactive. A uranium nucleus (mass 238 u, charge 92 e) at rest decays by emitting an alpha particle (mass 4 u, charge 2 e; essentially a helium nucleus), leaving behind a smaller (daughter) nucleus. This process can by modeled in the following way: the original uranium nucleus is basically a 7.4 fm radius sphere. Two protons and two neutrons “pinch off” to form the alpha particle, which is initially at rest and immediately adjacent to the remaining nucleons (the daughter). You can assume that at this moment their (center-to-center) separation is 7.4 fm. These two positively charged particles will then repel, “emitting” the alpha.
a. What is the mass (in u) and charge (in e) of the daughter nucleus? What is the name of this chemical element?
b. What is the total electrostatic potential energy of the alpha and daughter right after forming?
c. Find the speed of the daughter and the alpha after they have moved far apart. [Hint: this is a collision problem (an explosion).]
Please show work, and any explanations are greatly appreciated!
Explanation / Answer
a) immediatelt after seperation the daughter nucleus will have a mass of 234 and charge of 90
The nucleus is Thorium 234Th90
b) charge on Th = 92e and on alpha - 2e
distance of seperation d = 7.4 fm
elctrostatic potential energy Ue = kq1*q2/d2
= 9.0e+9*90.0*1.6e-19*2.0*1.6e-19/(7.4e-15)
= 5.60e-12 J
c) The enrgy is shared by the dauther and the alpha inverse to thier masses
energy of Th = 5.60e-12*4/238 = 9.57e-14J
when they are far apart the total potential energy is converted to thie KE
mv2 /2 = 9.57e-14 J
speed of the daughter v = (2*9.57e-14/234*1.66e-27)1/2
= 7.0 e+5 m/s
energy of alpha = 9.57e-12*234/238 = 9.41e-12 J
speed of the alpha v = (2*9.41e-12/4*1.66e-27)1/2 = 5.32e+7 m/s
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