A tritium nucleus explodes by beta decay. An electron leaves with
1x108m/sec.
Its
mass
is
9x10‐31kg.
What
is
the
recoil
velocity
if
the
3He+
ion
mass
is
3x
1.66x10‐27kg.
What
is
the
kinetic
energy
of
both
particles
in
Joules
and
in
electron volts
(1eV
=
1.16x
‐19J)?
If this were really a two-particle "explosion", then the recoil momentum of the 3He++ particle would be equal and opposite to that of the electron. One could use that fact and the mass to calculate the 3He++ velocity.
Actually, beta decay is a three-particle process, with a variable and often large part of the momentum carried away by an antineutrino. Whoever made up the question should be aware of that.
See
http://hyperphysics.phy-astr.gsu.edu/Hbase/Nuclear/beta.html
To find the recoil velocity of the 3He+ ion, we can use the principle of conservation of momentum. The total momentum before the decay must be equal to the total momentum after the decay.
Given:
Mass of the electron (me) = 9x10^-31 kg
Mass of the 3He+ ion (mion) = 3x1.66x10^-27 kg
Velocity of the electron (ve) = 1x10^8 m/s
Let's denote the recoil velocity of the 3He+ ion as vr.
Conservation of momentum equation:
me * ve = mion * vr
Now we can solve for vr:
vr = (me * ve) / mion
Substituting the given values:
vr = (9x10^-31 kg * 1x10^8 m/s) / (3x1.66x10^-27 kg)
Calculating vr:
vr = 6x10^-39 / (4.98x10^-27)
vr = 1.20x10^-12 m/s
Therefore, the recoil velocity of the 3He+ ion is approximately 1.20x10^-12 m/s.
To find the kinetic energy of both particles, we can use the equations:
For the electron (Ke):
Ke = (1/2) * me * ve^2
For the 3He+ ion (Kion):
Kion = (1/2) * mion * vr^2
Substituting the given values:
Ke = (1/2) * 9x10^-31 kg * (1x10^8 m/s)^2
Kion = (1/2) * 3x1.66x10^-27 kg * (1.20x10^-12 m/s)^2
Calculating Ke and Kion:
Ke = 4.5x10^-15 J
Kion = 1.99x10^-29 J
To convert the energy to electron volts (eV):
1 eV = 1.16x10^-19 J
Converting Ke and Kion:
Ke (eV) = 4.5x10^-15 J / (1.16x10^-19 J/eV)
Ke (eV) = 3.88x10^4 eV
Kion (eV) = 1.99x10^-29 J / (1.16x10^-19 J/eV)
Kion (eV) = 1.72x10^-10 eV
Therefore, the kinetic energy of the electron is approximately 3.88x10^4 eV, and the kinetic energy of the 3He+ ion is approximately 1.72x10^-10 eV.