A uranium nucleus
238
U may stay in one piece
for billions of years, but sooner or later it decays into an α particle of mass 6.64×10
−27
kg
and
234
Th nucleus of mass 3.88 × 10
−25
kg,
and the decay process itself is extremely fast
(it takes about 10
−20
s). Suppose the uranium
nucleus was at rest just before the decay.
If the α particle is emitted at a speed of
2.35×10
7 m/s, what would be the recoil speed
of the thorium nucleus
momentum is conserved.
initial momentum=finalmomentum
0=massThrium*velocitythorium+ massalpha*velocityalpha
To find the recoil speed of the thorium nucleus, we can apply the principle of conservation of momentum. This principle states that the total momentum before an event is equal to the total momentum after the event. In this case, we will consider the uranium nucleus and the emitted alpha particle as a system.
Given:
Mass of uranium nucleus (m1) = 238 U = 238 x 1.67 × 10^-27 kg
Mass of alpha particle (m2) = 6.64 x 10^-27 kg
Velocity of alpha particle (v2) = 2.35 x 10^7 m/s
Mass of thorium nucleus (m3) = 234 Th = 234 x 1.67 × 10^-27 kg
Recoil speed of thorium nucleus (v3) = ?
Using the conservation of momentum, we can write:
Total momentum before decay = Total momentum after decay
(mass of uranium nucleus x velocity of uranium nucleus) = (mass of alpha particle x velocity of alpha particle) + (mass of thorium nucleus x velocity of thorium nucleus)
(m1 x 0) = (m2 x v2) + (m3 x v3)
As the uranium nucleus is at rest just before the decay, its velocity is zero.
Simplifying the equation:
0 = (m2 x v2) + (m3 x v3)
Now, we can solve for the recoil speed of the thorium nucleus (v3):
v3 = -(m2 / m3) x v2
Substituting the given values:
v3 = -((6.64 x 10^-27 kg) / (234 x 1.67 × 10^-27 kg)) x (2.35 x 10^7 m/s)
Calculating:
v3 = -1.57971 x 10^6 m/s
Therefore, the recoil speed of the thorium nucleus is approximately -1.58 x 10^6 m/s. Note that the negative sign indicates that the thorium nucleus moves in the opposite direction of the emitted alpha particle.