The carbon isotope 14C is used for carbon dating of objects. A 14C nucleus can change into a different kind of element, a neighbor on the periodic table with lower mass, by emitting a beta particle – an electron or positron – plus a neutrino or an anti-neutrino. Consider the scenario where 14C ( mass of 2.34 x 10 -26) decays by emitting an electron and anti neutrino. The electron has a mass of 9.11x 10-31 kg and a speed of 5.5 x107 m/s. While the anti neutrino has a momentum of 1.0x10-24 kg-m/s. If the electron and anti neutrino are emitted at right angles from each other, calculate the recoil speed of the nucleus.
To calculate the recoil speed of the nucleus, we can use the principle of conservation of momentum. According to this principle, the total momentum before the decay must equal the total momentum after the decay.
Before the decay, the carbon nucleus is at rest, so its momentum is zero.
After the decay, we have two particles, the electron and the anti-neutrino, moving at right angles to each other. The momentum of the electron can be calculated using the formula:
Momentum = mass x velocity
The momentum of the anti-neutrino is given as 1.0x10^-24 kg-m/s.
Since the electron and the anti-neutrino are emitted at right angles, their momenta are perpendicular to each other. Therefore, we can use the Pythagorean theorem to find the total momentum after the decay.
Let's calculate the momentum of the electron first:
Momentum of electron = (mass of electron) x (speed of electron)
= (9.11x10^-31 kg) x (5.5x10^7 m/s)
= 5.01x10^-23 kg-m/s
Now, using the Pythagorean theorem:
Total momentum after decay = sqrt((momentum of electron)^2 + (momentum of anti-neutrino)^2)
= sqrt((5.01x10^-23 kg-m/s)^2 + (1.0x10^-24 kg-m/s)^2)
= sqrt(3.0472x10^-44 + 1.0x10^-48)
= sqrt(3.1472x10^-44) kg-m/s
≈ 1.774x10^-22 kg-m/s
Since the momentum before the decay is zero, the total momentum after the decay must also be zero. Therefore, the momentum of the nucleus in the opposite direction must be equal in magnitude to the total momentum after the decay.
Recoil momentum of nucleus = -1 x Total momentum after decay
= -1 x 1.774x10^-22 kg-m/s
≈ -1.774x10^-22 kg-m/s (negative as it has opposite direction)
Finally, we can calculate the recoil speed of the nucleus using the formula:
Momentum = mass x velocity
recoil speed of the nucleus = recoil momentum of nucleus / (mass of nucleus)
= -1.774x10^-22 kg-m/s / (2.34x10^-26 kg)
= -1.774x10^4 m/s (negative indicates opposite direction)
Therefore, the recoil speed of the nucleus is approximately 1.774x10^4 m/s in the opposite direction to that of the electron and anti-neutrino.