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 1.0 x107 m/s. While the anti neutrino has a momentum of 8.5x10-24 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 14C nucleus, we need to use the principle of conservation of momentum. According to this principle, the total momentum before the decay is equal to the total momentum after the decay.
Before the decay:
The momentum of the electron is given by the product of its mass and velocity:
Momentum of electron = mass of electron x velocity of electron
= (9.11 x 10^-31 kg) x (1.0 x 10^7 m/s)
= 9.11 x 10^-24 kg-m/s
The momentum of the anti-neutrino is given directly as 8.5 x 10^-24 kg-m/s.
After the decay:
The recoil speed of the nucleus is denoted as v_nucleus.
Since the electron and the anti-neutrino are emitted at right angles to each other, their momenta are orthogonal (perpendicular) to each other. So the vector sum of their momenta is equal to the momentum of the nucleus.
Vector sum of momenta before decay = Momentum of nucleus after decay
Using the Pythagorean theorem for vector addition, we have:
(Momentum of electron)^2 + (Momentum of anti-neutrino)^2 = (Momentum of nucleus)^2
(9.11 x 10^-24 kg-m/s)^2 + (8.5 x 10^-24 kg-m/s)^2 = (mass of nucleus x recoil speed of nucleus)^2
Solving for the recoil speed of the nucleus:
(recoil speed of nucleus)^2 = [(9.11 x 10^-24 kg-m/s)^2 + (8.5 x 10^-24 kg-m/s)^2] / (mass of nucleus)^2
(recoil speed of nucleus)^2 = (83.0921 x 10^-48 kg^2-m^2/s^2 + 72.25 x 10^-48 kg^2-m^2/s^2) / (2.34 x 10^-26 kg)^2
(recoil speed of nucleus)^2 = 155.3421 x 10^-48 kg^2-m^2/s^2 / 5.4756 x 10^-52 kg^2
(recoil speed of nucleus)^2 = 28.39984 x 10^4 m^2/s^2
Recoil speed of nucleus = sqrt(28.39984 x 10^4) m/s
Recoil speed of nucleus ≈ 5329.10 m/s
Therefore, the recoil speed of the 14C nucleus after emitting the electron and anti-neutrino is approximately 5329.10 m/s.