Question

The carbon isotope C-14 is used for carbon dating. C-14 (mass 2.34 x 10-26 kg) decays by beta-decay, in which the nucleus emits an electron (the beta particle) and a subatomic particle called a neutrino. In one such decay, the electron and neutrino are emitted at right angles to each other. The electron (mass 9.11 x 10-31 kg) has a speed of 5.0 x 107 m/s and the neutrino has a momentum of 8.0 x 10-24 kg m/s. What is the recoil speed of the daughter nucleus?

Answer 1

Step-by-step explanation:

We shall apply the law of conservation of momentum to calculate the momentum and the speed of daughter nucleus .

Since the velocity of electron is very high we shall apply relativistic formula to calculate its momentum .

=
`\frac{mv}{\sqrt{1-((v)/(c))^2 } }`

`= \frac{9.11 * 10^(-31)* 5* 10^7}{\sqrt{1-((5*10^7)/(3* 10^8))^2 } }`

45.55 x 10⁻²⁴ x 1.176

= 53.58 x 10⁻²⁴ .

momentum of neutrino = 8 x 10⁻²⁴ . They are perpendicular to each other so total momentum

= √ [( 53.58 x 10⁻²⁴ )²+(8 x 10⁻²⁴)²]

= 54.17 x 10⁻²⁴

Hence the momentum of recoiled daughter nucleus will be same but in opposite direction

velocity of recoil = momentum / mass

= 54.17 x 10⁻²⁴ / 2.34 x 10⁻²⁶

= 23.15 x 10² m /s