Question

Thorium \(^{230}_{90}Th\) decays to radium \(^{226}_{88}Ra\) by a emission. The masses of the neutral atoms are 230.033134 u for \(^{230}_{90}Th\) and 226.025410 u for \(^{226}_{88}Ra\). If the parent thorium nucleus is at rest, what is the kinetic energy of the emitted \(\alpha\) particle? (Be sure to account for the recoil of the daughter nucleus.)

Short answer

The kinetic energy of the emitted α particle is approximately 4.685 MeV.

Step-by-step solution

Write the Nuclear Decay Equation

The decay equation for thorium-230 to radium-226 along with emission of an alpha particle (\([^4_2 He, \alpha\)) can be written as:\[^{230}_{90}Th \rightarrow \ ^{226}_{88}Ra + \ ^{4}_2He\]

Calculate the Mass Defect

The mass defect \(\Delta m\) is the difference in mass between the original nucleus and the decay products. Calculate \(\Delta m\) using the formula:\[\Delta m = m_{Th} - (m_{Ra} + m_{He})\]Given masses: - \(m_{Th} = 230.033134\ u\)- \(m_{Ra} = 226.025410\ u\)- \(m_{He} = 4.002603\ u\)Substituting the values:\[\Delta m = 230.033134\ u - (226.025410\ u + 4.002603\ u) = 0.005121\ u\]

Convert Mass Defect to Energy

Use Einstein's mass-energy equivalence formula \(E=\Delta m \cdot c^2\) to convert the mass defect to energy:\[E = 0.005121\ u \cdot 931.5\ \text{MeV/u} = 4.769\ \text{MeV}\]This energy is released during the decay process.

Calculate Recoil Energy of Daughter Nucleus

In nuclear decay, conservation of momentum must be considered. The recoil energy \(E_{recoil}\) of the radium daughter can be calculated using conservation of momentum. If \(E_{\alpha}\) is the kinetic energy of the alpha particle:\[E_{recoil} = \frac{E_{total} \cdot m_{He}}{m_{Ra} + m_{He}}\]Substitute the mass and total energy:\[E_{recoil} = \frac{4.769\ MeV \cdot 4.002603\ u}{226.025410\ u + 4.002603\ u} \approx 0.084\ \text{MeV}\]

Calculate Kinetic Energy of Alpha Particle

Finally, subtract the recoil energy from the total energy to get the kinetic energy \(E_{\alpha}\) of the alpha particle:\[E_{\alpha} = E_{total} - E_{recoil} = 4.769\ MeV - 0.084\ MeV \approx 4.685\ MeV\]

Conclusion

The kinetic energy of the emitted \(\alpha\) particle is approximately \(4.685\ \text{MeV}\).

Key concepts

Alpha Particle Emission

Nuclear decay often involves the emission of particles from the nucleus. One common type of particle emitted is the alpha particle. An alpha particle is essentially a helium nucleus, consisting of two protons and two neutrons.
In the decay process, a heavier nucleus, such as thorium-230, transforms into a lighter nucleus, such as radium-226, by emitting an alpha particle. This emission of an alpha particle reduces the atomic mass and the atomic number of the original nucleus.
Alpha decay is a spontaneous process and significantly alters the identity of the nucleus, resulting in a different element entirely. Notably, this type of decay is common among larger and heavier nuclei that need to shed mass to achieve stability.

Mass-Energy Equivalence

The concept of mass-energy equivalence is one of the most profound insights from Einstein's theory of relativity. It is encapsulated in the famous equation \(E = mc^2\). In the context of nuclear decay, mass-energy equivalence explains how mass is converted into energy during the decay process.
This principle is critical in understanding how the small differences in mass, known as the mass defect, result in the substantial energy release in nuclear reactions.
During the alpha decay of thorium-230, the mass difference between the original and resultant nuclei (thorium and radium, plus an alpha particle), is converted into energy which is shared between the decay products. This energy release drives the kinetic motion of emitted particles.

Kinetic Energy

Kinetic energy in nuclear physics often refers to the energy of the particles emitted during radioactive decay. In the case of alpha decay, the kinetic energy is calculated based on the energy release due to the mass defect.
This kinetic energy is crucial in understanding the speed and impact of the emitted alpha particle. It can further be distinguished by calculating the resultant motion of both the alpha particle and the recoil of the daughter nucleus, like in our example where we calculated the kinetic energy of the alpha particle to be approximately 4.685 MeV.
Understanding kinetic energy values helps researchers and physicists determine the dynamics of nuclear reactions and predict the behavior and range of emitted particles.

Momentum Conservation

In any physical process, the law of conservation of momentum holds, and nuclear decay is no exception. Even though energy is released, the total momentum before and after the decay process remains constant.
For thorium-230 undergoing alpha decay, the nucleus was initially at rest; hence its momentum was zero. When it decays, the emitted alpha particle and the new daughter nucleus move in opposite directions.
This ensures that the total momentum of the system still equals zero. Calculating the recoil energy of the daughter nucleus, as demonstrated, involves this conservation principle and is essential for keeping the entire decay system in balance. Understanding momentum conservation is crucial in predicting motion outcomes from nuclear reactions.