Assuming a classical picture of the atom, an electron is captured into a circular orbit about a spherically shaped singly-charged ion, thus neutralizing it. In the frame of reference of the ion, the neutral atom acquires a component of velocity that is
To find the component of velocity acquired by the neutral atom in the frame of reference of the ion, we can apply the principle of conservation of momentum.
In this scenario, an electron is captured into a circular orbit around a singly-charged ion. Before the capture, the ion is positively charged, and the electron is initially at rest. After the capture, the neutral atom is formed, and the electron starts orbiting the ion.
As the electron moves in a circular path, it experiences centripetal force due to the electrostatic attraction between the ion and the electron. According to Newton's second law, this force is provided by the electrostatic force between the charges, and it is the only force acting on the electron.
Since no external forces act on the system (the electron and ion), the total momentum of the system must be conserved. The electron's momentum changes from zero before capture to a non-zero value after capture due to its circular motion.
As a consequence of the conservation of momentum, the ion will acquire an equal and opposite momentum to that of the electron, ensuring the overall momentum of the system remains constant.
In the frame of reference of the ion, the ion remains stationary. Therefore, the component of velocity acquired by the neutral atom (ion-electron system) in this frame will be the same as the component of velocity acquired by the electron, but in the opposite direction.
To find this velocity, we can use the relationship between the centripetal force acting on the electron, the electrostatic force between the charges, and the resulting circular motion.
The centripetal force is given by the equation:
F = (mv^2)/r
where m is the mass of the electron, v is the velocity of the electron, and r is the radius of the circular orbit.
The electrostatic force between the charges is given by the equation:
F = (k*q*Q)/r^2
where k is the electrostatic constant, q is the charge of the electron, and Q is the charge of the ion.
Setting these two forces equal to each other, we have:
(mv^2)/r = (k*q*Q)/r^2
Simplifying the equation, we get:
v = sqrt((k*q*Q)/m*r)
This equation gives us the magnitude of the velocity acquired by the electron in the circular orbit. In the frame of reference of the ion, the neutral atom acquires the same magnitude of velocity but in the opposite direction.
Note: This explanation assumes a classical model of the atom, which is an oversimplification of the quantum mechanical behavior of electrons in atoms. In reality, the behavior of electrons is described by quantum mechanics and cannot be fully understood using classical concepts.