91Ó°ÊÓ

In a hydrogen atom, the electrostatic force is the attractive force between the negatively charged electron and the positively charged proton. This force acts like a binder, keeping the electron in place as it orbits around the proton. The strength of this force is determined by Coulomb's Law and is given by the formula:\[ F_{e} = \frac{k \cdot e^2}{r^2} \]Here:

• \( k \) is Coulomb's constant and has a value of \(8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2\).
• \( e \) is the charge of the electron or proton, amounting to \(1.6 \times 10^{-19} \, \text{C}\).
• \( r \) is the distance between the electron and proton, which is the given radius of the electron’s orbit.

The electrostatic force is essential in holding together atoms and is much stronger than the gravitational force at the atomic scale. This force also acts as the centripetal force that keeps the electron in a stable orbit around the proton.

Centripetal force is the force that keeps an object moving in a circular path, directed towards the center of the circle. In the context of a hydrogen atom, this force is what enables the electron to orbit the nucleus, which in this case, is the stationary proton.The expression for centripetal force in circular motion is:\[ F_{c} = \frac{m_e v^2}{r} \]Where:

• \( m_e \) is the mass of the electron, approximately \(9.11 \times 10^{-31} \, \text{kg}\).
• \( v \) is the velocity of the electron.
• \( r \) is the radius of the circular path that the electron follows.

The centripetal force necessary for the electron’s motion is exactly provided by the electrostatic force as per Step 3 of the solution. This relationship highlights the close interplay between the motion and forces acting on subatomic particles. Ensuring that these forces are in a delicate balance allows the electron to maintain a stable orbit, analogous to the way a planet orbits a star.

Electron velocity is a measure of how fast the electron orbits the proton in the hydrogen atom. By treating the electrostatic force as the source of the necessary centripetal force, we can derive the velocity equation from equating those two forces:\[ \frac{k \cdot e^2}{r^2} = \frac{m_e v^2}{r} \]Through rearranging and solving for velocity \( v \), we get:\[ v = \sqrt{\frac{k \cdot e^2}{m_e \, r}} \]This equation shows how the velocity depends on constants such as the charge \( e \), mass \( m_e \), and Coulomb’s constant \( k \). By substituting the known values into the formula, the velocity of the electron is calculated to be approximately \(2.18 \times 10^6 \, \text{m/s}\).Understanding electron velocity is crucial as it affects energy levels and the radii of electron orbits. High velocities imply more energetic states, which are reflected in the quantum mechanics that underpin atomic physics. The classical interpretation taught here provides a stepping stone towards the more comprehensive quantum mechanical descriptions.