Question

In the Bohr's atomic model, the electrostatic force of attraction between nuclear charge \(\left(Z_{\mathrm{e}}\right)\) and electron of charge \(e\) is balanced by the centripetal force acting towards the centre of atom. If \(\epsilon_{0}\) be the permittivity of vacuum and \(r\) be the radius of orbit in which electron is revolving, the speed of electron is (a) \(\sqrt{\frac{Z e^{2}}{\left(4 \pi \varepsilon_{0}\right) m r}}\) (b) \(\sqrt{\frac{\left(4 \pi \varepsilon_{0}\right) m r}{Z e^{2}}}\) (c) \(\sqrt{\left(4 \pi \varepsilon_{\mathrm{o}}\right) m r Z e^{2}}\) (d) \(\frac{e}{\sqrt{\left(4 \pi \varepsilon_{\mathrm{o}}\right) m r}}\)

Short answer

The speed of the electron is given by answer choice (a), \(\sqrt{\frac{Z e^{2}}{(4 \pi \varepsilon_{0}) m r}}\).

Step-by-step solution

Write down the force balance equation

According to Bohr's model, the electrostatic force of attraction (Coulomb's force) between the electron and the nucleus is equal to the centripetal force required to keep the electron in a circular orbit. Coulomb's force is given by \(F_e = \frac{Z e^2}{4 \pi \varepsilon_0 r^2}\) and centripetal force is given by \(F_c = \frac{m v^2}{r}\), where \(Z\) is the atomic number, \(e\) is the charge of electron, \(\varepsilon_0\) is the permittivity of vacuum, \(m\) is the mass of the electron, \(v\) is the speed of the electron, and \(r\) is the radius of the orbit. Set these forces equal to each other to find the speed of the electron.

Set the equations equal and solve for \(v\)

Setting the electrostatic force equal to the centripetal force gives us \(\frac{Z e^2}{4 \pi \varepsilon_0 r^2} = \frac{m v^2}{r}\). Multiplying by \(r\) and dividing by \(m\) on both sides yields \(\frac{Z e^2}{4 \pi \varepsilon_0 r m} = v^2\). Taking the square root of both sides will give us the speed \(v\) of the electron.

Conclude with the correct expression for \(v\)

After taking the square root, we obtain the expression for \(v\) as \(v = \sqrt{\frac{Z e^2}{4 \pi \varepsilon_0 m r}}\), which corresponds to the answer choice (a).

Key concepts

Coulomb's force

Coulomb's force, named after physicist Charles-Augustin de Coulomb, describes the force between two charged particles. It is a central concept in understanding the behavior of electrons within an atom.

In the context of Bohr's atomic model, Coulomb's force explains the attraction between the positively charged nucleus and a negatively charged electron. The strength of this electrostatic force can be calculated with the formula: \(F_e = \frac{Z e^2}{4 \pi \varepsilon_0 r^2}\), where \(Z\) represents the number of protons in the nucleus (atomic number), \(e\) is the charge of the electron, \(\varepsilon_0\) is the permittivity of vacuum, and \(r\) is the radius of the electron's orbit.

Understanding this force helps explain why electrons remain in specific orbits around a nucleus, rather than simply flying away due to their kinetic energy.

Centripetal force

When an object moves in a circular path, there must be a force that keeps it from flying out straight, in line with its inertia. This force is known as the centripetal force and for an electron in an atom, it's the electrostatic attraction — the Coulomb's force — that provides this necessary centripetal pull.

To maintain a circular orbit, as per Bohr's model, the centripetal force on the electron is given by: \(F_c = \frac{m v^2}{r}\), where \(m\) is the mass of the electron, \(v\) is its orbit speed, and \(r\) is the radius of its circular orbit. This force is always directed towards the center of the orbit and is crucial for understanding the electron's stable orbit within atoms.

Electron orbit speed

The electron orbit speed is an important factor that dictates the stability of an electron's orbit within the Bohr model of the atom. Using the balance of forces, the orbit speed \(v\) is derived by equating the centripetal force and Coulomb's force. Following the steps from the solution, we obtain the formula for the speed of the electron: \(v = \sqrt{\frac{Z e^2}{4 \pi \varepsilon_0 m r}}\).

This formula indicates that the speed of an electron in orbit is determined by the charge of the electron, the mass of the electron, the permittivity of vacuum, the atomic number of the element, and the radius of the orbit. This orbital speed is constant for an electron in a stable orbit, and changes only when the electron jumps to a different orbit, releasing or absorbing energy in the process.

Permittivity of vacuum

The permittivity of vacuum, often denoted by \(\varepsilon_0\), is a physical constant that represents the ability of a vacuum to permit electric field lines. It is fundamental to the calculations involving electrostatic forces in the vacuum of space, which is the setting for Bohr's atomic model.

The value of \(\varepsilon_0\) is crucial when using Coulomb's law as it appears in the denominator of the force equation: \(F_e = \frac{Z e^2}{4 \pi \varepsilon_0 r^2}\). Its importance lies in the fact that it affects the magnitude of the electric force and therefore the calculated speed of the electron in its orbit. A correct understanding of \(\varepsilon_0\) is critical for interpreting and applying the electrostatic relationships in atomic physics.