Question

How does the attractive force that the nucleus exerts on an electron change with the principal energy level of the electron?

Short answer

The attractive force between the nucleus and an electron can be determined using Coulomb's Law and is given by \(F = \dfrac{k_e Z e^2}{r^2}\). Using the Bohr Model, the distance between the nucleus and the electron at the \(n^{th}\) energy level, \(r_n\), is given by \(r_n = a_0 n^2\). Substituting this expression into Coulomb's Law, we obtain a relationship between the attractive force and the principal energy level: \(F_n = \dfrac{k_e Z e^2}{(a_0 n^2)^2}\). As the principal energy level 'n' increases, the attractive force between the nucleus and the electron decreases.

Step-by-step solution

Coulomb's Law

Coulomb's Law states that the force between two charged particles is proportional to the product of their charges and inversely proportional to the square of the distance between them. For a nucleus with charge \(Q\) (in terms of elementary charge, i.e., \(Q = Ze\), where \(Z\) is the atomic number, and \(e\) is the elementary charge) and an electron with charge \(-e\), we can write the attractive force as: \[F = \dfrac{k_e Z e^2}{r^2}\] Here, \(k_e\) is the Electrostatic Constant, and \(r\) is the distance between the nucleus and the electron.

Bohr Model

The Bohr Model provides a simple representation of the atom in which the electrons are assumed to have circular orbits around the nucleus. In this model, the distance between the nucleus and the electron in the \(n^{th}\) orbit is given by: \[r_n = a_0 n^2\] where \(a_0 \approx 5.29 \times 10^{-11}\ m\) is the Bohr Radius, and \(n\) is the principal quantum number (energy level).

Force and Principal Energy Level Relationship

Now, we will use the expression for \(r_n\) from the Bohr Model and substitute it into Coulomb's Law to establish a relationship between the attractive force and the principal energy level: \[F_n = \dfrac{k_e Z e^2}{(a_0 n^2)^2}\] Note that as \(n\) increases, the distance between the nucleus and the electron increases, which results in a decrease in the attractive force between them.

Conclusion

Upon analyzing Coulomb's Law and substituting the expression for the distance from the Bohr Model, we can conclude that as the principal energy level 'n' of the electron increases, the attractive force between the nucleus and the electron decreases.