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 is given by Coulomb's Law and is inversely proportional to the square of the distance between them. In the Bohr model, the radius of the electron's orbit is related to the principal energy level (n) and thus, the force becomes inversely proportional to the fourth power of n: $F_n\propto \frac{1}{n^4}$. As the principal energy level increases, the attractive force decreases because the electron is farther away from the nucleus, and vice versa.

Step-by-step solution

Understanding the electrostatic force between the nucleus and electron

The electrostatic force between two charged particles is given by Coulomb's Law: $$F=k\frac{q_1{q}_2}{r^2}$$ where F is the force, k is the electrostatic constant, q1 and q2 are the charges of the particles (nucleus and electron), and r is the distance between them. In an atom, the nucleus has a positive charge due to the protons, and the electron has a negative charge. This causes an attractive force between them. Moreover, as the distance between the nucleus and electron increases, the force they experience decreases.

Understanding the principal energy level in a Bohr atom

Niels Bohr proposed a model of the atom in which the electrons are arranged in circular orbits around the nucleus. The principal energy level (also known as the principal quantum number, n) describes the energy and distance of an electron from the nucleus. As the value of n increases, the electron gets farther away from the nucleus, and its energy level increases. The radius of the electron's orbit in the Bohr model is given by: $$r_n=a_0{n}^2$$ where $r_n$ is the radius of the orbit of the electron, $a_0$ is the Bohr radius (approximately $5.29\times {10}^{-11}m$), and n is the principal energy level.

Analyzing the relationship between force and principal energy level

From the formula for the force, we need to determine how the force changes as we increase the principal energy level (n). Since the radius of the orbit ($r_n$) is related to n, we can express the force in terms of n: $$F_n=k\frac{q_1{q}_2}{r_n^2}$$ Now, we substitute the radius formula into the force formula: $$F_n=k\frac{q_1{q}_2}{(a_0{n}^2{)}^2}$$

Observing the trend between force and principal energy level

From the expression for $F_n$, we can see that the attractive force between the nucleus and electron is inversely proportional to the square of the principal energy level: $$F_n\propto \frac{1}{n^4}$$ As the principal energy level increases, the attractive force decreases. This means that when an electron moves to a higher energy level (a larger value of n), the attractive force exerted by the nucleus on the electron becomes weaker because the electron is farther away from the nucleus. Conversely, when the electron moves to a lower energy level (a smaller value of n), the force increases since the electron is closer to the nucleus.

Key concepts

Electrostatic Force

Imagine two objects lightly touching each other, then suddenly springing apart—that's akin to the basic principle of electrostatic force. In scientific terms, it's a force that acts between two charged particles. Charged particles can be positive or negative, much like the two different ends of a magnet.

Electrostatic force is a crucial concept when understanding the atomic structure, especially the interactions between atomic nuclei (which contain positively charged protons) and negatively charged electrons. The very fabric of the material world depends on these interactions. According to the principles of electrostatics, opposite charges attract and like charges repel each other. The strength of this force depends on the amount of charge and the distance between the charged particles.

For example, the more charged a particle is, the stronger the electrostatic force it will exert on another charged particle. Additionally, the closer two charged particles are to each other, the stronger the force of attraction or repulsion between them.

Bohr Atomic Model

The Bohr atomic model is like a tiny solar system with electrons orbiting the atomic nucleus, much as planets circle the sun. This model was proposed by Niels Bohr in 1913. In this model, electrons move in specific circular orbits or shells around a nucleus, similar to the way satellites orbit the Earth. Each of these orbits corresponds to a particular energy level.

The key factor here is that these energy levels are quantized, meaning that the electrons can only occupy certain distances from the nucleus—no in-betweens. These distances are defined by the principal energy levels, or principal quantum numbers, represented by the symbol 'n'. For instance, when an electron is at the first energy level (n=1), it's closest to the nucleus, and as it jumps to higher levels (n=2, n=3, and so on), it moves farther away.

Understanding the Bohr model is essential for grasping how atoms absorb and emit light, as these processes involve electrons moving between energy levels. When an electron drops to a lower energy level, it releases energy in the form of light, and when it jumps to a higher level, it absorbs light energy.

Coulomb's Law

Much like how gravity dictates that what goes up must come down, Coulomb's Law helps us understand the 'push' and 'pull' between electric charges. Named after Charles-Augustin de Coulomb, who formulated it in the 18th century, this law provides a formula to calculate the electrostatic force between two charges.

The law states that the force is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. The mathematical expression is: $$F=k\frac{q_1{q}_2}{r^2}$$where 'F' stands for the force, 'k' is Coulomb's constant, 'q1' and 'q2' represent the charges, and 'r' is the distance between the charges.

As you study further in physics, you'll find that Coulomb's Law lays the groundwork for electric fields, potential energy, and even impacts our understanding of molecular bonding. It's remarkable how this law, despite being centuries old, still accurately describes the force that holds atoms together in the vast expanse of the universe.