In molecules of gaseous sodium chloride, the chloride ion has one more electron than proton, and the sodium ion has one more proton than electron. These ions are separated by about \(0.24 \mathrm{nm}\). How much work would be required to increase the distance between these ions to \(1.0 \mathrm{~cm} ?\)

To fully grasp the exercise involving the separation of ions, it is essential to understand Coulomb's Law. This foundational principle in physics describes the electrostatic interaction between electrically charged particles. The law states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

The equation for Coulomb's Law is beautifully simple: \[ F = k \frac{q_1 \cdot q_2}{r^2} \]Where F is the electrostatic force between the charges, k is Coulomb's constant (approximately \(8.99 \times 10^9 \mathrm{N \cdot m^2 / C^2}\)), q_1 and q_2 are the amounts of the charges, and r is the distance separating them. Understanding Coulomb's Law is crucial because it lets you determine the strength of the force that holds ions together or pushes them apart, which is at the heart of the original exercise.

The electrostatic force is a fundamental aspect of the interaction between two charged particles. Derived from Coulomb's Law, this force can either be attractive or repulsive depending on the charges of the interacting particles: opposite charges attract, while like charges repel.

Electrostatic force is what keeps electrons in atoms orbiting around the nucleus, and it's also what makes it possible for ions of opposite charge to form stable compounds. In the original exercise, the electrostatic force is what initially holds the sodium and chloride ions together, and overcoming this force requires work.The significance of electrostatic force in the context of the exercise is that you are effectively doing work against this force to separate the ions, which is why understanding how this force behaves with changing distance is essential to solving the problem.

Energy calculations are a vital part of physics, especially when considering work done on or by a system. Work is defined as the process of transferring energy to an object via the application of a force that causes the object to move.

In the context of our exercise, the work required to separate the ions is directly related to the electrostatic potential energy, which is energy stored due to the positions of charged objects relative to each other. The formula for work done in moving a charge in an electric field is:\[ W = \int F \, dr \]In this scenario, we are integrating the electrostatic force over the distance the ions are separated. The negative sign indicates that work is done against the electrostatic force. It’s the integration and understanding of how energy changes with the distance in an electrostatic field that allow us to calculate the work needed to increase the distance between the ions.

The elementary charge is a constant that represents the smallest unit of electric charge that is possible and is denoted by the symbol \(e\). This value is fundamental in physics because it is the charge of a single proton (or the negative of the charge of a single electron), and it plays a vital role in various physical phenomena and equations, including Coulomb's Law.

The value of the elementary charge is approximately \(1.6 \times 10^{-19}\) Coulombs (C). In our original problem, the charges of the sodium and chloride ions are multiples of the elementary charge, and this constant is critical for calculating the force and work associated with moving charged particles across a distance.