Question

A sodium ion, \(\mathrm{Na}^{+}\), with a charge of \(1.6 \times 10^{-19} \mathrm{C}\) and a chloride ion, \(\mathrm{Cl}^{-}\), with charge of \(-1.6 \times 10^{-19} \mathrm{C}\), are separated by a distance of \(0.50 \mathrm{nm}\). How much work would be required to increase the separation of the two ions to an infinite distance?

Short answer

The work required to increase the separation between the Sodium ion and the Chloride ion to an infinite distance is approximately \(2.87\times 10^{-19} \mathrm{J}\).

Step-by-step solution

Convert distance from nm to meters

First, we'll convert the given distance between the two ions from nanometers (nm) to meters (m). Given distance = \(0.50 \mathrm{nm}\) 1 nm = \(10^{-9} \mathrm{m}\) Distance in meters = \(0.50 \mathrm{nm} \times 10^{-9} \mathrm{m/nm} = 0.50 \times 10^{-9} \mathrm{m}\)

Calculate electrostatic force (Coulomb's Law)

We can now calculate the electrostatic force between the Sodium and Chloride ions using Coulomb's Law: Coulomb's Law: \(F = k \frac{q_1 q_2}{r^2}\) where: - \(F\) is the electrostatic force, - \(k\) is the electrostatic constant, which is \(8.99 \times 10^{9} \mathrm{N m^2 C^{-2}}\), - \(q_1\) is the charge of the Sodium ion (\(1.6 \times 10^{-19}\mathrm{C}\)), - \(q_2\) is the charge of the Chloride ion (\(-1.6 \times 10^{-19}\mathrm{C}\)), - \(r\) is the distance between the ions which is \(0.50 \times 10^{-9} \mathrm{m}\).

Calculate potential energy

Now that we have the electrostatic force, we can compute the potential energy of the current setup. The potential energy associated with the electrostatic force is given by the following formula: \(PE = k \frac{q_1 q_2}{r}\) We can simply insert the known values into this formula to find the potential energy: \(PE = (8.99 \times 10^{9} \mathrm{N m^2 C^{-2}})(1.6 \times 10^{-19} \mathrm{C})(-1.6 \times 10^{-19} \mathrm{C})/(0.50 \times 10^{-9} \mathrm{m})\)

Calculate the work done

The work done is equal to the change in potential energy when the distance between the ions is increased to infinity. When the distance between the ions is infinite, the potential energy becomes zero. Therefore, the work done is equal to the difference between the potential energy at infinity and the potential energy at the initial distance: Work done = \(PE_\infty - PE_{initial}\) Since \(PE_\infty = 0\), the work done is simply equal to the negative of the initial potential energy: Work done = \(-PE_{initial}\) We can now substitute the initial potential energy from step 3 to get the work done: Work done = \(-(-8.99 \times 10^{9} \times 1.6 \times 10^{-19} \times 1.6 \times 10^{-19} / (0.50 \times 10^{-9}))\)

Final calculation and result

Now, we just need to compute the final value for the work done: Work done = \(8.99 \times 10^{9} \times (1.6 \times 10^{-19})^2 / (0.50 \times 10^{-9})\) Work done = \(2.87\times 10^{-19} \mathrm{J}\) Thus, the work required to increase the separation between the Sodium ion and the Chloride ion to an infinite distance is approximately \(2.87\times 10^{-19} \mathrm{J}\).

Key concepts

Coulomb's Law

In the realm of electrostatics, one of the fundamental principles used to understand the interactions between charged particles is Coulomb's Law. This law quantitatively describes the force between two charges. It states that the electrostatic force (\( F \)) between two point charges varies directly as the product of their magnitudes (\( q_1 \) and \( q_2 \)) and inversely as the square of the distance (\( r \)) between them. The law is mathematically represented as:

• \( F = k \frac{q_1 q_2}{r^2} \)
• Where \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^{9} \mathrm{N m^2 C^{-2}} \).

Coulomb's Law not only tells us the magnitude of the force, but it also indicates whether the force is attractive or repulsive. If the charges have opposite signs, like in our example of sodium and chloride ions, the force is attractive, pulling the ions toward each other. Conversely, if the charges are of the same sign, the force is repulsive.

Potential Energy

Potential energy in electrostatic scenarios is critical for understanding how much energy a charged pair stores due to its position relative to another charge. The potential energy (\( PE \)) between two point charges is given by the formula:

• \( PE = k \frac{q_1 q_2}{r} \)
• Where \( k \) is the electrostatic constant.

In our example, since the sodium ion \( \mathrm{Na}^+ \) and the chloride ion \( \mathrm{Cl}^- \) have charges of opposite signs, the potential energy is negative. This indicates a bound system where energy is required to separate the charges to an infinite distance.
In simple terms, potential energy measures the stability of the charge configuration: the more negative the potential energy, the more stable the system and the more work required to pull the charges apart.

Work Done

In physics, when we talk about work done concerning electrostatic forces, we refer to the energy needed to move a charge within an electric field. In this exercise, we determine the work done to separate the sodium and chloride ions from a finite separation to an infinite one.

• The work done is the change in potential energy as the distance changes:
• Work Done = \( PE_\infty - PE_{initial} \)
• The potential energy at infinite distance (\( PE_\infty \)) is zero because the charges no longer exert a force on each other.

Thus, the work done is effectively equal to the negative of the initial potential energy, i.e., \(-PE_{initial}\). In this case, moving the ions infinitely apart requires \( 2.87 \times 10^{-19} \mathrm{J} \) of energy. This energy expenditure is necessary to overcome the attractive forces between the oppositely charged ions.