Problem 17 - Chemistry The Central Science

Question

A sodium ion, \(\mathrm{Na}^{+},\) with a charge of \(1.6 \times 10^{-19} \mathrm{Cand}\) 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?

Step-by-Step Solution

Answer

The work required to increase the separation of the two ions to an infinite distance is \(4.5744 \times 10^{-19} J\).

Step 1: Recall the formula for electric potential energy

The formula for electric potential energy between two charged particles is given by: \[U = \frac{k * q_1 * q_2}{r}\] where: - \(U\) is the electric potential energy, - \(k\) is the electrostatic constant \((k \approx = 8.99 \times 10^9 N.m^2.C^{-2})\), - \(q_1\) and \(q_2\) are the charges of the particles, and - \(r\) is the separation distance between the particles. We will use this formula to calculate the electric potential energy of the ions at their given separation distance.

Step 2: Determine the electric potential energy at 0.50 nm separation

Given the charges of Na+ as \(q_1 = 1.6 \times 10^{-19} C\) and Cl- as \(q_2 = -1.6 \times 10^{-19} C\), and the separation distance of 0.50 nm \((0.50 \times 10^{-9} m = 5.0 \times 10^{-10} m)\), we can plug the values into the formula: \[U_1 = \frac{8.99 \times 10^9 * (1.6 \times 10^{-19}) * (-1.6 \times 10^{-19})}{5.0 \times 10^{-10}}\] Now, calculate \(U_1\): \[U_1 = -4.5744 \times 10^{-19} J\]

Step 3: Determine the electric potential energy at an infinite separation

When the separation distance between the ions is infinitely large, the electric potential energy becomes zero: \[U_2 = 0 J\]

Step 4: Calculate the work required to increase the separation

The work required to change the electric potential energy from \(U_1\) to \(U_2\) is equal to the difference in electric potential energy: \[W = U_2 - U_1\] Plug in the values of \(U_1\) and \(U_2\): \[W = 0 - (-4.5744 \times 10^{-19})\] Now, calculate the work: \[W = 4.5744 \times 10^{-19} J\]

Step 5: Final Answer

The work required to increase the separation of the two ions to an infinite distance is \(4.5744 \times 10^{-19} J\).

Key Concepts

Coulomb's Law

At the heart of understanding electrostatic interactions is Coulomb's Law. It lays the groundwork for us to determine the magnitude of the electric force between two charges. According to this fundamental principle, the force (\f\(F\f\)) between two point charges is directly proportional to the product of their charges (\f\(q_1\f\) and \f\(q_2\f\)) and inversely proportional to the square of the distance (\f\(r\f\)) that separates them.
Formulated by Charles-Augustin de Coulomb in the 18th century, the law is mathematically expressed as:\f[F = k \frac{|q_1 * q_2|}{r^2}\f]where \f\(k\f\) is the electrostatic constant (approximately \f\(8.99 \times 10^9 N.m^2.C^{-2}\f\)). This fundamental relationship is essential for calculating forces and subsequent potential energies in electrostatic systems, such as the one involving the sodium and chloride ions in the exercise.
Moreover, Coulomb's Law helps us understand that like charges repel each other, while opposite charges attract each other with a force that diminishes with distance but never quite reaches zero, which aligns with our intuition about how charged particles behave in the real world.

Charge Separation

The concept of charge separation refers to the distance between two oppositely charged particles, like the sodium (\f\(\text{Na}^+\f\)) and chloride (\f\(\text{Cl}^-\f\)) ions in our example. This distance is crucial because it determines the strength of the electrostatic force and the associated electric potential energy between them.

Importance in Calculations

When calculating electric potential energy, even small changes in charge separation can lead to significant differences in the energy values because of the inverse square relationship outlined by Coulomb's Law. In real-world applications, managing charge separation is vital in fields such as chemistry, where bond lengths determine molecular structures and energy profiles.
Additionally, understanding the principles of charge separation allows us to predict how much work is needed to bring charges closer together or to move them further apart, which can be seen in processes like the dissociation of ionic compounds or the operations of batteries and capacitors.

Work and Energy in Electrostatics

Within the realm of electrostatics, work is related to the movement of charges in an electric field. The equation for work (\f\(W\f\)) is essential because it connects the dots between force, distance, and energy. Where electric potential energy measures the potential for charges to do work because of their positions, work itself is the energy transfer that occurs when a charge is moved against an electric field.
The work required to move a charge from one point to another can be calculated as the change in electric potential energy. As demonstrated in the solution to our exercise, moving the sodium and chloride ions from a separation of 0.5 nm to an infinite distance involves work that equals the negative of the initial electric potential energy because \f\(U_2\f\) at an infinite distance is zero.

Conservation of Energy

It's also key to remember that the work-energy principle stems from the conservation of energy, which states that energy cannot be created or destroyed, only transformed. Therefore, the work done in an electrostatic context is a transformation of electric potential energy into another form, like kinetic energy or thermal energy, depending on the situation and systems involved.