Question

Exercise 29 notes that more energy is required to ionize sodium than is retrieved by adding that electron to an isolated chlorine atom, but the \(\mathrm{NaCl}\) bond represents a lower energy because the attracting ions draw close together. Quantif ying the ener \(8 y\) -lowering effect of having alternating plus and minus charges can be rather involved for a 3D lattice, but a one-dimensional calculation is instructive. Consider an int inite line of point charges alternating between te and - \(e\). with a unif orm spacing between adjacent (opposite) charges of \(a\). (a) The electrostatic potential energy per ion is the same for a given pusitive inn as for a given negative ion. Why? (b) Calculate the electrostatic potential energy per ion. For simplicity, assume that a positive charge is at the origin. The following power series expansion will be helpful: \(\ln (1+x)=-\sum_{n=1}^{\infty}(-x)^{n} / n\).

Short answer

The electrostatic potential energy per ion for this one-dimensional array of alternating charges is \(-4ke \ln 2\).

Step-by-step solution

Equal Electrostatic Potential Energy

The electrostatic potential energy per ion is the same for a given positive ion as for a negative ion because in the line, each ion is surrounded by equal and opposite charges in a symmetrical arrangement. If the positive charge is at the origin (as the problem assumes), the next positive charge will be two units away, the next four units away, and so on. The same is true in the negative direction, with the first negative charge being one unit away, the next three units away, etc. The symmetry of the distribution results in an equal magnitude of electrostatic potential energy, but opposite sign, for positive and negative ions.

Calculate the electrostatic potential energy per positive ion

To find the electrostatic potential energy per ion, consider the sum of the electric potentials on our chosen positive ion, each from other ions at various distances. Recall that the electric potential energy from each point charge is given by \(U=-\frac{kq}{r}\) where \(k\) is Coulomb’s constant, \(q\) is the charge of the ion (either \(+e\) or \(-e\)), and \(r\) is the distance from the center charge. Since every \(a\) breaks symmetry, only the terms that mirror each other around the central point (origin) will contribute to the sum. Hence, calculating for only one side (say positive direction), and then multiplying the result by 2 should give the total potential energy. By substituting \(r=na\) and summing over all \(n\), you find the total potential energy is \(-2k\sum_{n=1}^{\infty} (-1)^n\frac{qe}{n \cdot a}\)

Apply the power series expansion

Next, apply the given power series expansion, \(\ln(1+x)=-\sum_{n=1}^{\infty}(-x)^n/n\). Setting \(x = -1\) yields \(\ln 2 = \sum_{n=1}^{\infty} 1/n\), which allows you to simplify the series in the potential energy equation: \(-2ke\sum_{n=1}^{\infty} (-1)^n/n = -2ke (\ln 2 + Sum_{n=even}1/n)\). Because the even terms in the series sum to another natural logarithm of 2, the total potential is \(-4ke \ln 2\)

Key concepts

Ionization Energy

Ionization energy is the amount of energy required to remove an electron from an atom or ion in its gaseous state. It reflects the difficulty of removing electrons from atoms, as they are attracted to the nucleus by the electromagnetic force. Elements such as sodium (Na), referenced in the textbook exercise, have a lower ionization energy compared to other elements, such as noble gases, because of their single valence electron which is further away from the nucleus and less tightly bound due to lesser effective nuclear charge.

This concept is crucial to understanding why energy must be added to ionize sodium, as opposed to releasing energy when electrons are added to form negative ions, such as the chloride ion (Cl-). When forming NaCl, the energy needed to ionize sodium is more than compensated for by the energy released when chlorine gains an electron and both ions attract each other to form an ionic bond.

Coulomb's Law

Coulomb's Law describes the force between two point charges. The law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Expressed as an equation, the law is written as: \( F = k \frac{|q_1 q_2|}{r^2} \), where \( F \) is the force, \( k \) is Coulomb's constant (\( 8.9875 \times 10^9 \, \text{Nm}^2/\text{C}^2 \)), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them.

This principle is pivotal in explaining interactions in ionic compounds like NaCl, where the electrostatic potential energy is calculated using the product of the charges of the respective ions and the separation between them, as demonstrated in the exercise solution.

Power Series Expansion

The power series expansion is a way of expressing a function as an infinite sum of terms calculated from the function's derivatives at a single point. In particular, the textbook exercise uses the series expansion of the natural logarithm \( \ln(1+x) \). This series is defined as \( \ln(1+x) = -\sum_{n=1}^{\infty}(-x)^n/n \).

Application in Electrostatics

By using this expansion, one can simplify complex functions into a series that are often easier to compute, especially when dealing with inverse relations or calculating potentials at various points in a field. The solution in the textbook demonstrates how applying the series to the electrostatic potential energy scenario can help simplify and solve for the potential energy per ion in a one-dimensional ionic lattice.

Electric Potential

Electric potential is the potential energy per unit charge at a point in a static electric field. The potential energy is due to the positional relationship of different charges in the field. It is essentially the work done by an external force to bring a charge from infinity to that point without any acceleration.

Using the concept of electric potential, one can understand the potential energy for a system of charges easily. For a point charge, the electric potential (V) is given by \( V = \frac{U}{q} = -\frac{kq}{r} \), where \( U \) is the electric potential energy, \( q \) is the test charge, and \( r \) is the distance from the point charge generating the electric field.

In the context of the exercise, the calculation of the electrostatic potential energy per ion in an infinite line of alternating point charges uses the concept of electric potential, illustrating how energy levels change based on the spatial arrangement of ions. This understanding is essential when considering the stability and formation of ionic compounds, such as sodium chloride.