The net potential energy between two adjacent ions, EN, may...

The net potential energy between two adjacent ions, $E_{N}$, may be represented by the sum of Equations $2.8$ and $2.9$; that is, $$ E_{N}=-\frac{A}{r}+\frac{B}{r^{n}} $$ Calculate the bonding energy $E_{0}$ in terms of the parameters $A, B$, and $n$ using the following procedure: 1. Differentiate $E_{N}$ with respect to $r$, and then set the resulting expression equal to zero, because the curve of $E_{N}$ versus $r$ is a minimum at $E_{0 \text { - }}$ 2. Solve for $r$ in terms of $A, B$, and $n$, which yields $r_{0}$, the equilibrium interionic spacing. 3. Determine the expression for $E_{0}$ by substituting $r_{0}$ into Equation $2.11$.

The net potential energy between two adjacent ions, $E_{N}$, may be represented by the sum of Equations $2.8$ and $2.9$; that is,
$$
E_{N}=-\frac{A}{r}+\frac{B}{r^{n}}
$$
Calculate the bonding energy $E_{0}$ in terms of the parameters $A, B$, and $n$ using the following procedure:
1. Differentiate $E_{N}$ with respect to $r$, and then set the resulting expression equal to zero, because the curve of $E_{N}$ versus $r$ is a minimum at $E_{0 \text { - }}$
2. Solve for $r$ in terms of $A, B$, and $n$, which yields $r_{0}$, the equilibrium interionic spacing.
3. Determine the expression for $E_{0}$ by substituting $r_{0}$ into Equation $2.11$.

Key Concepts

Bonding Energy Calculation

Bonding energy is defined as the potential energy evaluated at the equilibrium interionic spacing. It quantifies the energy required to separate the ions from this equilibrium state and is derived by substituting the equilibrium separation into the original potential energy expression. This calculation provides a measure of the strength of the ionic bond.

Equilibrium Interionic Spacing

This key concept involves finding the specific distance between ions (denoted as the equilibrium distance) at which the net force is zero, meaning that the attractive and repulsive forces are balanced. The equilibrium spacing is obtained by solving the equation formed by setting the derivative of the potential energy function with respect to distance to zero.

Ionic Potential Energy

This concept refers to the energy associated with the interaction between ions in a solid, which typically includes both attractive and repulsive contributions. The attractive part is usually modeled inversely with distance (reflecting electrostatic Coulomb attraction), while the repulsive part accounts for the short-range repulsion due to overlapping electron clouds and is often represented as an inverse power of the distance.

Differentiation to Find Minimum Energy

To determine the equilibrium position of interacting ions, the total potential energy is differentiated with respect to the separation distance. Setting this derivative to zero locates the critical point where the potential energy is minimized, corresponding to the stable equilibrium state. This mathematical procedure is essential in identifying the distance where forces balance.

Transcript

00:01 In this question, we are asked to determine an expression for the bonding energy e0 of two adjacent ions, and we are provided instructions on how to do this.

00:15 So step one is to differentiate the given expression, which happens to be equation 2 .11 that was referenced later in the question.

00:28 So we're going to differentiate with respect to r, this expression, negative a over r plus b over r to the n.

00:38 I find it helpful to go ahead and write this as negative a times r to the negative 1 plus b times r to the negative n.

00:52 And so that makes it easier to see that when we differentiate, we will take the coefficient negative a times the exponent negative one times r to the negative second because we decrease our exponent by one.

01:10 Plus again our coefficient b times our exponent negative n times r to the negative n minus 1 because we decrease our exponent by 1 and so this gives me a times r to the negative second second minus b times n times r to the negative quantity n plus one.

01:42 And so if we go ahead and write this in like a regular format, we'll have a over r squared minus b times n over r to the n plus 1 and the second step that we are given is to set this coefficient or sorry a derivative equal to 0 because of course finding the minimum of a function which is going to represent our bonding energy in this case is found by finding the derivative and setting it equal to 0 so we will be able to add that b n over over r to the n plus one term to both sides.

02:33 And so now i'm going to collect all of my r terms on the same side.

02:38 So i'm going to multiply both sides by r squared and then divide both sides by b times n.

02:48 And remembering my rules for coefficients...

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