For a K^+-Cl^-ion pair, attractive and repulsive energies EA...

For $\mathrm{a} \mathrm{K}^{+}-\mathrm{Cl}^{-}$ion pair, attractive and repulsive energies $E_{A}$ and $E_{R}$, respectively, depend on the distance between the ions $r$, according to $$ \begin{aligned} E_{A} &=-\frac{1.436}{r} \\ E_{R} &=\frac{5.86 \times 10^{-6}}{r^{9}} \end{aligned} $$ For these expressions, energies are expressed in electron volts per $\mathrm{K}^{+}-\mathrm{Cl}^{-}$pair, and $r$ is the distance in nanometers. The net energy $E_{N}$ is just the sum of the preceding two expressions. (a) Superimpose on a single plot $E_{N}, E_{R}$, and $E_{A}$ versus $r$ up to $1.0 \mathrm{~nm}$. (b) On the basis of this plot, determine (i) the equilibrium spacing $r_{0}$ between the $\mathrm{K}^{+}$and $\mathrm{Cl}^{-}$ions, and (ii) the magnitude of the bonding energy $E_{0}$ between the two ions. (c) Mathematically determine the $r_{0}$ and $E_{0}$ values using the solutions to Problem $2.14$ and compare these with the graphical results from part (b).

For $\mathrm{a} \mathrm{K}^{+}-\mathrm{Cl}^{-}$ion pair, attractive and repulsive energies $E_{A}$ and $E_{R}$, respectively, depend on the distance between the ions $r$, according to
$$
\begin{aligned}
E_{A} &=-\frac{1.436}{r} \\
E_{R} &=\frac{5.86 \times 10^{-6}}{r^{9}}
\end{aligned}
$$
For these expressions, energies are expressed in electron volts per $\mathrm{K}^{+}-\mathrm{Cl}^{-}$pair, and $r$ is the distance in nanometers. The net energy $E_{N}$ is just the sum of the preceding two expressions.
(a) Superimpose on a single plot $E_{N}, E_{R}$, and $E_{A}$ versus $r$ up to $1.0 \mathrm{~nm}$.
(b) On the basis of this plot, determine (i) the equilibrium spacing $r_{0}$ between the $\mathrm{K}^{+}$and $\mathrm{Cl}^{-}$ions, and (ii) the magnitude of the bonding energy $E_{0}$ between the two ions.
(c) Mathematically determine the $r_{0}$ and $E_{0}$ values using the solutions to Problem $2.14$ and compare these with the graphical results from part (b).

Key Concepts

Potential Energy Functions

Potential energy functions describe how the energy of a system varies as a function of a spatial parameter, such as the distance between interacting particles. In the context of ionic interactions, the total potential energy is often modeled as the sum of attractive and repulsive contributions, each having a distinct mathematical form. These functions are used to analyze the stability and behavior of the system, such as the formation of ionic bonds or crystal lattices.

Attractive Forces

Attractive forces arise from interactions that lower the potential energy of a system when particles are brought closer together. In ionic compounds, these attractions are often based on electrostatic interactions between oppositely charged ions, and are typically represented by an inverse-distance relationship. These forces drive the formation of a bond as they provide the necessary energy decrease to overcome other competing interactions.

Repulsive Forces

Repulsive forces occur when particles are pushed apart by short-range interactions, often due to electron cloud overlap or other quantum mechanical effects. These forces are typically modeled by a steeply rising potential as the interparticle distance decreases, such as by an inverse power law with a high exponent. They serve to prevent particles from collapsing into each other, thereby establishing an equilibrium spacing in the system.

Equilibrium Spacing

The equilibrium spacing is the distance at which the total potential energy of the system is minimized. This occurs when the attractive and repulsive forces balance each other out, often determined mathematically by setting the derivative of the total potential energy with respect to distance to zero. At this point, the system is in a stable configuration, and any small displacement would result in forces that restore the equilibrium.

Bonding Energy

Bonding energy refers to the amount of energy required to break the bond between two interacting particles at equilibrium. It is equivalent to the depth of the potential energy well and represents the stability of the bond. The more negative the bonding energy, the stronger and more stable the bond between the particles.

Graphical Analysis of Energy Curves

Graphical analysis involves plotting the individual attractive and repulsive energy functions, as well as their sum (the net energy), as a function of the interionic distance. This visual representation is useful for identifying the equilibrium spacing, which corresponds to the minimum of the net energy curve, and for understanding the relative contributions of the attractive and repulsive forces at different distances.

Mathematical Determination of Equilibrium

The equilibrium conditions for systems governed by potential energy functions are mathematically determined by calculating the derivative of the net energy with respect to distance and finding where this derivative equals zero. This process identifies the equilibrium spacing and allows for the computation of the bonding energy by evaluating the net energy at that distance. This analytical approach complements graphical methods and provides precise quantitative insights.

Transcript

00:01 In this question we are going to explore the attractive and repulsive and net energies between a sodium ion and a chlorine ion and then we are going to determine the bonding radius and bonding energy so our first instruction was to make a single plot with the these uh the attraction energy the repulsion energy and the net energy up to a radius of one nanometer and i had to play with the values to see what made the most sense let me insert i'm going to pause the recording and insert a table from excel that shows you uh the first values that i thought i would so you can see here now to the right uh i wasn't sure what values exactly i would need so so knowing that i wanted to have a good number of data points, i started at 0 .05 nanometers.

01:01 And i use that to calculate all of these energies.

01:04 And i want you to notice a problem.

01:07 We have, starting at 0 .05 nanometers, i have an energy of something on the order of 187 ,000.

01:18 And then when i use 0 .1, i still have something on the order of 717.

01:28 And then at 0 .15, i have a total energy over here of 18 .99.

01:38 And then all of a sudden, when i start plotting at 0 .20 nanometers, now i have a bunch of values that are much closer to each other, and i'm going to be able to read better off of my graph using the shape of the graph if i don't have such a drastically hugely scaled vertical axis.

02:08 So in fact, the data points that i actually used to make my plot that i'm about to upload an image of, i actually, let me pull over really quickly.

02:24 I actually, i'll show you real quick.

02:26 Quick.

02:26 I used, i started at 0 .2 and then i went up by every hundredth all the way up to one nanometer just so that i could get, you know, a really good graph without a lot of interpolation by excel, but i could get a really good smooth curve that i would be able to interpret.

02:53 And i'm going to to pause the recording again and upload that image.

02:57 All right.

02:58 And as you can see here, my blue curve is my attraction energy.

03:08 My orange curve is my repulsion energy.

03:12 And this gray curve is the net energy so for part b to get a value of r naught and the binding energy so what's the minimum r it's right over here so for part b r naught the minimum radius is going to occur per, i won't say equals, i should say, is at the minimum of the net energy function.

03:59 And so where's that minimum? about right in here.

04:09 All right.

04:10 And then we need to take that up.

04:15 I actually used a straight edge on my computer screen to do this.

04:19 But you know, you could like like upload this in powerpoint, make a nice straight line.

04:24 I am not going to mess with that here in onenote, but you can see we're going to pass between this marking, which is 0 .22, and this marking, which is 0 .24.

04:43 So according to the graph, half, my radius here is going to be approximately 0 .23 nanometers...

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