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, since the curve of \(E_{N}\) versus \(r\) is a minimum at \(E_{0}\) 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 substitution of \(r_{0}\) into Equation 2.11

Short Answer

Expert verified
Answer: \(E_{0} = -A \left(\frac{n-1}{nB}\right)^{\frac{1}{n-1}}+B\left(\frac{n-1}{nB}\right)\)

Step by step solution

01

Differentiate the net potential energy equation

First, we differentiate the net potential energy equation \(E_{N}\) with respect to \(r\). The equation is: $$E_{N} = -\frac{A}{r}+\frac{B}{r^{n}}.$$ To differentiate it with respect to \(r\), we use the power rule and get: $$\frac{dE_{N}}{dr} = \frac{A}{r^{2}}-\frac{nB}{r^{n+1}}.$$ Since the curve of \(E_{N}\) versus \(r\) is at a minimum at \(E_{0}\), we set the derivative equal to zero: $$0 = \frac{A}{r^{2}}-\frac{nB}{r^{n+1}}.$$
02

Solve for equilibrium interionic spacing (\(r_{0}\))

Now, we need to solve the above equation for \(r\) in terms of \(A, B,\) and \(n\), which gives us the equilibrium interionic spacing \(r_{0}\). To do this, we can first isolate \(A\) and then manipulate the equation as follows: $$A = r^{2}\left(\frac{nB}{r^{n+1}}\right)$$ $$A = \frac{nB r^{2}}{r^{n+1}}$$ $$A r^{n-1} = nB r^{2-n}$$ Now we isolate \(r\): $$r^{n-1} = \frac{nB}{A}$$ $$r_0 = \left(\frac{nB}{A}\right)^{\frac{1}{n-1}}$$
03

Determine the expression for \(E_{0}\)

Finally, we substitute the expression for \(r_{0}\) back into the net potential energy equation, \(E_{N}\), to find the expression for \(E_{0}\): $$E_{0} = -\frac{A}{r_{0}}+\frac{B}{r_{0}^{n}} = -A \left(\frac{n-1}{nB}\right)^{\frac{1}{n-1}}+B\left(\frac{n-1}{nB}\right)$$ Thus, the bonding energy, \(E_{0}\), in terms of the parameters \(A, B,\) and \(n\) is: $$E_{0} = -A \left(\frac{n-1}{nB}\right)^{\frac{1}{n-1}}+B\left(\frac{n-1}{nB}\right)$$

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Most popular questions from this chapter

Compute the percentage ionic character of the interatomic bond for each of the following compounds: \(\mathrm{MgO}, \mathrm{GaP}, \mathrm{CsF}, \mathrm{CdS},\) and \(\mathrm{FeO}\)

The net potential energy \(E_{N}\) between two adjacent ions is sometimes represented by the expression $$E_{N}=-\frac{C}{r}+D \exp \left(-\frac{r}{\rho}\right)$$ in which \(r\) is the interionic separation and \(C\) \(D,\) and \(\rho\) are constants whose values depend on the specific material. (a) Derive an expression for the bonding energy \(E_{0}\) in terms of the equilibrium interionic separation \(r_{0}\) and the constants \(D\) and \(\rho\) using the following procedure: 1\. Differentiate \(E_{N}\) with respect to \(r\) and set the resulting expression equal to zero 2\. Solve for \(C\) in terms of \(D, \rho,\) and \(r_{0}\) 3\. Determine the expression for \(E_{0}\) by substitution for \(C\) in Equation 2.12 (b) Derive another expression for \(E_{0}\) in terms of \(r_{0}, C,\) and \(\rho\) using a procedure analogous to the one outlined in part (a).

For an \(\mathrm{Na}^{+}-\mathrm{Cl}^{-}\) ion pair, attractive and repulsive energies \(E_{A}\) and \(E_{R},\) respectively, depend on the distance between the ions \(r\) according to $$\begin{array}{l} E_{A}=-\frac{1.436}{r} \\ E_{R}=\frac{7.32 \times 10^{-6}}{r^{8}} \end{array}$$ For these expressions, energies are expressed in electron volts per \(\mathrm{Na}^{+}-\mathrm{Cl}^{-}\) pair, and \(r\) is the distance in nanometers. The net energy \(E_{N}\) is just the sum of the two expressions above. (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 \(\mathrm{Na}^{+}\) 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).

Without consulting Figure 2.6 or Table \(2.2,\) determine whether each of the electron configurations given below is an inert gas, a halogen, an alkali metal, an alkaline earth metal, or a transition metal. Justify your choices. (a) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{5}\) (b) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{7} 4 s^{2}\) (c) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6}\) (d) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{1}\) (e) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6} 4 d^{5} 5 s^{2}\) (f) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2}\)

Relative to electrons and electron states what does each of the four quantum numbers specify?

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