Question

For ionic bonding, the net potential energy between two adjacent ions, EN, may be represented by the sum of Equations 2.8 and 2.9 from the text; that is, The binding energy 0 represents that point where EN is minimized. Derive an expression for the bonding energy in terms of the parameters A, B, and n. (Hint: first differentiate and derive an expression for r0, the equilibrium interionic spacing, in terms of A, B, and n. r

Answer 1

- [ A/[A/nB]^1/1-n + B/ [A/nB]^n/1-n].

Step-by-step explanation:

The mathematical representation for the net potential energy as described in the Question above is given below as;

En = -A/r + B/r^n.

Therefore, let's call the equation above equation (1). Hence, there is the need to differentiate equation (1) above wrt r.

(NB: wrt = with respect to)

Thus, [dEn/ dr] = 0. -------------------------(2).

d [ - A/r + B/r^n]/ dr = 0. -------------------(3).

A/r^2 - nB/r^n+1 = 0 ------------------------(4).

r^2/r^n+1 = A/nB ----------------------------(5).

r^1-n = A/nB -----------------------------------(6).

(1 - n )ln r= ln A/nB ------------------------(7)

ln r = 1/1 - n ln [A/nB] ---------------------(8).

r = e^ln(A/nB)^1/1-n ----------------------(9).

r = [A/nB]^1/1-n. ---------------------------(10).

Thus, put the value in (10) above that is r = [A/nB]^1/1-n into equation (1).

Hence, the bonding energy = - [ A/[A/nB]^1/1-n + B/ [A/nB]^n/1-n].