The bonding force F, between atoms may be expressed approximately as follows:

F(r) = A/r^M - B/r^N (N > M) -----------(1)
Where r, is the center-to-center spacing between atoms and A, B, M, and N are constants that vary according to the type of bond.
A/r^M represents the attractive force while B/r^N represents the repulsive force.

(a) Express the equilibrium spacing r₀ in terms of A, B, M, and N.
(b) Derive another form for F(r) in which the only constants are r₀, A, M, and N.
(c) From the equation for F(r) derived in (b) calculate the following:
(i) The spacing r₁ for which F is maximum
(ii) The value of the maximum force F_max.

The strings:

(a) S₇P₇A₇₂ (Dynamic - Equilibrium)

(b) S₇P₃A₃₁ (Force - Pull)

(c)(i) S₇P₁A₁₂ (Containership -Length)
(c)(ii) S₇P₃A₃₁ (Force-Pull)
The math:

Pj Problem of Interest is of type force (electrostatic). Problem is primarily of type force since interatomic force underlies problems (a), (b) and (c).

F(r) = A/r^M - B/r^N (N > M) -----------(1)

At the equilibrium spacing r₀, the resultant interatomic force is zero, that is the attractive force balances the repulsive force.
So, F(r₀) = A/(r₀)^M - B/(r₀)^N = 0
So, A/(r₀)^M = B/(r₀)^N

So,(r₀)^N-M = B/A -------(2)

So, equilibrium spacing, r₀ = (B/A)^(1/N-M)-------(3)

From equation (2) B = A(r₀)^N-M

Let A(r₀)^N-M replace B in equation (1) :

Then, F(r) = A/r^M[1 - ((r₀)^N-M/r^N-M)]-------(4)

(c)(i) Let r₁ be the spacing for which F is maximum
Then the derivative of F at r₁, dF(r₁)/dr = 0.

Using F(r) of equation (4):

dF(r)/dr = -MAr^-(M+1) + NA(r₀)^N-M(r^-(N+M))

dF(r₁)/dr = 0, implies:

NA(r₀)^N-M((r₁)^-(N+M)) = MA(r₁)^-(M+1)

So, (N/M)(r₀)^N-M = [(r₁)^-M]/[(r₁)^-N]

So,(N/M)(r₀)^N-M = (r₁)^N-M

So, The spacing r₁, for which F is maximum:

r₁ = r₀(N/M)^(1/N-M).

(c)(ii) Substitute the expression for r₁ in equation 4 to get the expression for the value of maximum Force, F_max.

The attractive forces in interatomic bonds are primarily electrostatic. So, the expression A/r^M for the attractive force is usually assumed to be the same as the forces between electric charges. Consequently, Coulomb's Law applies and the force is inversely proportional to the square of the spacing. So M is usually = 2. The value of N is more dependent on the type of bond. For metallic bonds, N ranges from 7 to 10. For ionic and covalent bond, N ranges from 10 to 12.

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