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The bonding force F, between atoms may be expressed approximately as follows:
F(r) = A/rM - B/rN (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/rM represents the attractive force while B/rN represents the repulsive force.
In general, the potential energy, U(r) between atoms is defined as the work capacity of interatomic forces for a given reference frame.
Hence, U(r) = ∫F(r)dr = ∫(A/rM - B/rN)dr -----------(2)
(a) Determine the expression for U(r) by integrating equation (2).
(b) Show that the curve U(r) in figure 10.8 has a minimum at equilibrium spacing, r0.
(c) What is the significance of U(ro)?
(d) Explain the meaning of the area under the curve U(r) from r0 to infinity.
The strings:
S7P3A31 (Force - Pull)
The math:
Pj Problem of Interest is of type force (electrostatic). Energy is the capacity for work. It is force that is the doer of the work. So energy and work problems are of type force.
(a) U(r) = ∫F(r)dr = ∫(A/rM - B/rN)dr
= -(A/M-1)(1/rM-1) + (B/N-1)(1/rN-1)
= -(a/rM-1) + b/(1/rN-1) + C -------(3)
Where a, b and C are constants and a = A/M-1, b = b/N-1
If we set our reference spacing, r = ∞ (infinity) then U(∞) = 0
So, C = 0 from equation (3)
So, U(r) = -a/rm + b/rn (n > m) -------(4)
Where, m = M -1 and n = N -1.
So, the potential energy between atoms is the sum of the attractive energy, -a/rm and the repulsive energy, b/rn.
(b) U(r) = ∫F(r)dr
So, dU(r)/dr = F(r)
At the equilibrium spacing r0, F(r) = 0.
So, dU(r0)/dr = 0
So, the curve of U(r) is horizontal at the equilibrium spacing, r0
Furthermore, a second derivative of U(r) = d2U(r)/dr = dF(r)/dr
F(r)/dr gives a positive slope at equilibrium spacing,r0
So, U(r0) = Umin since d2U(r)/dr is positive
So, the curve of U(r) has its minimum at equilibrium spacing, r0 (fig.10.8).
(c) U(r0) = Umin = bonding energy between the atoms.
The bonding energy is the energy necessary to separate the atoms completely.
(d) The area under the curve U(r) from r0 to infinity
= ∫r0∞F(r)dr = bonding energy
The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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