The energy-separation curve for two atoms, a distance, r, apart is: U(r)=−A/r^m+B/r^n

Derive and expression for the stiffness of the bond at the equilibrium spacing, in terms of A, B, m, n, and r0.
S=dF/dr at r=r0:

The energy-separation curve for two atoms, a distance, r, apart is: U(r)=−A/r^m+B/r^n

Derive and expression for the stiffness of the bond at the equilibrium spacing, in terms of A, B, m, n, and r0.
S=dF/dr at r=r0:

To derive the expression for the stiffness of the bond at the equilibrium spacing, we need to compute the derivative of the energy function with respect to the distance, r, and evaluate it at the equilibrium spacing, r0.

Given: U(r) = -A/r^m + B/r^n

First, let's find the derivative of the energy function U(r) with respect to r:

dU/dr = d/dx (-A/r^m) + d/dx (B/r^n)

To differentiate -A/r^m, we use the power rule:

d/dx (-A/r^m) = A * m * r^(m-1)

To differentiate B/r^n, we again use the power rule:

d/dx (B/r^n) = -B * n * r^(n-1)

Combining these derivatives, we have:

dU/dr = A * m * r^(m-1) - B * n * r^(n-1)

Next, we evaluate this derivative at the equilibrium distance, r0:

dU/dr |r=r0 = A * m * r0^(m-1) - B * n * r0^(n-1)

Now, the stiffness of the bond, S (which represents the force constant or spring constant) can be defined as the negative of the second derivative of the potential energy function with respect to distance:

S = -d^2(U)/dr^2

To find the second derivative, we differentiate the expression we obtained earlier, dU/dr, with respect to r:

d^2(U)/dr^2 = d/dx (A * m * r^(m-1) - B * n * r^(n-1))

For the first term, we use the power rule again:

d/dx (A * m * r^(m-1)) = A * m * (m-1) * r^(m-2)

For the second term, we have:

d/dx (-B * n * r^(n-1)) = -B * n * (n-1) * r^(n-2)

Combining these derivatives, we have:

d^2(U)/dr^2 = A * m * (m-1) * r^(m-2) - B * n * (n-1) * r^(n-2)

Now, evaluating this second derivative at the equilibrium distance, r0:

d^2(U)/dr^2 |r=r0 = A * m * (m-1) * r0^(m-2) - B * n * (n-1) * r0^(n-2)

Therefore, the expression for the stiffness of the bond at the equilibrium spacing, in terms of A, B, m, n, and r0 is:

S = -[A * m * (m-1) * r0^(m-2) - B * n * (n-1) * r0^(n-2)]