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:
This is wrong answer for S:
Am/r^(m+1) - Bn/r^(n+1)
still looking for what F is.
F is dU/dr, but can't be "r" in answer, only A, B, m, n and r0
Well, if F is dU/dr, and U is a function of r, then there will certainly be r in the answer.
But, you want F(r) at r=r0, so geez, just plug in r0!
Am/r0^(m+1) - Bn/r0^(n+1)
There are other ways of expressing that, but that's another exercise.
I tried this yesterday and doesn't work
To derive the expression for the stiffness of the bond at the equilibrium spacing, we need to find the derivative of the energy-separation curve with respect to distance, r, and evaluate it at the equilibrium spacing, r0.
Given the energy-separation curve: U(r) = -A/r^m + B/r^n
1. First, find the derivative of U(r) with respect to r:
dU/dr = d/dx (-A/r^m) + d/dx (B/r^n)
= A * m * r^(-m-1) - B * n * r^(-n-1)
= Am/r^(m+1) - Bn/r^(n+1)
2. Next, evaluate the derivative at the equilibrium spacing, r0:
S = dU/dr evaluated at r=r0
= Am/r0^(m+1) - Bn/r0^(n+1)
So, the expression for the stiffness, S, at the equilibrium spacing, r=r0, is given by:
S = Am/r0^(m+1) - Bn/r0^(n+1)
This expression represents the stiffness of the bond at the equilibrium spacing in terms of the constants A, B, exponents m and n, and the equilibrium spacing r0.