The energy-separation curve for two atoms, a distance, r, apart is: U(r)=−Arm+Brn
1) Derive an expression for the equilibrium spacing, r0, as a function of A, B, m, and n. Explicitly indicate multiplication with a * symbol.
r0 = ??
2) Derive and expression for the stiffness of the bond at the equilibrium spacing, in terms of A , B , m , n , and r0 .
S=dFdr at r=r0 :
Derive and expression for the stiffness of the bond at the equilibrium spacing, in terms of A , B , m , n , and r0 .
S=dFdr at r=r0 :
To derive the expression for the equilibrium spacing, r0, as a function of A, B, m, and n, we need to find the point where the energy-separation curve is at its minimum.
1) Equilibrium spacing, r0:
The equilibrium spacing occurs when the energy is at its minimum. In other words, it's the value of r where the derivative of U(r) with respect to r is equal to zero:
dU/dr = 0
To find the derivative, we differentiate the equation U(r) with respect to r term by term:
dU/dr = -(Amr^(m-1)) + (Brn-1)
Setting this derivative equal to zero, we can solve for r:
-(Amr^(m-1)) + (Brn-1) = 0
Amr^(m-1) = Brn-1
Dividing both sides of the equation by Am:
r^(m-1) = (B/A) * r^(n-1)
Now, we can isolate r:
r0^(m-1) = (B/A) * r0^(n-1)
Dividing both sides by r0^(n-1):
r0^(m-1-n+1) = (B/A)
Simplifying:
r0^(m-n) = (B/A)
Finally, taking the (m-n)th root of both sides:
r0 = (B/A)^(1/(m-n))
So, the expression for the equilibrium spacing, r0, as a function of A, B, m, and n is:
r0 = (B/A)^(1/(m-n))
2) Stiffness of the bond, S, at equilibrium spacing:
To find the stiffness of the bond at the equilibrium spacing, we need to calculate the derivative of the force with respect to r at r=r0.
S = dF/dr at r=r0
The force is the negative derivative of the energy:
F = -dU/dr
Differentiating the equation U(r) with respect to r:
F = Amr^(m-1) - Brn-1
Now, let's differentiate F with respect to r:
dF/dr = d/dx (Amr^(m-1) - Brn-1)
dF/dr = Am(m-1)r^(m-2) - Bn(n-1)r^(n-2)
Now, we can evaluate dF/dr at r=r0:
S = dF/dr at r=r0 = Am(m-1)r0^(m-2) - Bn(n-1)r0^(n-2)
So, the expression for the stiffness of the bond at the equilibrium spacing, in terms of A, B, m, n, and r0 is:
S = Am(m-1)r0^(m-2) - Bn(n-1)r0^(n-2)