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The potential energy between two atoms in a molecule given by U(x)=a/x12 – b/x6 where a & b are positive constants and x is distance between atoms . The atom is in stable equilibrium when x is equal to what ?
11 months ago

1)dU/dr = -12a/(r^13) +6b/(r^7) = 0 or
12 a = 6b*(r^6)b or r = {(2)^(1/6)}*(a/b)^(1/6)} at equilibrium.
differentiating once again we find that d^2/U/dr^2 > 0 for r = {(2)^(1/6)}*(a/b)^(1/6)
Umin = a/{4*(a/b)^2} - b/{2*(a/b)} = b^2/(4a) - b^2/(2a) = -(b^2)/4a
So required minimum energy = (b^2)/4a

2)and 3)
{(2)^(1/6)}*(a/b)^(1/6)} = 1.13*10^(-10) or
(a/b) =(1/2) (1.13)^6 * 10^-60 = 1.041*10^-60 ----------------- 1, and

(b^2)/4a = 1.54*10^-18 or
b = 4*1.54*(10^-18)*(a/b) ---------------------- 2; substituting from 1 fro a/b, weget]
b = 6.16*(10^-18)*1.041*10^-60 = 6.142*10^-78 J m^6 and
a = 6.675*10^-138 j m^12
11 months ago
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