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The potential energy between two atoms in a molecule given by U(x)=a/x 12 – b/x 6 where a & b are positive constants and x is distance between atoms . The atom is in stable equilibrium when x is equal to what ?

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 ?

Grade:11

1 Answers

Arun
25750 Points
5 years 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

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