What is the equilibrium distance between atoms in a two-atom molecule?

[getty102]

Homework Statement

One model for the potential energy of a two-atom molecule, where the atoms are separated by a distance r, is U(r)=U₀[(r₀/r)¹³-(r₀/r)⁹] where r₀=0.85nm and U₀=6.2eV. What is the distance between the atoms when the molecule is in stable equilibrium?

Homework Equations

The Attempt at a Solution

I set U(r)=0 b/k it's stable equilibrium and solved for r which gave me r₀ which is incorrect. Any ideas?

 

[tiny-tim]

hi getty102!

U(r)=U₀[(r₀/r)¹³-(r₀/r)⁹]
…
I set U(r)=0 b/k it's stable equilibrium and solved for r which gave me r₀ which is incorrect. Any ideas?

the zero of potential energy is arbitrary …

in this case, it has been deliberately set to 0 at r = r_o​

what is the connection between potential energy and force?

 

[getty102]

F(r_eq) = -[U₀'[(r_o/r_eq)¹³-(r₀/r_eq)⁹][((-13r₀¹³)/r_eq¹⁴)+(9r₀⁹)/r_eq¹⁰] ?

 

[tiny-tim]

(just got up :zzz:)

F(r_eq) = -[U₀'[(r_o/r_eq)¹³-(r₀/r_eq)⁹][((-13r₀¹³)/r_eq¹⁴)+(9r₀⁹)/r_eq¹⁰] ?

yes, except U_o is a constant, and your U_o' should just be U_o

and now what is the connection between force and equilibrium? ​

(btw, we don't usually write "_eq" until the last line of the proof, it's too confusing)

 

[getty102]

When F(r) = 0, stable equilibrium?

 

[tiny-tim]

When F(r) = 0, stable equilibrium?

well, equilibrium of some sort (whether it's stable is another question)

so you're looking for U' = 0, and for U'' … … … ? ​