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Question

The potential energy variation between two atoms as a function of the distance between them is given by U(x)=U0[(ax)122(ax)6], U0,a>0.
Find the equilibrium position and state the nature of equilibrium (stable or unstable).

A
x0=2a and stable
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B
x0=a and stable
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C
x0=2a and unstable
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D
x0=a and unstable
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Solution

The correct option is B x0=a and stable
For equilibrium, dUdx=0
dUdx=U0[12a12x13+12a6x7]
dUdx=0 at x=x0
12a12x70=12a6x130
a6=x60
x0=a

For checking the nature of equilibrium,
For stable equilibrium, the Ux curve should be concave upwards, so that Umin exists at equilibrium position.


d2Udx2>0 (for stable equilibrium)
Simillarly, d2Udx2<0 (for unstable equilibrium)

Now, differntiating dUdx w.r.t x
d2Udx2|x=x0=U0[12×13a12x14012×7a6x80]
Now, put x0=a
d2Udx2|x=a=U0a2[12×1312×7]
d2Udx2|x=a=U0a2[15684]=72U0a2>0
(As U0 & a both are positive values)
d2Udx2|x=a is positive.
It is a stable equilibrium

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