The potential energy variation between two atoms as a function of the distance between them is given by U(x)=U0[(ax)12−2(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 Bx0=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 U−x curve should be concave upwards, so that Umin exists at equilibrium position.
Now, differntiating dUdx w.r.t x ⇒d2Udx2|x=x0=U0[12×13a12x140−12×7a6x80]
Now, put x0=a d2Udx2|x=a=U0a2[12×13−12×7] d2Udx2|x=a=U0a2[156−84]=72U0a2>0
(As U0 & a both are positive values) ⇒d2Udx2|x=a is positive. ∴It is a stable equilibrium