If the potential energy of a gas molecule is

U= M/r⁶- N/r¹²

M&N being positive constant, then the potential energy at equilibrium must be

a: zero. b: M²/4N. c:NM²/4. d:MN²/4

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Solution

You know that the molecule will be in equilibrium if its potential energy U is a minimum.

ForFor this,
dU/dr = 0.

Therefore,

U = M/r6 - N/r12 ------(1)

du/dr = -6Mr-7 + 12Nr^-13 = 0
from which r⁶ = 2N/M. -----(2)

Squaring both side,

Therefore, r¹² = 4N²/M^{2 ------(3)}

Substituting (2) and (3) in (1)
U = M/(2N/M) – N/(4N²/M²)
= M²/2N – M²/4N
= M²/4N
[Option (b)].

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U = \frac{M}{r^{6}} - \frac{N}{r^{12}}

U = - \frac{A}{r^{6}} + \frac{B}{r^{12}},

U = \frac{A}{r^{12}} - \frac{B}{r^{6}}

ϕ (r) = - [2 {(\frac{r_{o}}{r})}^{6} - {(\frac{r^{o}}{r})}^{12}]

ϕ_{o}

_{o}

U (r) = \frac{a}{r^{12}} - \frac{b}{r^{6}}

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