The potential energy function for the force between two atoms in a diatomic molecule is approximately given by U(x) = ax¹² - b/x⁶, where a and b are constants and x is the distance between the atoms. If the dissociation energy of the molecule is D = [U(x = ∞) - Uat equilibrium], D is (a) b²/2a (b) (b²)/(12a) (c) b²/4a (d) b²/6a

Answer: (c). Equilibrium: dU/dx = 0 ⇒ x⁶ = 2a/b. Ueq = ax¹² - b/x⁶ = (ab²)/(4a²) - b²/2a = -b²/4a U(∞) = 0, so D = 0 - (-b²/4a) = b²/4a So the answer is (c) b²/4a.

The potential energy function for the force between two atoms in a… | JEE Main 2010 (AIEEE)

Q14JEE Main 2010 · AIEEEDifficult

Potential Energy & Equilibrium

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U(x) = \frac{a}{x^{12}} - \frac{b}{x^6},$ where $a$ and $b$ are constants and $x$ is the distance between the atoms. If the dissociation energy of the molecule is

$D = \left[U(x = \infty) - U_{at\ equilibrium}\right], \quad D$ is

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Answer: (c)

Equilibrium: $\frac{dU}{dx} = 0 \Rightarrow x^6 = \frac{2a}{b}$ .
$U_{eq} = \frac{a}{x^{12}} - \frac{b}{x^6} = \frac{ab^2}{4a^2} - \frac{b^2}{2a} = -\frac{b^2}{4a}$
$U(\infty) = 0$ , so $D = 0 - \left(-\frac{b^2}{4a}\right) = \frac{b^2}{4a}$
So the answer is (c) $\frac{b^2}{4a}$ .

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