Potential Energy & Equilibrium: Practice Problem & Solution
The potential energy between two atoms, in a molecule, is given by \(U(x) = \frac{a}{x^{12}} - \frac{b}{x^6}\) where \(a\) and \(b\) are positive constants and \(x\) is the distance between the atoms. The atom is in stable equilibrium, when: (1995)
Short Solution (Best for NEET):
To solve this problem, we apply the core principles of Potential Energy & Equilibrium. Understanding the underlying formula is key to arriving at the correct answer below:
For equilibrium, the force is zero: \(F = -\frac{dU}{dx} = 0\). Given \(U(x) = ax^{-12} - bx^{-6}\), then \(\frac{dU}{dx} = -12ax^{-13} + 6bx^{-7}\). Setting \(\frac{dU}{dx} = 0\) gives \(\frac{12a}{x^{13}} = \frac{6b}{x^7} \Rightarrow 12a = 6bx^6 \Rightarrow x^6 = \frac{12a}{6b} = \frac{2a}{b}\). So, \(x = (\frac{2a}{b})^{1/6}\). For stable equilibrium, \(\frac{d^2U}{dx^2} > 0\), which holds true for this value of \(x\).
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