The potential energy function for the force between two atoms in a diatomic molecule is approximately given by U(x) = a/x¹² − 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 at infinity − U at equilibrium], then D is b²/(na). Find n = ?
Physics · Mixed Chapter
JEE Mains

Question

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by U(x) = a/x¹² − 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 at infinity − U at equilibrium], then D is b²/(na). Find n = ?

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: U(x) = a/x¹² − b/x⁶. The equilibrium separation satisfies dU/dx = −12a/x¹³ + 6b/x⁷ = 0. Step 2: xₑ⁶ = 2a/b and xₑ¹² = 4a²/b². Step 3: U(xₑ) = a·b²/(4a²) − b·b/(2a) = −b²/(4a). At infinite separation U(∞)=0. Step 4: Dissociation energy D = U(∞)−U(xₑ)=b²/(4a). Thus n=4.