Equilibrium Separation Condition

The equilibrium separation ($R$) for atoms in a crystal occurs when the total energy ($E$) of the system is at its minimum. Mathematically, this minimum occurs when the derivative of the energy function with respect to the inter-atomic spacing is equal to zero ($ \frac{dE}{dR} = 0 $).

Calculating Equilibrium Spacing

The given total energy function is: $E(R) = \frac{0.5}{R^{12}} - \frac{1}{R^6}$

Rewrite the energy function using negative exponents for easier differentiation: $E(R) = 0.5 R^{-12} - R^{-6}$

Now, differentiate $E(R)$ with respect to $R$: $ \frac{dE}{dR} = \frac{d}{dR} (0.5 R^{-12}) - \frac{d}{dR} (R^{-6}) $ $ \frac{dE}{dR} = 0.5 \times (-12) R^{-12-1} - (-6) R^{-6-1} $ $ \frac{dE}{dR} = -6 R^{-13} + 6 R^{-7} $

Set the derivative equal to zero to find the equilibrium condition: $ -6 R^{-13} + 6 R^{-7} = 0 $

Rearrange the equation: $ 6 R^{-7} = 6 R^{-13} $ Divide both sides by 6: $ R^{-7} = R^{-13} $

To solve for $R$, multiply both sides by $ R^{13} $: $ R^{13} \times R^{-7} = R^{13} \times R^{-13} $ $ R^{13-7} = R^{13-13} $ $ R^6 = R^0 $ $ R^6 = 1 $

Since $R$ represents the inter-atomic spacing, it must be a positive value. Taking the sixth root of both sides gives: $ R = 1 $

The equilibrium separation between the atoms is 1.00 Angstroms.