Q1(b). Residual entropy of CO and N₂O

What residual entropy is

1. The third law sets S=0 at T=0 for a perfect crystal: one arrangement only, W=1, S=k1=0.
2. The calorimetric (third-law) entropy, ∫ Cₚ dln T plus transition terms, is found to be smaller than the statistical (spectroscopic) entropy for some substances. The difference is the entropy the solid already had at 0 K and that calorimetry cannot see: the residual entropy Sₘ(0).
3. It arises when the solid is disordered at 0 K: several arrangements have the same energy and the disorder is frozen in, not removed by cooling.

Why CO and N₂O

1. Both molecules are linear, nearly symmetrical in shape and have a very small dipole moment. The energy of a CO molecule lying as CO or as OC in the lattice is almost the same; likewise NNO and ONN.
2. On solidifying, each molecule takes either orientation at random. On further cooling the energy difference is too small to drive reordering, and molecules cannot turn over in the rigid solid: the random arrangement is frozen in.
3. For N molecules with two orientations each, W=2ᴺ: 3pt] S=k2ᴺ=Nk2; for one mole Sₘ(0)=R2=8.314×0.6931, Sₘ(0)=5.76J K⁻¹ mol⁻¹
4. Measured for CO: about 5 J K⁻¹ mol⁻¹, close to R2. The small shortfall shows that a little ordering does take place.

Other cases: ice, with disorder in the hydrogen positions, Rln32=3.37 J K⁻¹ mol⁻¹ (measured 3.4); FClO₃, four orientations, R4=11.5 (measured 10.1). A molecule such as N₂ or CO₂, symmetrical end to end, has no such choice and no residual entropy.

Application. residual entropy is a check on the third law: tabulated standard entropies of CO, N₂O and H₂O use the statistical value, or add Sₘ(0) to the calorimetric value, so that ΔᵣS° and equilibrium constants come out right.