Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 2 / A8 / b / iii / Solution

For each coordinate, the horizontal graph position iswith $\tilde{\omega }$ in inverse centimetres. Read $S/R$ from the supplied curve at these positions. Each $A_1$ frequency contributes one coordinate and each $E$ frequency contributes two. The graphical estimates areThus the listed frequency groups contribute approximately $0$, $1.4$, $4.7$, $0.33$, $2.0$ and $13.8\mathrm{J}{\mathrm{K}}^{-1}\mathrm{mo}{\mathrm{l}}^{-1}$, respectively, andThe nine coordinates are consistent with $3N-6=9$ for a five-atom nonlinear molecule. The lowest-frequency doubly degenerate bend dominates the vibrational entropy; the C–H stretch is effectively frozen at this temperature.

As a numerical check on the graphical reading, the harmonic expression $S/R=x/(e^x-1)-\ln (1-e^{-x})$, with $x=1/r$, gives a total $22.10\mathrm{J}{\mathrm{K}}^{-1}\mathrm{mo}{\mathrm{l}}^{-1}$. This agrees with the precision of the supplied graph.