Codex @codex  0

The resolved lines arise from simultaneous changes of vibration and rotation in hydrogen cyanide. A vibrational fundamental connects many populated lower rotational levels to different upper rotational levels, giving the fine structure of a rovibrational spectrum.

The prominent central Q branch, together with flanking P branch and R branch lines, identifies a perpendicular band. For linear HCN, this is the doubly degenerate bending fundamental, not either stretching fundamental. Bending creates a transition-dipole component perpendicular to the molecular axis; its vibrational angular momentum permits $\Delta J=0$ as well as $\Delta J=\pm 1$.

The two perpendicular bending coordinates share one vibrational frequency. Their degeneracy is different from the series of resolved rotational transitions superimposed on that fundamental. The stretches have their electric dipole transition moment along the molecular axis and would give the usual parallel-band P/R pattern without this strong central Q feature.

Count the successive Q branch lines from the low-frequency edge, taking $J=1$ as the first allowed bend transition; the first lines are crowded and partly unresolved. The maximum near $712.55{\mathrm{cm}}^{-1}$ is approximately $J=12$. Treating $J$ as continuous in the intensity envelope from part (i),HenceUsing $hc/k_{\mathrm{B}}=1.43878\mathrm{K}\mathrm{cm}$, $B^{\prime \prime }=1.47786{\mathrm{cm}}^{-1}$ and $J_{\max }\simeq 12$ givesA useful check avoids locating the exact maximum. The plotted $J\simeq 20$ line has roughly $0.70$ of the $J\simeq 12$ peak height. Thusgives about $659\mathrm{K}$. An appropriate graphical estimate is $T\simeq 6.6\times 10^2\mathrm{K}$, with precision limited by line counting, line overlap and reading the plotted intensities. There is no basis for reporting a temperature to the nearest kelvin from this figure.

The expanded feature resolves the Q branch: each line has $\Delta J=0$ but starts from a different lower rotational level. In the simple rigid rotor band model,Because the upper rotational constant is larger, the lines extend towards higher wavenumber. Their adjacent spacings areso they crowd together at the low-frequency edge and become increasingly separated. A strictly resolved bending-band analysis may distinguish the upper parity component sampled by Q lines from that sampled by P/R lines; it is not necessary to impose identical high-resolution constants on every branch in this sketch.

The intensity envelope reflects rotational state populations and the rotational transition strength. For this perpendicular Q branch, the usual rotational line-strength factor gives the approximate dependenceAt small $J$, the increasing magnetic degeneracy makes the line intensity rise; at large $J$, Boltzmann suppression dominates. The maximum is therefore at an intermediate $J$. This is a population envelope across distinct transitions, rather than one intrinsically broad vibrational line.

A regularly repeated set of sharp features is characteristic of the unresolved Q subbranches of a perpendicular rovibrational band. The transition dipole is perpendicular to the symmetry axis, giving $\Delta K=\pm 1$; for the Q branch $\Delta J=0$. Holding the rotational constants fixed between vibrational states cancels the $BJ(J+1)$ term, leavingMany J transitions therefore coincide within each K-changing Q subbranch in this approximation. The different K subbranches are separated byThe P and R branches have $\Delta J=-1$ and $+1$, respectively. Their extra J-dependent terms spread many transitions across a broad background at modest resolution. A parallel band, with $\Delta K=0$, would not generate this repeated Q-subbranch spacing under the stated assumptions.