atoms are interchanged by three proper rotations, so $σ = 3$ . The three mirror operations do not increase the rotational symmetry number to six. Isotopic identity matters for this count.

Using the supplied wavenumber form of

k_{B}

θ_{a} = θ_{b} = \frac{0.110}{0.695} = 0.15827 K, θ_{c} = \frac{0.0593}{0.695} = 0.085324 K .

All are far below

298 K

, justifying the high-temperature nonlinear rigid rotor expression. Then

q_{rot} = \frac{π}{3} [\frac{29 8 ^{3}}{( 0.15827 ) ^{2} ( 0.085324 )}]^{1/2} ≃ 6.5741 \times 1 0^{4},

and

S_{rot, m} = R [ln q_{rot} + 3/2] ≃ 104.71 J K^{- 1} mo l^{- 1} .

The calculation treats the molecule as rigid and separates electronic, vibrational and nuclear-spin contributions from the requested rotational term.

iii

Solution

For each coordinate, the horizontal graph position is

r = \frac{T}{θ _{vib}} = \frac{( 298 ) ( 0.695 )}{ω} = \frac{207.11}{ω},

with

ω

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 are

ω / cm^{- 1} 30346803631220774261 degeneracy 111222 r 0.0683 0.3046 0.5706 0.1698 0.2676 0.7935 S / R per coordinate 0 0.17 0.56 0.02 0.12 0.83 S / R including degeneracy 0 0.17 0.56 0.04 0.24 1.66

Thus the listed frequency groups contribute approximately

0

1.4

4.7

0.33

2.0

and

13.8 J K^{- 1} mo l^{- 1}

, respectively, and

S_{vib, m} ≃ 2.67 R ≃ 22 J K^{- 1} mo l^{- 1} .

The nine coordinates are consistent with

3 N - 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