Combination differences
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص458-459
2025-12-06
32
Combination differences
The rotational constant of the vibrationally excited state, B1 (in general, Bv), is in fact slightly smaller than that of the ground vibrational state, B0, because the anharmonicity of the vibration results in a slightly extended bond in the upper state. As a result, the Q branch (if it exists) consists of a series of closely spaced lines. The lines of the R branch converge slightly as J increases; and those of the P branch diverge:
vP(J) = v−(B1+B0) J+(B1−B0)J2
vQ(J) = v+(B1−B0) J(J+1)
vR(J) = v+(B1+B0) (J+1) +(B1−B0) (J+1)2
To determine the two rotational constants individually, we use the method of com bination differences. This procedure is used widely in spectroscopy to extract information about a particular state. It involves setting up expressions for the difference in the wavenumbers of transitions to a common state; the resulting expression then depends solely on properties of the other state. As can be seen from Fig. 13.36, the transitions vR(J − 1) and vP(J + 1) have a common upper state, and hence can be anticipated to depend on B0. Indeed, it is easy to show from eqn 13.63 that
vR (J − 1) −vP (J +1) =4B0(J+ 
)
Therefore, a plot of the combination difference against J + 1–2 should be a straight line of slope 4B0, so the rotational constant of the molecule in the state v = 0 can be determined. (Any deviation from a straight line is a consequence of centrifugal distortion, so that effect can be investigated too.) Similarly, #R(J) and #P(J) have a common lower state, and hence their combination difference gives information about the upper state:
vR(J) − vP(J) = 4B1(J + )
The two rotational constants of 1H35Cl found in this way are B0 = 10.440 cm−1 and B1 =10.136 cm−1.
Fig. 13.36 The method of combination differences makes use of the fact that some transitions share a common level.
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