Xjyuk

My understanding is that a stronger bond has a higher wavenumber in IR spectrum. But why does the C–H vibration have a higher wavenumber than the C=O vibration? The latter is a double bond, so I think it should be a stronger bond than the C–H single bond.

A property of the harmonic oscillator is that the oscillation frequency, $omega$, is dependent not only on $k$ (the spring constant) but also on the mass $m$ of the object:

$$omega = sqrtfrackm$$

We can crudely model a chemical bond as a two-body harmonic oscillator, which largely obeys the same rule, except that the mass must be replaced with the reduced mass $mu$, defined by:

$$frac1mu = frac1m_1 + frac1m_2$$

where $m_1$ and $m_2$ are the masses of the two bodies. So, for a C–H bond, we have

$$frac1mu_ceCH = frac1pu12 u + frac1pu1 u implies mu_ceCH = pu0.923 u$$

and for a C=O bond, we have

$$frac1mu_ceCO = frac1pu12 u + frac1pu16 u implies mu_ceCO = pu6.86 u$$

From J. Chem. Phys. 1946, 14 (5), 305–320 the force constant $k$ for the C=O bond in formaldehyde is approximately $2.46$ times that for the C–H bond in methane (the units are in the old cgs system, so I am lazy to quote the actual values).

So, we can come up with a very rough theoretical estimate. The wavenumber used in IR spectroscopy, denoted by $barnu$, is directly proportional to the (angular) frequency $omega$.

$$beginalign
fracbarnu_ceCHbarnu_ceCO &= fracomega_ceCH / 2pi comega_ceCO / 2pi c \
&= fracomega_ceCHomega_ceCO \
&= sqrtfrack_ceCHk_ceCO cdot fracmu_ceCOmu_ceCH \
&= sqrtfrac12.46 cdot fracpu6.86 upu0.923 u \
&= 1.74
endalign$$

This is not too far from the experimental value of $pu2900 cm-1/pu1730 cm-1 = 1.68$ (using values in the middle of the typical ranges for alkanes and aldehydes).

According to Skoog, Analytical Chemistry:

Using classical mechanics, assuming a diatomic molecule, the frequency of vibration $nu$ may be described by
$$nu = frac12pi sqrtfrackmu $$

where $k$ represents the force constant of this bond, and $mu$ the reduced mass of the particles bond together, defined as

$$mu = fracm_1 cdot m_2 m_1 + m_2$$

An equivalent quantum chemical / mechanial description includes $c$, the speed of light, yielding a result expressing the radiation in wavenumbers:
$$barnu = frac12pi cdot c sqrtfrackmu $$

So (1), with the same reduced mass $mu$, the observed frequency increases with increasing force constant $k$. And (2) $nu$ equally increases while lowering the reduced mass $mu$.

Skoog further mentions as typical force constant of single bonds the range of $3 times 10^2 ldots 8 times 10^2,puN m^-1 $ with an average of $5 times 10^2,puN m^-1$; while stating $1 times 10^3,puN m^-1$ and $1.5 times 10^3,puN m^-1$ for typical double, and triple bonds, respecitively.

Consequently, for a carbonyl $ceC=O$ double bond, assuming as the mass $m_1$ of a carbon atom of

$$m_1 = frac12 times 10^-3,pukg / mol 6.0 times 10^23,puatoms / mol times 1,puatom = 2.0 times 10^-26,pukg$$

and for oxygen a mass $m_2 = 2.7 times 10^-26,pukg / atom$,
yields a reduced mass $mu = 1.1 times 10^-26,pukg$. Using the above mentioned equation determines as vibration frequency (in wavenumbers) a value of

$$barnu = 5.3 times 10^-12,pus cdot cm^-1 sqrtfrac1 times 10^3,puN cdot m^-11.1 times 10^-26,pukg = 1.6 times 10^3pucm^-1$$.