# Period of Vibrational Modes

I am trying to understand more about the period of vibrational nodes. Currently, I am interested in the wavelength of the $\ce{Si-O}$ bond stretch in silica. I think this value is roughly $1000\mathrm{~cm^{-1}}$.

I am somewhat confused on how to find the period of vibrational modes. I am new to this subject. When I look up the frequency, like I said, I get units of $\mathrm{cm^{-1}}$, but I want units of $\mathrm{s^{-1}}$.

Any pointers on how I can get this value?

Is this the way to calculate it:

$$T = \frac{1}{\nu}$$

where

$$\nu = c \bar{\nu}$$

with $T$, $\nu$, and $\bar{\nu}$ being the period, frequency, and wavenumber respectively?

With this I get:

\begin{align} T &= \frac{1}{c\bar{\nu}} \\ &= \frac{1}{(3 \times 10^{10}\mathrm{~cm~s^{-1}})(1000\mathrm{~cm^{-1}})} \\ &= 33.33\mathrm{~fs} \end{align}

Also, as temperature increases, wouldn't the period decrease? But in the literature, I find that as temperature increases, the wavenumber decreases. Thus, the period increases. Why would the period increase? I would think at a higher temperature, bond stretching would occur more quickly. Is this not the case?

• $P=\lambda/c$. Assuming $\lambda=1*10^{-5}$ (i.e. 1000 wavelengths in 0.01 m) and $c=3*10^8$ m/s have $3.(3)*10^{-14}$ s perios.... yeah your calculation is indeed correct. – permeakra Jul 23 '15 at 16:14
• @permeakra OK, so here is what I am confused about: As temperature increases, wouldn't the period decrease? But in the literature, I find that as temperature increases, the wavelength decreases. Thus, the period increases. Why would the period increase? I would think at higher temperature, bond-stretching would occur more quickly. Is this not the case? – Jackson Hart Jul 23 '15 at 16:21
• Molecular vibrations are usually approximted as harmonic oscillator. In it no matter how much energy of oscillation is, the period is the same, only aplitude differs. (in truth the approximation breaks at higher vibration levels, but lower ones usually are good enough, and they are the only ones active in IR spectra) – permeakra Jul 23 '15 at 17:33
• @permeakra Interesting, but I have recently read a paper where the frequency (cm^-1) decreased as a function of temperature. If energy does not matter, why would it decrease? – Jackson Hart Jul 23 '15 at 18:38
• (un)harmonic oscillator approximation ignores other energy levels in the molecule. However, change of temperature may change relative occupation of other energy level, say, rotational levels in gas phase or electronic states in paramagnetic molecules. Change of relative occupation of rotational levels will move and change apparent IR peak position, see wikipedia://Rotational–vibrational+coupling. In solutions change of dielectric constant of solven also may have influence. Unfortunately, can't point the exact reason without the article at hand. – permeakra Jul 23 '15 at 20:34