# Can we determine the amplitude of vibration of O-H bonds in water?

This is going to be a long shot, and I might be wrong in any and all of my assumptions, be warned (... and please correct me). I might also be trivially right, I have no idea.

At equilibrium, each degree of freedom has the kinetic energy:

$$E = \frac{1}{2}k_\mathrm{B}T$$

So, I have water at, say, $300\ \mathrm K$. Water has three atoms, so 9 degrees of freedom (3 per atom, because space is three-dimensional). One of them is for a $\ce{O-H}$ stretching. I know that usually, you would need to divide into modes of oscillations, but let's say we combine half the energy of the symmetric and antisymmetric stretchings, giving that one $\ce{O-H}$ stretching has a full $\frac{1}{2}k_\mathrm{B}T$ in it. So we have an average, roll of drums,

$$E = \frac{1}{2}(1.3806488 \times 10^{-23})(300) = 2.071\times 10^{-21}\ \mathrm J$$

in a $\ce{O-H}$ stretching.

NOW, let's say this stretching behaves like an harmonic oscillator, its' energy is given by the amplitude of oscillation:

$$E = \frac{1}{2}kA^2$$

where $k$ is the stiffness constant of the chemical bond.

That's nice, but we have two unknowns, $k$ and $A$. So let's go further. If we know the absorption band's wavenumber, we can say that the amplitude corresponds to this wavelength, $2.734\ \mathrm{\mu m}$ (source = Wikipedia). This would mean that we have a stiffness constant:

$$k = 5.5412 \times 10^{-10}\ \mathrm{N/m}$$

I know that, although it was fun to blindly math around, this result is wrong. Just the fact that the stiffness constant would depend on temperature doesn't sound right. Neither does the bit about wavelength = amplitude.

My question is: is it possible to know the amplitude of oscillation/stiffness constant of a $\ce{O-H}$ bond?

My question is: is it possible to know the amplitude of oscillation/stiffness constant of a O−H bond?

Yes.

Your approach is along the right lines but -

The Hooke's Law formula you used

$$E = \frac{1}{2}kA^2$$

is for the stretching of a spring. Hooke's law changes if the spring has mas(ses) attached at the end(s), now it is a harmonic oscillator. In this case Hooke's Law becomes

$$E = \frac {h}{2 \pi} \sqrt{\frac{k}{\mu}}$$

where k is the force constant you are interested in and $\mathrm{\mu}$ is the reduced mass of the system.

Since

$$E=h\nu$$

The frequency, which can be experimentally measured, is given as

$$\mathrm{frequency} = \frac {1}{2 \pi} \sqrt{\frac{k}{\mu}}$$

The frequency is independent of the amplitude, affected only by the masses and the stiffness of the spring.

If the masses of the atoms at the ends of the spring are given as A and B, then

$$\mu =\frac{A \cdot B}{A+B}$$

So for an $\ce{O-H}$ bond $\mathrm{\mu=\frac{1 \cdot 16}{1+16}}\ \mathrm{u}$. If we measure the frequency of the $\ce{O-H}$ stretch absorption, then we can calculate the force constant.

Here is a link to a one-page example that shows the actual computation and explains the specific units to be used very nicely.

You also ask about the amplitude of oscillation. If, by this you mean the bond length, then yes this can also be calculated from spectra (relatively easy in simple diatiomic cases, but more difficult in complex polyatomic cases. The methodology involves assuming a rigid rotor and then using the Schrodinger equation to solve for the energy states and extracting the bond length. A nice example using $\ce{HCl}$ can be found here, look down the page for the section entitled "Bond Length of HCl".

• What I meant with "amplitude" is the deviation from the mean bond length. Your document is very helpful, just to make it clear, though, they deduce the "spring constant" from the wavenumber of the stretching mode's absorption band and the bond length from the wavenumber of a rotational mode's absorption band. Can we return to my idea with HCl at 300K and find the amplitude $A = \sqrt{\frac{2E}{k}} = \sqrt{\frac{2(2.071\times 10^{-21}J)}{481N/m}} = 2.93fm$? Which would seem to fit with the $127fm$ bond length of HCl. Aug 10 '15 at 0:15

Finally, I have a day off to read that question and response. Half-way reading it I was already a bit suspicious mostly because of the half-jolly and half-formal style OP used. :) Don't get me wrong, in principle I have nothing against such style. But the thing is that usually when such style is used some nonsense is eventually introduced at some point, and this nonsense is not so easily detectable since it is buried under all these jolly informalities.

That is exactly what happened this time too.

we can say that the amplitude corresponds to this wavelength

Wait, what? Amplitude has no relation whatsoever to wavelength.

My question is: is it possible to know the amplitude of oscillation/stiffness constant of a $\ce{O-H}$ bond?

Yes, you can calculate the force constants of molecular vibrations (as well as frequencies). These calculations are routinely done in the domain of quantum chemistry these days.

• Perhaps you could expand a bit upon both of your responses. At the moment they are more or less unqualified statements with no explanation.
– bon
Aug 9 '15 at 20:35