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Dam7

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- TL;DR Summary
- I'm trying to understand the article 'Onsager, Lars. Electric moments of molecules in liquids. Journal of the American Chemical Society, 1936, 58: 1486-1493' in which the author investigates the dielectric constant for a polar liquid or solution.

The model that he uses is a dielectric in which there is a spherical cavity with a dipole at its center. The dipole ##\vec{m}## has a component due to a permanent dipole and a component due to an induced dipole (because of polarization).

In order to obtain the dipole moment in the cavity, the author gets the following formula:

$$\left(1-\frac{2(\epsilon-1)\alpha}{(2\epsilon+1)a^3}\right)\vec{m}=\mu_0\hat{u}+\frac{3\epsilon}{2\epsilon+1}\alpha\vec{E}$$

where ##\epsilon## is the dielectric constant of the dielectric outside the cavity, ##a^3## is the radius of the cavity (that should be at the order of molecule dimensions), ##\hat{u}## is the direction of the permanent dipole and ##\alpha## should be the polarizability of the dipole.

Now, he substitutes the following (and here my doubts start):

$$\alpha=\frac{n^2-1}{n^2+2}a^3$$

where ##n## is the 'internal refractive index' (as he says). I think that ##n## is simply the refractive index of the solute.

This is the first doubt: what is exactly ##n##? (I will explain the reason for this doubt soon)

So if the solute is different from the solvent I have that ##\epsilon\neq n^2## because ##\epsilon## is the dielectric constant of the solvent and ##n## is the refractive index of the solute.

The second doubt is: is the latter equation that I wrote the Clausius-Mossotti equation?

Otherwise, I don't understand where it comes from despite the fact that the entire publication seems to be critical of the Clausius-Mossotti equation for polar stuff.

After substituting, mediating the direction of the permanent dipole with Boltzmann distribution, and using the relation ##\vec{P}=N\langle\vec{m}\rangle##, he gets:

$$\vec{P}=N\left(\frac{\epsilon(n^2+2)^2(2\epsilon+1)}{9kT(2\epsilon+n^2)^2}\mu_0^2+\frac{\epsilon(n^2+2)}{2\epsilon+n^2}\alpha\right)\vec{E}$$

where ##\vec{P}## is the polarization, ##N## is the numerical density, ##k## is the Boltzmann constant, ##T## is the temperature and ##\mu_0## is the permanent dipole moment in the vacuum.

The third doubt is: what kind of polarization is ##\vec{P}##?

In my opinion, it is the polarization only of the solute because, in order to get the total polarization, we would also take into account the polarization of the dielectric, right?

At this point, the author wants to investigate the dielectric constant of a pure liquid.

In order to do that, the author uses the latter formula with the formula ##(\epsilon-1)\vec{E}=4\pi\vec{P}## and gets:

$$\epsilon-1=4\pi N\frac{\epsilon(n^2+2)^2(2\epsilon+1)}{9kT(2\epsilon+n^2)^2}\mu_0^2+\frac{3\epsilon(n^2-1)}{2\epsilon+n^2}$$

The last doubt is: why the author continues to write ##\epsilon## and ##n^2## also for a pure liquid? Aren't the solute and the solvent the same species and ##\epsilon=n^2##?

But I'm quite sure that for the author the two are different also because at a certain point, he writes ##\frac{\epsilon-1}{\epsilon+2}-\frac{n^2-1}{n^2+2}## that should be zero!

Thanks in advance to everybody that will answer. I'm sorry for the length but I wanted to be as clear as possible.

In order to obtain the dipole moment in the cavity, the author gets the following formula:

$$\left(1-\frac{2(\epsilon-1)\alpha}{(2\epsilon+1)a^3}\right)\vec{m}=\mu_0\hat{u}+\frac{3\epsilon}{2\epsilon+1}\alpha\vec{E}$$

where ##\epsilon## is the dielectric constant of the dielectric outside the cavity, ##a^3## is the radius of the cavity (that should be at the order of molecule dimensions), ##\hat{u}## is the direction of the permanent dipole and ##\alpha## should be the polarizability of the dipole.

Now, he substitutes the following (and here my doubts start):

$$\alpha=\frac{n^2-1}{n^2+2}a^3$$

where ##n## is the 'internal refractive index' (as he says). I think that ##n## is simply the refractive index of the solute.

This is the first doubt: what is exactly ##n##? (I will explain the reason for this doubt soon)

So if the solute is different from the solvent I have that ##\epsilon\neq n^2## because ##\epsilon## is the dielectric constant of the solvent and ##n## is the refractive index of the solute.

The second doubt is: is the latter equation that I wrote the Clausius-Mossotti equation?

Otherwise, I don't understand where it comes from despite the fact that the entire publication seems to be critical of the Clausius-Mossotti equation for polar stuff.

After substituting, mediating the direction of the permanent dipole with Boltzmann distribution, and using the relation ##\vec{P}=N\langle\vec{m}\rangle##, he gets:

$$\vec{P}=N\left(\frac{\epsilon(n^2+2)^2(2\epsilon+1)}{9kT(2\epsilon+n^2)^2}\mu_0^2+\frac{\epsilon(n^2+2)}{2\epsilon+n^2}\alpha\right)\vec{E}$$

where ##\vec{P}## is the polarization, ##N## is the numerical density, ##k## is the Boltzmann constant, ##T## is the temperature and ##\mu_0## is the permanent dipole moment in the vacuum.

The third doubt is: what kind of polarization is ##\vec{P}##?

In my opinion, it is the polarization only of the solute because, in order to get the total polarization, we would also take into account the polarization of the dielectric, right?

At this point, the author wants to investigate the dielectric constant of a pure liquid.

In order to do that, the author uses the latter formula with the formula ##(\epsilon-1)\vec{E}=4\pi\vec{P}## and gets:

$$\epsilon-1=4\pi N\frac{\epsilon(n^2+2)^2(2\epsilon+1)}{9kT(2\epsilon+n^2)^2}\mu_0^2+\frac{3\epsilon(n^2-1)}{2\epsilon+n^2}$$

The last doubt is: why the author continues to write ##\epsilon## and ##n^2## also for a pure liquid? Aren't the solute and the solvent the same species and ##\epsilon=n^2##?

But I'm quite sure that for the author the two are different also because at a certain point, he writes ##\frac{\epsilon-1}{\epsilon+2}-\frac{n^2-1}{n^2+2}## that should be zero!

Thanks in advance to everybody that will answer. I'm sorry for the length but I wanted to be as clear as possible.