Clausius-Mossotti equation
The Clausius-Mossotti equation links the macroscopically measurable quantity of the permittivity number with the microscopic (molecular) quantity of electrical polarizability . It is named after the two physicists Rudolf Clausius and Ottaviano Fabrizio Mossotti and reads:
It is
- the molar polarization (its unit is that of a molar volume , e.g. m ^{3} / mol)
- the molar mass (in kg / mol)
- the density (in kg / m ^{3} )
- the Avogadro constant .
The equation applies to non-polar substances without a permanent dipole moment , i.e. i.e., there are only induced dipoles ( displacement polarization ). The Debye equation is used for substances with permanent dipoles, which takes into account the polarization of the orientation as well as the displacement polarization .
Derivation
The macroscopic polarization is the sum of all induced dipoles divided by the observed volume (the polarization corresponds to a dipole):
where the particle number density , polarizability , local electric field strength at the location of the atom / molecule.
The macroscopically measurable quantities electrical susceptibility or the permittivity number establish the connection between the polarization and the E-field:
The equation is obtained by equating:
In order to be able to make further statements, the local field must be determined.
Side note: For dilute gases the induced dipoles do not influence each other, the local field is equal to the applied external field and from this:
For a dielectric with a higher density, the local field is unequal to the applied external field, since induced dipoles in the vicinity also build up an electric field.
- : externally applied electric field + polarization field generated on the dielectric surface (de-electrification field),
- : Field of polarization charges on the surface of a fictitious sphere around the molecule under consideration ( Lorentz field )
This results in a local E-field of:
Inserted into the above equation:
Switching provides:
Or. by dissolving:
Now one can express the particle density in terms of macroscopically measurable quantities ( density , molar mass and Avogadro's constant ):
Insertion gives the Clausius-Mossotti equation:
Or. by dissolving:
Lorentz – Lorenz equation
The Lorentz – Lorenz equation is another form of the Clausius-Mossotti equation that results from this when the result of the electromagnetic wave equation is used. The Lorentz-Lorenz equation got its name from the Danish mathematician and scientist Ludvig Lorenz , who published it in 1869, and the Dutch physicist Hendrik Lorentz , who independently derived it and published it in 1878.
The Lorentz-Lorenz equation is therefore:
Like the Clausius-Mossotti equation, the equation is also approximately valid for homogeneous solids and liquids.
For most gases the following applies , which is why it is approximate that
and with the help of
This formula can be used for gases under normal pressure. The refractive index of the gas can then Hilfer the Molrefraction as
are expressed with the pressure of the gas , is the gas constant, and the (absolute) temperature, which together determine the particle number density . Accordingly , with the molar concentration. If one substitutes for the complex refractive index , with the absorption index , the result is:
Accordingly, the imaginary part, i.e. the absorption index, is proportional to the molar concentration
and thus to the absorbance . Accordingly, Beer's law can be derived from the Lorentz-Lorenz equation. The change in the refractive index in dilute solutions is therefore also approximately proportional to the molar concentration.
literature
- Richard P. Feynman, Robert B. Leighton, Matthew Sands: Lectures on Physics, Volume II . Definitive ed. Addison-Wesley, 2005, ISBN 0-8053-9047-2 .
Individual evidence
- ^ Thomas Günter Mayerhöfer, Jürgen Popp: Beyond Beer's law: Revisiting the Lorentz-Lorenz equation . In: ChemPhysChem . n / a, n / a, May 12, 2020, ISSN 1439-4235 , doi : 10.1002 / cphc.202000301 .
- ↑ Thomas G. Mayerhöfer, Alicja Dabrowska, Andreas Schwaighofer, Bernhard Lendl, Jürgen Popp: Beyond Beer's Law: Why the Index of Refraction Depends (Almost) Linearly on Concentration . In: ChemPhysChem . tape 21 , no. 8 , April 20, 2020, ISSN 1439-4235 , p. 707-711 , doi : 10.1002 / cphc.202000018 .