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: ${\displaystyle \varepsilon _{\mathrm {r} }}$ ${\displaystyle \alpha }$

${\displaystyle P_{m}={\frac {\varepsilon _{\mathrm {r} }-1}{\varepsilon _{\mathrm {r} }+2}}{\frac {M_{m}}{\rho }}={\frac {N_{\mathrm {A} }}{3\,\varepsilon _{0}}}\alpha }$

It is

• ${\displaystyle P_{m}}$the molar polarization (its unit is that of a molar volume , e.g. m ³ / mol)
• ${\displaystyle M_{m}}$the molar mass (in kg / mol)
• ${\displaystyle \rho }$the density (in kg / m ³ )
• ${\displaystyle N_{\mathrm {A} }}$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): ${\displaystyle {\vec {P}}}$ ${\displaystyle {\vec {p}}_{\text{ind}}}$

${\displaystyle {\vec {P}}=N{\vec {p}}_{\text{ind}}=N\alpha {\vec {E}}_{\text{lokal}}}$

where the particle number density , polarizability , local electric field strength at the location of the atom / molecule. ${\displaystyle N}$ ${\displaystyle \alpha }$ ${\displaystyle {\vec {E}}_{\text{lokal}}}$

The macroscopically measurable quantities electrical susceptibility or the permittivity number establish the connection between the polarization and the E-field: ${\displaystyle \chi }$ ${\displaystyle \varepsilon _{\mathrm {r} }}$

${\displaystyle {\vec {P}}=\chi \varepsilon _{0}{\vec {E}}=\left(\varepsilon _{\mathrm {r} }-1\right)\varepsilon _{0}{\vec {E}}}$

The equation is obtained by equating:

${\displaystyle \left(\varepsilon _{\mathrm {r} }-1\right)\varepsilon _{0}{\vec {E}}=N\alpha {\vec {E}}_{\text{lokal}}}$

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: ${\displaystyle {\vec {E}}_{\text{lokal}}={\vec {E}}}$

${\displaystyle \left(\varepsilon _{\mathrm {r} }-1\right)={\frac {N}{\varepsilon _{0}}}\alpha }$

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.

${\displaystyle {\vec {E}}_{\text{lokal}}={\vec {E}}+{\vec {E}}_{\text{L}}}$

${\displaystyle {\vec {E}}}$: externally applied electric field + polarization field generated on the dielectric surface (de-electrification field),

${\displaystyle {\vec {E}}_{\text{L}}={\vec {P}}/(3\varepsilon _{0})}$: 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:

${\displaystyle {\vec {E}}_{\text{lokal}}={\vec {E}}+{\frac {1}{3\varepsilon _{0}}}{\vec {P}}={\vec {E}}+{\frac {\left(\varepsilon _{\mathrm {r} }-1\right)\varepsilon _{0}}{3\varepsilon _{0}}}{\vec {E}}={\frac {\varepsilon _{\mathrm {r} }+2}{3}}{\vec {E}}}$

Inserted into the above equation:

${\displaystyle \left(\varepsilon _{\mathrm {r} }-1\right)\varepsilon _{0}{\vec {E}}=N\alpha {\frac {\varepsilon _{\mathrm {r} }+2}{3}}{\vec {E}}}$

Switching provides:

${\displaystyle {\frac {\varepsilon _{\mathrm {r} }-1}{\varepsilon _{\mathrm {r} }+2}}={\frac {N\alpha }{3\varepsilon _{0}}}}$

Or. by dissolving: ${\displaystyle \varepsilon _{r}}$

${\displaystyle \varepsilon _{\mathrm {r} }=1+\chi _{e}=1+{\frac {3N\alpha }{3\varepsilon _{0}-N\alpha }}}$

Now one can express the particle density in terms of macroscopically measurable quantities ( density , molar mass and Avogadro's constant ): ${\displaystyle N}$ ${\displaystyle \rho }$ ${\displaystyle M_{m}}$ ${\displaystyle N_{\mathrm {A} }}$

${\displaystyle N={\frac {N_{A}\rho }{M_{m}}}}$

Insertion gives the Clausius-Mossotti equation:

${\displaystyle {\frac {\varepsilon _{\mathrm {r} }-1}{\varepsilon _{\mathrm {r} }+2}}{\frac {M_{m}}{\rho }}={\frac {N_{\mathrm {A} }}{3\varepsilon _{0}}}\alpha }$

Or. by dissolving: ${\displaystyle \varepsilon _{\mathrm {r} }}$

${\displaystyle \varepsilon _{\mathrm {r} }=1+\chi _{e}=1+{\frac {3N_{\mathrm {A} }\rho \alpha }{3M_{m}\varepsilon _{0}-N_{\mathrm {A} }\rho \alpha }}}$

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. ${\textstyle \varepsilon _{\mathrm {r} }=n^{2}}$

The Lorentz-Lorenz equation is therefore:

${\displaystyle {\frac {n^{2}-1}{n^{2}+2}}={\frac {N\alpha }{3\varepsilon _{0}}}}$

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 ${\displaystyle n^{2}\approx 1}$

${\displaystyle n^{2}-1\approx {\frac {N\alpha }{\varepsilon _{0}}}}$

and with the help of ${\displaystyle {n^{2}-1}\approx 2(n-1)}$

${\displaystyle n-1\approx {\frac {N\alpha }{2\varepsilon _{0}}}}$

This formula can be used for gases under normal pressure. The refractive index of the gas can then Hilfer the Molrefraction as ${\displaystyle n}$ ${\displaystyle A}$

${\displaystyle n\approx {\sqrt {1+{\frac {3Ap}{RT}}}}}$

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: ${\displaystyle p}$ ${\displaystyle R}$ ${\displaystyle T}$ ${\displaystyle N}$ ${\displaystyle N=N_{\mathrm {A} }\cdot c}$ ${\displaystyle c}$ ${\displaystyle n}$ ${\displaystyle m=n+ik}$ ${\displaystyle k}$

${\displaystyle m\approx 1+c{\frac {N_{\mathrm {A} }\cdot \alpha }{2\varepsilon _{0}}}}$

Accordingly, the imaginary part, i.e. the absorption index, is proportional to the molar concentration

${\displaystyle k\approx c{\frac {N_{\mathrm {A} }\cdot \alpha ''}{2\varepsilon _{0}}}}$

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

1. ^ 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 .
2. ↑ 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 .