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}}}$

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

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

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:

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

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

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

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.

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

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

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 .