The runaway temperature of the polar Clausius-Mossotti relation came from the small-…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electric fields in matter · Electric fields in matter

The Boundary-Value Problem, Ch Molecular polarisability, permanent dipoles and the Clausius-Mossotti relation

Problem

The runaway temperature

T_c

of the polar Clausius-Mossotti relation came from the small-

x

form of the Langevin function. This problem keeps the full function and asks what the same model predicts below

T_c

. A liquid has number density

N_v

of molecules with electronic polarisability

\alpha

and permanent moment

p

. No field is applied, and each molecule feels the Lorentz local field

E_{\rm loc}=P/3\varepsilon_0

of a uniform polarisation

P

of the liquid. The mean moment of one molecule along

\mathbf{E}_{\rm loc}

\alpha E_{\rm loc}+pL(x)

, with

L(x)=\coth x-1/x

and

x=pE_{\rm loc}/k_BT

. ((a)) Show that a self-sustaining polarisation requires

L(x)=\frac{T}{3T_c}\,x,\qquad T_c=\frac{N_vp^2}{9\varepsilon_0k_B\left(1-N_v\alpha/3\varepsilon_0\right)},

and that the polarisation it gives is

P=P_sL(x)

with

P_s=N_vp/(1-N_v\alpha/3\varepsilon_0)

. ((b)) Given that

L(x)/x

falls steadily from

1/3

x\to0

towards zero as

x

grows, show that a solution with

x\ne0

exists only when

T<T_c

. ((c)) Just below

T_c

, use

L(x)\approx x/3-x^3/45

to show that

P/P_s\approx\sqrt{5(1-T/T_c)/3}

. ((d)) For liquid water take

N_v=3.34\times10^{28}\,\mathrm{m^{-3}}

p=6.2\times10^{-30}\,\mathrm{C\,m}

and

\alpha/4\pi\varepsilon_0=1.5\times10^{-30}\,\mathrm{m^3}

. Find

T_c

. Then solve the equation of part (a) at

T=293\,\mathrm{K}

for

x

P

and

E_{\rm loc}

, using the large-

x

form

L(x)\approx1-1/x

and checking afterwards that it was accurate. Say what these numbers imply about the model.