problems 4-7 is a practical problem related to the
Clausius-Mossotti Equation (5-10); the aim here is to
practice some spreadsheet work with real data.
Tabulated data reports the dielectric constant (K for us but
labeled as "epsilon" in this text) for air and pentane at
various pressures. Higher pressure will compress the
material, so there are more molecules to polarize per m^3 and
hence a larger dielectric constant. While the dielectric constant
should increase due to compression, to the extent that the
molecules remain independent when at high pressure, the
molecular property of polarizability, "alpha" (Eq. 5-8),
given by the Clausius=-Mossotti Equation should be constant.
The Clausius-Mossotti Equation requires us to know N (number
of molecules per m^3) in addition to K to calculate alpha.
The ideal gas law says:
P/NkT = 1
where P=pressure (in pascal), k=Boltzmann's constant,
T=temperature in K, but high pressure air is not exactly
ideal. The problem suggests looking up the density of high
pressure air in the American Institute of Physics Handbook,
but I'll save you that work. Using WAPP I've found that
P/NkT = 1.0005 -0.5209E-03*x+0.3021E-05*x^2
where x=pressure in *atm*.
(You should find that the corrections are quite
small, even at 100 atm; at T=200 the corrections are
significantly larger ~20%.). Once you've determined the rhs
you can use P (now in pascal) to calculate N. (N at STP is
known as Loschmidt number: 2.69e25 m^(-3).)
Plan: enter the values for K into a spreadsheet, calculate
alpha for the given pressures and discuss: how constant is
alpha for air. (For me the result at 60 atm seems suspicious.
Why?)
For pentane (a petroleum-like liquid) N can be calculated from its density and molar mass.
For pentane, I'm not sure I trust the result at 12000 atm. Why?
Note the "epsilon" in these problems is the dielectric constant we've called K.
SO use Eq. 5-10 to calculate alpha at different pressures...spreadsheet is the way-to-go.
According to our theory alpha is an atomic constant: it should not depend (much) on pressure.
Since it should all be in a spreadsheet, I see no reason to test the simple version
of Clausius-Mossotti for pentane: use the correct formula (5-10) to find alpha.
Eq. 5-13 relates (through very approximate theory) alpha to the radius of an atom.
Use this crude approximation to determine a "radius" for air and pentane. (Of course
neither are spheres so the result is only order-of-magnitude.)