Lab Manual- Refractive Index

16
III. REFRACTIVE INDEX OF HCl
A. Theory.
One of the most interesting results of Maxwell’s electromagnetic theory of light is the
relationship κ = n2
, where κ is the dielectric constant (or the relative permittivity, the ratio of the
permittivity of the material to that of the vacuum) and n is the index of refraction. The physical
reason for this relationship can be understood without going into the details of the
electromagnetic theory. The refractive index of a medium is the ratio of the speed of light in a
vacuum to its speed in the medium, n = c/cm. Light always travels more slowly through a
material substance than it does through a vacuum so n > 1. A light wave is a rapidly alternating
electric and magnetic field which acts to polarize the dielectric (an electrical insulator that may
be polarized by an applied electric field, when positive charges move toward the field and
negative charges shift in the opposite direction) through which it passes, pulling the electrons
back and forth in rapid alternation. The greater the polarizability of the molecules, the greater is
the field induced in opposition to the applied field, and the greater therefore is the “resistance” to
the transmission of the light wave. We know that increasing the polarization increases the
dielectric constant and hence we might expect n and κ to be related. The detailed theory1
leads to
the relation κ = n2
. Note that n is complex, with its real part and imaginary part affecting the
refraction and absorption through the medium in different ways.
If n is measured at high frequencies (in the infrared region), then the polarization
contribution (Pµ) due to reorientation of any permanent dipole moment is lost because the
molecule cannot rotate fast enough to oppose the external electric field. Only the atomic (Pa) and
electronic (Pe) polarization remain. These can be further separated by going to visible
frequencies (even higher frequencies) where Pa is lost because the heavy nuclei cannot move fast
enough in response to the external electric field. Under these conditions (which are in fact the
most convenient experimentally), the equation for the total molar polarization (PM) reduces to
0 0M e a e MP P P P P Rµ= + + » + + = where RM is the molar refraction
so 2 2
( 1)/( 2) ( 1)/( 2)M MP R V V n nk k» = - + = - + (6)
Here V = molar volume = RT/P for an ideal gas = V/(N/NA) = (NA×V)/N. Note that RM has the
dimensions of volume/mole and NA is Avogadro’s constant. And since n is generally very close
to 1, we can derive (you can prove it in your report) !" ∝ $ ∙ ('(
-1) ,.
CF 10/2020

17
Some insight into the polarizability of an atom can be gained from the following simple
model. If Z electrons are uniformly distributed about a nucleus of charge eZ , but confined to a
spherical volume of radius 0r , the effect of an external field 0E would be to shift the center of
the spherical cloud of electrons a distance r from the nucleus in order to oppose the applied field.
This crude picture of a polarized atom is shown below.
In such a situation the induced dipole moment ( µ ) is the product of the charge eZ and
the displacement distance r between the nucleus and the center of the cloud of electrons
0eZ r Eµ a= × º × (7)
where the latter equality is the defining relation of the polarizability a .
The field from a spherically symmetrical shell of charge is zero inside the shell so that the
nucleus does not feel a force from the electrons beyond r. The electrons within the radius r act
as if they were concentrated at the center of the cloud where r = 0. The number of electrons
within this sphere is 3
0( / )Z r r , because the density of charge has been assumed to be uniform
within the sphere. Therefore, from Coulomb’s law, the force on the nucleus is
2 3 2
1 2 0/ [ ][ ( / ) ]/F q q r eZ eZ r r r= × = - (8)
This force on the nucleus is balanced by that produced by the external field 0F eZ E= - × (note
the unit of N/C is the electric field strength), so that equating these forces we obtain
3
0 0/E eZ r r= × (9)
r0
Nucleus
r
Electron
cloud
No field Field

18
From this result, from Eq. 7, and from the definition of polarizability, a , we obtain
3
0 0/ E ra µº = (10)
This remarkably simple result states that the induced polarizability is proportional to the volume
of the atom, hence the unit of polarizability is volume. On a molar basis, RM = 4πNAα/3.
For molecules, the molar refraction is approximately the sum of the refractions of the
electron groups within it. For example, assuming that RM(H) = 1/2RM(H2), one obtains a value of
1.03 cm3
/mol for the H atom. By extension, using this and the observed RM = 6.70 cm3
/mol for
CH4, one obtains RM(C) ≈ 2.60 cm3
/mol. Using a similar procedure on many compounds,
average atomic values of RM have been deduced and tables are available.2,3
(Additional
corrections are necessary for compounds with multiple π bonds since these are especially
polarizable and might deviate significantly from the simple summation of atomic volumes.) The
atomic value for Cl is 5.84 cm3
/mol. So RM(HCl)=6.87 cm3
/mol. Pay attention to the units.
The RM values are usually based on index of refraction measurements using the yellow
sodium D line (doublet emission at 589.0, 589.6 nm) so they are often given the symbol RD. In
the absence of absorption, n and RM vary only slowly with wavelength in the visible region and
hence it is acceptable to use a He-Ne laser (632.8 nm) to measure n, as we do in this experiment.
B. Interferometry.
Fig. 3. Michelson interferometer setup for index of refraction measurement
Count by Photodiode
as the cell is evacuated

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In this experiment we will measure the index of refraction (n) of HCl by means of
a Michelson interferometer (essentially the same configuration as the famous LIGO:
Laser Interferometer Gravitational-Wave Observatory) as shown above.
Light from a Helium-Neon (He-Ne) laser enters from the left (when facing the
Screen). The purpose of the lens is to provide a diverging source for clear fringes. The
incident beam impinges upon the beamsplitter B, which is a mirror that has been partially
coated with aluminum so that half of the light is reflected while the other half passes
through. The reflected portion, which we shall call Beam 1, traverses Path 1 and is
reflected back to B by mirror M1. Half of Beam 1 is transmitted through B to a screen E.
The other portion is reflected from B, as shown by the dashed line and is lost.
The transmitted portion of the original beam, which we shall call Beam 2, travels
through a gas cell to M2 and half of this is reflected to E. If the difference in path length
that Beams 1 and 2 must travel to get to E is a whole number of wavelengths they will
combine in phase so that a maximum in intensity will occur. However, if the difference is
an odd integer multiple of half wavelengths the two beams will be 180° out of phase and
destructive interference will occur, resulting in a minimum in intensity. Since the light
source is a diverging one, these interferences are revealed as a set of concentric circular
intensity maxima and minima. If the (reflective) mirrors are not exactly flat or
perpendicular the fringes will depart from their circular appearance.
We now proceed to evacuate a portion of path 2 by pumping out the gas from the
cell chamber. Since the light will travel faster along the evacuated portion the effective
path length along Path 2 will be shortened, hence the difference in path lengths traveled
by Beams 1 and 2 will change. This will result in a shift in the interference fringes. If the
evacuation is done slowly enough the experimenter can count the number of fringes
formed in the center as the pattern shifts. In this way one can determine by what number
of wavelengths the effective path length changes and so can calculate the change in the
index of refraction as the chamber is evacuated.
When light from the monochromatic source travels through the evacuated
chamber its speed is c = 2.99792458 ´ 1010
cm/sec (exact, you can use ca. 2.998). With a
gas in the chamber the light will travel more slowly through it. Since the frequency of the
incoming light is the same whether in a gas or in a vacuum, the wavelength

20
(velocity/frequency) will be shorter when the light is traveling in a gas than it would be in
a vacuum. Let f = frequency of the light (in vacuum or gas, unit is s-1
) and d = length
(standard unit is m) of the vacuum chamber. Since the light travels through the vacuum
chamber twice, once in each direction, a central ray travels a distance 2d through the
chamber (measure the inner wall-to-wall distance). When the chamber is evacuated the
light travels a certain number of wavelengths distance through it. Let this number be Ne,
Ne = 2d/λ = 2df/c where λ = wavelength in vacuum = c/f (11)
With a gas in the chamber the light will travel a greater number of wavelengths distance
through it since the wavelength will be shorter in a medium other than vacuum. Let this
number be Ng,
Ng = 2d/λg = 2df/cg = (2df/c)·n (12)
where λg = cg/f and cg = c/n = velocity of light in the gas, and n is the index of refraction.
Combining (11) and (12), we get Ng = n·Ne.
As the gas is pumped from the chamber the fringe pattern shifts due to the
decrease in the number of wavelengths distance that the light beam travels through it.
The number of fringes formed in the center as the chamber is evacuated from atmospheric
pressure (or any pressure) gives the change in the number of wavelengths. Let it be DN,
DN = Ng − Ne = (n−1)·Ne = (n−1)·2d/λ or n−1 = DN·λ/2d (13)
For error analysis of the observed n value, it comes from the inaccuracy in
measuring DN and d (essentially half of the smallest increment of your ruler).
C. Experimental Procedure.
Attach the glass cell through tubing to the HCl vacuum line. Move the
interferometer assembly into place nearby and place the gas cell in path 2. (Be careful not
to touch the front surface mirrors. Do not wipe them off if they appear dirty – rinse with

21
100% ethanol or methanol and gently wipe with a cotton swab or lens paper.) DO NOT
LOOK INTO THE LASER BEAM! Your eye level should always be above the laser
beam path (safety rule). Protecting your eyes is crucial in laser experiments. To align the
interferometer, remove the diverging lens and send the beam through the two paths and
superimpose the two spots on the screen by adjusting the horizontal and vertical screws
on the mirrors. Now insert the diverging lens. Interference fringes should be visible on
the screen about one foot away. By further fine adjustment of the mirrors we can obtain
clear, circular fringes. The cell may have to be rotated slightly so that the reflections from
the glass windows do not interfere with the fringe pattern. Circular fringes are not a
necessity but are desirable since their larger spacing eases the task of counting. It should
be possible to count each new fringe as it emerges from the center as the cell is emptied.
Position a photodiode at this center point and connect the output to the y input of the
software x-y chart recorder to conveniently record the fringe count as the cell is emptied.
Open the valve to the cell slightly to slowly evacuate any air in the cell and
observe the recorder response. Experiment with air fillings and evacuations to obtain a
feel for the optimum valve opening procedure and recorder scales. Then fill the cell with
approximately one atmosphere HCl and record the exact pressure and room temperature
T(K). (Take care with the gas cylinder valve during this process. Add the gas slowly.)
Next close the HCl gas cylinder valve and its line to the manifold and then open the cell
valve to slowly condense the HCl in the pump cold trap. Count the fringes recorded as
the cell is evacuated. To help distinguish the real fringes from noise, let the detecting
system record the signal a bit longer before your start to fill the cell or after you complete
the evacuation of the cell. The accurate starting and ending points for your fringe
counting should be achieved.
You need to correct the reading to STP conditions (0°C and 1 atm) by multiplying
by the pressure-temperature factor [760/p(torr)]·[T(K)/273] and use STP parameters for T
and P in Eq. 6 (Why? Comment in your report. Hint: check the number of gaseous
molecules using ideal gas law). Or you can simply use your measured (n–1) value and
room temperature with your individual pressure reading (note that this route does not
allow you to directly average the DN values because they are associated with different
pressures while the room temperature remains the same). In that case you need to

22
calculate the n value and use it in Eq. (6) to get the RM value individually, and then
compare between your measurements. Which method is more accurate? Include that
in your data analysis and show your work. Repeat this procedure until consistent
results are obtained for four measurements (even though your pressure reading and DN
value will vary between individual measurement; after STP correction the DN values
become directly comparable which should be close to 140). Average these and compute
the standard deviation of the data points. Measure the length of the gas cell in cm unit
with a ruler (note the smallest measuring increment for error analysis). Do not include the
window thickness.
D. Data analysis.
From your measurements and the wavelength of the light source (632.8 nm, no
error), calculate (n–1) and RM at 0°C and 1 atm. Give the standard deviation of RM.
Compare RM with the value of obtained by adding atomic refractivities. To a first
approximation, RM is independent of temperature or physical state and it provides a rough
measure of the actual total volume (without free space) of the molecules in one mole.
Calculate an effective spherical radius for the HCl molecule from your measured RM
value. Compare it with the literature re value of about 1.27 Å (you will determine a more
accurate value of re from your HCl rotational analysis in part II). Summarize your results
on the attached report form. Is it cool that you can measure the actual molecular size from
a simple optical experiment?
References
1. See, for example, “Electricity and Magnetism”, G. P. Harnwell. McGraw-Hill
(1949). p. 26.
2. Handbook of Chemistry and Physics (under Molar Refraction of Organic
Compounds).
3. “Dielectric Behavior and Structure”, C. P. Smyth, McGraw-Hill (1955).

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Index of refraction report form
Date: _______
Name: __________________________________________________
Partners:______________________________________________________________
Show all data and sample calculations in your report.
Note that the values must correspond to STP conditions (STP = 0°C, 1 atm) — See lab
manual for conversion from your experimental DN values at room temp. and pressure.
[Average values]
DN (# Fringes)(STP) = Standard Dev.____________
(n–1) (STP) = Standard Dev.____________
RM (STP) = cm3
/mol Standard Dev.____________
(Note: you have to use STP correspondingly in Eq. 6 when you correct your DN)
RM (your T and P) = cm3
/mol Standard Dev.____________
RM = cm3
/mol (from sum of atomic refractivities)
Radius of HCl from your RM (STP) = Å (to be compared with the
literature value that is commonly reported at STP conditions)

Lab Manual- Refractive Index

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Lab Manual- Refractive Index