Investigating the variation of refractive index with solute

Investigating the variation of refractive index with solute concentration
- a possible approach
NB. A Google search with the key words “variation refractive index solute
concentration” throws up several useful sites. The approach outlined below derives
from one of them.
Consider a cell made out of glass as
shown in the plan view. This could
be made from microscope slides
and a suitable adhesive. It is
advantageous to keep the cell small.
If a beam of light is allowed to pass
through the cell it will emerge
deviated because of the
combination of the oblique partition
and the differential refractive index.
Consider the light ray ABCD in the
second diagram. It is incident normally on the first glass face and passes into liquid 1
of refractive index n. In the analysis which follows, the parallel-sided glass pieces are
ignored as (if they are indeed parallel-sided) they will have no effect on the final
emergent angle .
In the diagram: 
= the angle of the oblique partition
= the angle of incidence on the liquid-liquid interface at B
= the angle of refration at the liquid-liquid interface at B
= the angle of emergence of the ray from the cell at C.
By considering a triangle defined by the ray BC, the oblique glass
partition and the right-hand end of the cell, it is straight-forward to
show that the angle of incidence at C is  - .
Applying Snell’ Law at B and C respectively:
n sin   n  n  sin 
 n  n  sin      1sin 
(as the refractive index of the air = 1.00)
If  and n are small, then so will ,  -  and  be and , so we can
apply the small-angle approximation to these equations which then
n   n  n 
 n  n       
The subtraction of one equation from the other and a small amount
of manipulation leaves:
   n
So a measurement of  and a knowledge of  allow n to be found.
Practical details
It is probably easiest to use a laser pointer as the light source (monochromatic and
well-collimated) and determine  by measuring the displacement of the light spot at
the end of a long laboratory.
e.g. for  = 50 and n = 0.01, a 2.5 mm displacement is obtained from a 3 m light
Clearly for n = 0.1 then the displacement would be 25 mm, which should be easily
NB. Laser precautions.
Gareth Kelly
January 2003