This is a lab question...I don't understand how to do it.
A solution consisting of isobutyl bromide and isobutyl chloride is found to have a refractive index of 1.4194 at 20 degrees celsius. The refractive indices at 20 degrees celsius of isobutyl bromide and isobutyl chloride are 1.4368 and 1.3785, respectively. Determine the molar composition (in percent) of the mixture by assuming a linear relation between the refractive index and the molar composition of the mixture.
From poking around the internet I got that I should be using this equation:
(1.4194)(100) = 1.4368x + 1.3785y
and
x + y = 100
Obviously x + y = 100 is used because we are looking at % composition and it has to add up to 100. But I don't really understand the first equation.
Let X = percent bromide
and Y = percent chloride
Then 1.4368(X) + 1.3785(Y) = mixture*100
You can do it another way.
1.4368-1.3785 = 0.05830 difference between the two pure materials.
So how far up the curve have we moved in going from 1.3785 (pure chloride) to 1.4194 for the mixture.
1.4194-1.3785 = 0.04090.
[0.04090/0.05830]*100 = 70.15% bromide and if I solve the two equations you posted I get 70.15% Br.
OR you can do it a third way.
Just graph it on the x axis moving left to right 100 mole percent Cl (0%Br) to 0%Cl (100 mole % Br).
25% bromide and 75% chloride
The first equation is derived from the assumption that there is a linear relationship between the refractive index and the molar composition of the mixture.
Let's break down the equation step by step to understand it better:
(1.4194)(100) = 1.4368x + 1.3785y
1.4194 is the refractive index of the mixture,
1.4368 is the refractive index of isobutyl bromide,
1.3785 is the refractive index of isobutyl chloride,
x represents the molar composition of isobutyl bromide (in percent),
y represents the molar composition of isobutyl chloride (in percent).
The equation can be interpreted as follows: the refractive index of the mixture is equal to the weighted sum of the refractive indices of the individual components.
By multiplying the refractive indices by their respective molar compositions (x and y) and summing them, we can calculate the refractive index of the mixture. The equation is set equal to (1.4194)(100) since we want the refractive index on one side and the molar compositions on the other side.
The second equation, x + y = 100, represents the fact that the molar compositions of the two components must add up to 100% since it is a binary mixture.
Using these two equations together, we can solve for the values of x and y, which represent the molar compositions of isobutyl bromide and isobutyl chloride in the mixture, respectively.
I hope this clarifies the first equation and its significance in determining the molar composition of the mixture.