Suppose that a 2.50x10^-6M solution of red dye No. 2 had an absorbance of 0.585 at its wavelength of maximum absorbance. A second solution of red dye No. 2 is too concentrated to read its absorbance so 5.00 mL of the concentrated dye solution was diluted with water to a volume of 150 mL. The diluted solution has an absorbance of 0.289, measured under identical conditions. What is the molarity of dye in the concentrated solution?

For the first one,

Absorbance = kc
A = 0.585, solve for k, c =- 2.50 x 10^-6M

Use k in the second run of the concd stuff.
A = kc
You will get a concn that must be adjusted because of the dilution. How much was it diluted. From 5 to 150 mL; therefore, the concn will whatever you calculate x the factor 150/5.
Or you can look at it another way. You can say that the absorbance would have been higher than 0.289 by 0.289 x (150/5), then work A = kc the normal way.

To determine the molarity of the dye in the concentrated solution, we need to use the concept of Beer-Lambert Law, which states that the absorbance (A) of a sample is directly proportional to the concentration (C) of the absorbing species and the path length (l) of the sample.

The equation for Beer-Lambert Law is:
A = εcl

Where:
A = Absorbance
ε = Molar absorptivity (a constant specific to the absorbing species and wavelength)
c = Concentration in Molarity
l = Path length

We can use this information to set up a ratio between the two solutions:

A1 / A2 = (εc1l) / (εc2l)

Since the path length (l) and molar absorptivity (ε) are identical for both solutions, they cancel out:

A1 / A2 = c1 / c2

Now we can substitute the given values:

A1 = 0.585 (absorbance of the first solution)
A2 = 0.289 (absorbance of the diluted solution)
c1 = unknown (molarity of the concentrated solution)
c2 = 2.50x10^-6 M (molarity of the diluted solution)

Plugging in these values, we get:

0.585 / 0.289 = c1 / 2.50x10^-6

Now we can solve this equation for c1:

c1 = (0.585 / 0.289) * (2.50x10^-6)

Calculating this, we find:

c1 ≈ 5.03x10^-6 M

Therefore, the molarity of the dye in the concentrated solution is approximately 5.03x10^-6 M.