How do you determine concentration of a sample using wavelength and absorbency? Say our wavelength for a sample of Red 40 is 508.50 nm and absorbance is 0.283 when we use 6ml water and 4 ml Red 40. What would be our concentration and the slope ε? This is really frustrating me so any help would be appreciated.
I don't know anything about Red 40, but you will need the extinction coefficient. Perhaps you determined this from a plot of concentration vs absorbance in the lab?
I am going to attempt to answer you question with reference to NADH, which has an accepted extinction coefficient of 6220 M-1.cm-1 at 340 nm.
Say I have a solution of NADH in a test-tube, I take 4 ml of this and dilute with water to 10 ml. I then take this diluted solution and measure its absorbance at 340 nm in a spectrophotometer using a cuvette with a path-length of 1 cm. The value I obtain is 0.4. (Edit. In the above measurement, I take care to 'blank' the spectrophotometer against water).
What is the concentration of NADH in the original solution?
From the Beer-Lambert law (where c is concentration)
0.4 = 6220 x c x 1
c = 0.64 x 10-6 M = 0.64 µ M
But the solution on which the measurement was made was diluted 4 in 10 or 1 in 2.5
Therefore the concentration of NADH in the original solution is
c = 2.5 x 0.64 x 10-6 M = 160.77 x 10-6 M = 160.77 µ M
So, the concentration of NADH in the original solution is about 161 µ M (micro-Molar).
I am making a number of assumptions here, the main one being that the spectrophotometer obeys the Beer-Lambert law in the absorbance region of interest. You should determine that this is so by measuring the absorbance of series of NADH solutions and make sure that the resulting graph of absorbance vs concentration is linear and through the origin.
At the very least, doubling the concentration of NADH should exactly double the absorbance (0.8) and halving it should give exactly half the absorbance (0.2). In addition, measurement of water (or whatever you are 'blanking against) should give zero - that is the spectrophotometer should be properly 'blanked'. If not you will need to subtract this value.
So what happens if we do not dilute our original sample, but just measure the absorbance at 340nm directly?
A = 6220 x 160.77 x 10-6 = 1
The original solution is predicted to have an absorbance of 1, but IMO even with modern spectrophotometers you are 'pushing your luck' to assume that Beer-Lambert is obeyed at such a high absorbance. A 'rule-of-thumb' is to keep the absorbance below 0.8.
One final comment. It is very difficult to accurately determine an extinction coefficient from absorbance versus concentration plots, unless you accurately know the concentration of the original material. If this figure relies on a 'dry-weight' measurement, you will need to be sure that the compound is pure, for a start.
Reading your question again, I think we need a bit more experimental detail, and if you could supply it, I'll have another go, maybe.
NADH is widely used in the biochemical sciences where (among many other things) the relatively high extinction coefficient of 6220 M-1.cm-1 is exploited in continuous enzyme assays. If an enzyme produces or consumes NADH (or NADPH) then it is (usually) very easy to design an assay.
It would appear that the accepted extintion coefficient (Molar Absorption Coefficient) for Allura Red AC (Red 40) is about 25 000 M-1.cm-1 (or 25 000 mol-1 L cm-1) at about 505 nm. The best reference I can get it here. Azo dyes tend to have very high extinction coeffficients, so this 'looks about right'
The diluted solution has an absorbance of 0.283
Taking the above as the extinction coefficient, and in addition assuming the measurement was made in a cuvette with a path length of 1 cm (l =1), one can calculate from Beer-Lambert that the concentration of Allura Red in the diluted solution is about 11.32 x 10-6 M.
The dilution factor is 4 in 10, or 1 in 2.5. Therefore, the concentration of the original solution is about 28.3 x 10-6 M.