Calculating total fluorescence based on thin slices in a slab

Question

I am modeling an acid-base reaction that includes a fluorescence indicator. My simple model is a slab of acid next to a slab of base and so on. These slabs are very thin, typically less than 10 microns and are assumed to be laminar such that diffusion is the only method of mass transport. At the interface of acid and base the reaction occurs and the pH is 10^-7 (which is shown as zero on the attached figure).

The attached figure shows an example of the concentration gradients of acid and base at some time t. Because of symmetry only a half slab of acid is shown next to a half slab of base. I want to calculate the total fluorescence emitted from both half slabs together. With regards to the concentrations on the figure, for the acid slab the concentration is the hydrogen ion concentration. For the base slab the concentration is the hydroxyl ion concentration. I have an equation which gives me the fluorescence as a function of pH.

So the way I have been calculating the fluorescence is to get the pH for each slice shown in the figure and assume that the pH is constant for that slice (of course as the slice thickness is decreased this becomes a better assumption). Then the fluorescence is calculated for each slice based on the known pH and the equation.

Now comes my question: If I want to calculate the total fluorescence for both slabs, can I just simply add up the fluorescence from each slice in the acid slab and base slab? In other words, the total fluorescence emitted from both slabs together is the sum of the individual fluorescence for each slice. Is this the correct methodology?

Answer 1

You appear to be interested in monitoring pH gradients across the glass slab based on diffusion of an acid and base by monitoring the variation in fluorescence intensity or emission maximum shifts. Before you proceed further on this fancy experiment you will have to establish certain fluorescence facts. Hence control experiments are extremely necessary. A quick survey on Google Scholar will show that umbelliferone emission is pH dependent. I am sure you will have access to a good quality spectrofluorometer. Make sure you collect emission spectrum at the pH range of interest with identical concentrations of the dye in various pH solutions. By identical concentration of the dye in each pH solution, I mean prepare dye solutions as accurately as analytically possible. One could be prepare a stock solution of the dye and perhaps weigh the volume transferred just to be very accurate in each pH solution.

Once you collect emission spectra (fixed excitation), the emission maxima will/might change. The key observation here is that do emission intensities also change or not as a function of pH?

When the fluorescence intensity and emission are changing, it implies that we are not dealing with identical species of the dye, thus different "species" have different fluorescence quantum yields. It is then perhaps not kosher to add up fluorescence intensities. Alternatively, you might try even a fancier experiment and measure quantum yields as a function of pH.

Of course, the measuring instrument is oblivious to these things, it is just counting photons from the region you are collecting light from. In short, the experiment is not trivial and the deeper we explore, the more problems will be exposed. For example, is the detector uniformly sensitive to all the wavelengths of your interest or not.