Trust me, once I was an engineer...

Introduction

In this article I try to understand the rules behind the B&W developers, making use of some mathematical skills (while I still have them), having also in account that once I made much more complicated calculus, for instance with high voltage lines, electrically and mechanically. I can not do that anymore, I forgot everything, but here the equations are much easier to handle. This is only the first approach to the problem, more refinements will be done, this need some more time to treat data, although with help of 'modern technologies', understand computer.
I concentrate my attention only in developers MQ (Metol-hydroquinone) but the ideas may be used with other pair, if we know the equivalence to these two ones. For instance, Phenidone is said to be equivalent to Metol but 5 to 10 times stronger. When using Phenidone instead of Metol, we must reduce the quantity dividing by a number between 5 and 10.

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With a certain dosis of patience I was regarding the quantities involved in MQ developers, Ilford and Kodak developers mainly. I filled a spread sheet with recipes and ordered the data by increasing Metol amounts. It was not that difficult to observe the graphics obtained with the amounts of the components, having the Metol quantity as the independent variable. I came to the conclusion that the main components like Hydroquinone, Sodium Sulfite and Sodium Carbonate may be calculated with following rules:

The amount of hydroquinone is in most cases 1,82 times the amount of Metol.
The amount of Sodium Sulfite is 11,5 times the amount of Metol plus 30g, the minimum amount that such developers have in general. The amount of Sodium Carbonate is 45 g minus 6 times the amount of Metol, it decreases as Metol increases. Finally the amount of Potassium Bromide has no exact rule, imho, just use an amount between 0 and 3 g/L. All quantities are given for 1 Liter developer, to be used indiluted.

I am not 100% sure, of course, whether these coefficients apply to most of the recipes, but I will discuss some examples:

The developer D76 from Kodak has this composition (see my source here)

Metol - 2 g
Hydroquinone - 5g
Sodium Sulfite - 100 g
Sodium Carbonate - 0 g
Borax - 2g
Potassium Bromide - 0 g

With my equations, I would have following developer instead:

Metol - 2 g
Hydroquinone - 3,64 g
Sodium Sulfite - 53 g
Sodium Carbonate - 33 g
Borax - 0 g
Potassium Bromide - 0 to 3 g

But, if I take the amount of Hydroquinone as base (5 g), then I have another recipe:

Metol - 2,75 g
Hydroquinone - 5 g
Sodium Sulfite - 61,6 g
Sodium Carbonate - 28,5
Borax - 0 g
Potassium Bromide - 0 to 3 g

The last one is perahps 'more like' D76 than the first, with more Sodium sulfite; the Carbonate of the second calculated recipe is, in D76, replaced by more sulfite and borax.

Another example is D23 from Kodak, the simpliest developer with only 2 components (see my source here):

Metol - 7,5 g
Hydroquinone - 0 g
Sodium Sulfite - 100 g
Sodium Carbonate - 0 g
Potassium Bromide - 0 g

Here we could say, according to the equations proposed, the developer would be:

Metol - 7,5 g
Hydroquinone - 13,65 g
Sodium Sulfite - 116 g
Sodium Carbonate - 0
Potassium Bromide - 1 to 3 g

Note: In both cases, No Sodium Carbonate appears.

Here we find a big difference in the amount of Hydroquinone. The high level of Metol, let us say, from 5 g/L upwords, makes Hydroquinone superfluous and lead to very fine grain developer but low contrast. On the other extremity, when Metol is zero, Hydroquinone must be higher than a certain threeshold, I think about 5 g/L is a realistic value but I found developers with much more than that and they are for high contrast but not fine grain or general purpose developers.

Now, let me give two more examples, that are more like the equations. Agfa/Ansco 40:

Metol - 4.5 g
Hydroquinone - 7.5 g
Sodium Sulfite - 54 g
Sodium Carbonate - 46 g
Potassium Bromide - 3 g

With the equations I propose, we would calculate pro liter:

Metol - 4,5 g
Hydroquinone - 8,19 g
Sodium Sulfite - 81,75 g
Sodium Carbonate - 18 g
Potassium Bromide - 0 to 3 g

These last two are not the same, but there is, maybe, another relation to be considered, if we see more in detail the sum of Sodium Sulfite and Sodium Carbonate in both is equal 100 g, more or less. This sum increases with the amount of developing agents, from about 80 to 100.

Another example is following, GEVAERT G.214:

Metol 2 g
Hydroquinone 3 g
Sodium Sulfite (anhydrous) 25 g
Sodium Carbonate 16 g
Potassium Bromide 1 g

Calculated:

Metol - 2 g
Hydroquinone - 3,64 g
Sodium Sulfite - 53 g
Sodium Carbonate - 33 g
Potasium Bromide - 0 to 3 g

Note: The relation Sulfite to Carbonate is the same in both, which leads, perhaps to the same pH.

Again the result is not totally identical. In fact, unless we use very complicated equations, it is not easy to describe all MQ developers with a unique set of simple equations. These equations may be used only as starting points for a new developer to be tested. They are more or less accurate for Metol varying between 1 and 5, not less and not higher than these extremes. Because the number of combinations is endless, I can not give any warranty for the success of developers following the equations proposed:

Hydroquinone = 1,82*Metol

Sodium Sulfite = 30 + 11,5*Metol

Sodium Carbonate = 46 - 6*Metol

For Metol = 1 to 5