Just remember that it is not the characteristic curve I am fitting, but the (usually) first order reaction curve that is the same basic curve as the change in potential with time of an ideal capacitor being charged from 0 to a constant source through an ideal resistor. That curve seems to fit quite well the tables supplied by Kodak in publication J-109.
I don't yet know if we are talking about the same thing. The basic equation relating a gradient G at one temperature to the maximum gradient is assumed to be:
G = Gmax[(1 - e^(-kt)]
where e is the base of the natural logarithms. When we divide both sides by Gmax , we can plot the resulting equation on semi-log graph paper as Gmax - G on the log axis vs t on the linear axis. Since Gmax has been found to be independent of temperature, the constant k is the only other number we must know in order to define the value of G for any development time t at this temperature. If anyone is interested, I can provide more detail on the calculations, which includes how to generalize to a wide ranges of temperatures.
Certainly each film-developer combination has its own pecular characteristic curve shape, which makes automating the tabulating of development charts a more complicated business, but once the gradients are known, the development time-temperature relationships are much more amenable to generalization in form, which allows a certain degree of reduction of experiments neded for any one film-developer combination. That is not such a big deal for Kodal or Ilford, but for you and me the notion of being able to reduce the number of characteristic curves for one film by a factor of at least 4 is enticing.
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Well, we define gradient or slope of a curve at any point and time of development as dy/dx where y = density and x = Log E. That gives slope or contrast. Thus the curve of the print can be gotten from the curve of the negative and the curve of the print material by this: dy(p)/dx(p) = dy(f)/dx(f) * dy(m)/dx(m), where p = print, f = film and m = print material or paper. So, by doing this point by point or using the equation which I described before as a spline, you can take two curves and get the result when printing. This assumes no flare, and equivalent neutral density at each point from 400 - 700 nm. The first can be produced in the lab easily, and the second is assumed as any slight error is tiny. Silver itself is never exactly neutral as we all know.
I should have said simply that the relationship between time of development and the gradient or CI of a particular film-developer combination is a first order differential equation that can be defined by two constants. When the equation is arranged so that it can be plotted as a straight line on semi-log paper, the intercept is Gmax (or gamma infinity) and the slope of the line is k in the equation I posted above. This equation does not "know" how the values of G were measured or defined. Another way to define a first order system is that the rate of reaction at any time is proporional to the amount of material that remains to be acted upon. The solution of this equation represents experimental curves as satisfactorily as any equation containing only two arbitrary constants, and is quite precise enough, and its greater simplicity justifies its use, according to Hardy & Perrin.
Basically then, as I have contended, each film and developer combination has a different slope as the constants differ. This is from the laws of Kinetics and Thermodynamics cited above.
Basically then, as I have contended, each film and developer combination has a different slope as the constants differ. This is from the laws of Kinetics and Thermodynamics cited above.
PE
Yes, indeed. And the family of data that represents the gradient-time-temperature variation for any one of these film-developer combinations can be recreated satisfactorily by three properly chosen points from those data.
Linear Mode
