It's hard to tell what your equations are saying with all the undefined variables. However, the relationship is simple. Each f-stop change in aperture is a factor of 2 in light, so you need to adjust the exposure time by a factor of 2 to compensate. For example, f/2 and 1/100 second is the same exposure in terms of light level as f/4 and 1/25 second.
Note that f/4 is two f-stops smaller aperture than f/2 since f-stop numbers go in sqrt(2) sequence: f/2, f/2.8, f/4, f/5.6, f/8, f/11, etc.
Note that there are limits to this relationship. Film will generally have a non-linear response at very long exposure times. This is known as reciprocity law failure. Digital sensors tend to be linear with total light received, but pick up noise proportional to the exposure length. Therefore the signal to noise ratio goes down with exposure length. This is why many digital cameras will limit exposure length to around 30 seconds usually. If you want a longer exposure with one of those, you take a bunch of 30 second exposures and combine them in software.