I've had a look around for an answer specific to Raimi's trilogy for a while and can't find anything. However, to coincide with Raimi's trilogy and the change to have Peter have organic webs and not web shooters the comics also made that same change. To cut a long story short Peter is turned into a spider by Queen, births from the spider and ever since then his abilities were "replaced" and he uses organic webbing. He is once again back to web shooters since the "One More Day" arc though.
However, during his time with organic webs in the comics Marvel released Official Handbook of the Marvel Universe Spider-Man: Back in Black which had details of his new abilities. In the Spider-Man Update section and the abilities/accessories panel is the following quote which states his organic webs last for a week before decomposing.
Since the Queen's transformation, Parker can produce silk from glands within his forearms, limited by his body's health and nutrition. These organic webs have many similar properties to the artificial webbing though they require a week to decay rather than decomposing within two hours.
Click image to enlarge.
Considering we don't see lots of webbing hanging around in Raimi's trilogy and it is never mentioned as a plot point I'd imagine that the organic webbing either follows the normal 1-2 hours of the shooters or the updated week we see in the comics.
Of interest to note is that originally the Raimi trilogy was going to use web shooters and the prototypes were even displayed at E3. As this was the case it could be that they were going for the normal 1-2 hours and never changed it when they changed to organic webbing.
