We can answer this on two levels, the method of "breathing" by spiders and other arthropods and the way that foggers and mists actually affect the arthropod pest. The first is on breathing by bugs, and spiders have a somewhat different setup than insects do. Insects usually have "spiracles" that are small openings along their sides or somewhere else on their exoskeleton (on maggots they are on the end of the abdomen). These allow air to move into the body of the insect so that the air then is transported passively through smaller and smaller tubes to each area of the body, supplying the needed oxygen throughout. Some aquatic insects will have gills that remove the oxygen from the water, as with dragonfly larvae.
Spiders and scorpions have "book lungs" under their abdomen, and these are large folds that lie within a pocket, more or less like thick pages in a book. This maximizes the surface area that is in contact with air (oxygen) and surrounding the book lungs will be large quantities of the spider's blood, allowing the oxygen to move into the blood for transport throughout its body. So, neither spiders nor insects nor any other arthropod "breathes" actively in the manner of vertebrates. We actually pump the air in and the waste products out, and anything in that air (dust, vapors, droplets) gets pulled into our lungs as well. Particles and droplets are far less likely to move through the spiracles or into the book lungs of spiders, so getting the tiny droplets of a fog or a mist into a spider is not likely to happen. This is what foggers emit - droplets - and they may be very small in diameter but they are droplets nonetheless.
However, some insecticide active ingredients will vaporize to leave the vapors in the air, and this may be part of the mechanism of pyrethrum. Even though the droplets have evaporated the vapors remain to some extent, and thus we get some repellency to flies from the use of aerosol misting devices set up over doorways. For most other active ingredients we should not expect that vaporization, and the droplets from foggers go up into the air and finally settle down onto horizontal surfaces below, with the active ingredient still in that droplet and thus settling only on that surface once the water or other diluents evaporate. The tiny droplets from foggers go only where air movement carries them while they are still aloft, and for the smallest of droplets this could take an hour or so. But, they are incapable of entering sealed areas such as wall voids, drawers, sealed packages of food, closed closets, etc. This is why fogging the air is not going to control an infestation of food pest insects where the larvae are living and feeding inside closed packages of food.
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