This is actually a very difficult question, because you don't tell us what resources you have on hand to test with, nor what other chemicals this COULD be. The universe of "unknown white powder in bucket" is very large - it could be thousands of different things.
However, if I assume:
1. your chemical stockpile is composed of stuff you typically get at a grocery store or normal home chemicals
2. you don't have any analytical equipment
then, some thoughts:
a). soda ash is odorless. Is this odorless?
b). soda ash is a poorly crystalline white powder. Are there obvious crystals (like you might see in granular salt) or is it a fine poorly crystalline white powder?
c). soda ash, when dissolved in de-ionized, or maybe distilled, water, has a pH of around 11.5. If you make a saturated solution at standard temp (25 C), by adding soda ash to water until it won't dissolve any further, and then measure the pH, do you get a pH around that level?
d). soda ash is sodium carbonate. If you add white vinegar to a solution of this mystery solid, does it foam? That might be CO2 evolution (or some horribly poisonous gas if this WASN'T soda ash).
If you have a very accurate mass scale, and accurate volumetric pipettes, and can control temperature reasonably, you could make a saturated solution at 35C, then take a 1 ml or 10 ml sample, quickly weigh that and see whether it is close to the density of pure soda ash dissolved in water, (1.33 times the weight of water alone). The reason I suggest trying to do it this way (as opposed to trying to measure the density of the solid) is that soda ash is usually hydrated, so water molecules are attached to the solid that screw up direct measurements of mass/density and such.
35C is the max. solubility point, so if you could maintain temp. but filter or decant, then gently heat further (to say 50 C) while avoiding any water vaporizing (a watch glass on a beaker) if solids formed you might conclude your mystery solid had similar (and unusual) solubility compared to soda ash. Most chemicals have increasing solubility with temperature, but soda ash solubility increases up to around 35C and then decreases.
If you have pure ethanol, you could see whether the mystery powder dissolves. Soda ash doesn't dissolve in ethanol.
Those are just a few thoughts, for relatively easy and minimal equipment type tests. None are conclusive on their own, but in combination could allow a reasonable guess as to the identity of your mystery solid. I am sure smarter people could come up with better methods.
Best Regards, Gerald
