If that bubbling is occurring before the engine gets warmed up, you know it isn't boiling. That air can only come from one place, ... The combustion chamber, due to the leaking head gasket. The chemical test at the radiator, in this case, would be a verification test, not a diagnostic test.
I'm not convinced about a thermostat being heated in a pot of boiling water. No professional has the time to do that, and I did see a mechanic next to me get fired for doing that, (it was the straw that broke the camel's back after a series of things he did after numerous warnings). A wax pellet has to melt, a thermostatic spring has to relax, then you need the push of the coolant from the water pump to push the valve open. I'd be willing to bet a new thermostat also won't open because of the air that is accumulating. If I'm right, you can run a piece of small wire through the thermostat's opening to prevent it from sealing tightly. That will allow the air bubbles to wash through and the hot liquid will get to the thermostat and cause it to open fully. Instead, a lot of import engines use a thermostat with a tiny bleed hole already in the plate. They have a little weight dangling from it. That is there to allow any air to escape so the liquid can get to the sensing element. If your old thermostat had that bleed hole and the engine still was overheating, then it may indeed be defective.
If you do indeed need a new thermostat, they only cost a couple of bucks. If you can't afford that, how do you afford to put gas in the car? Dollar-wise you're better off in the long run by putting a thermostat back in. When it's missing, the engine will run too cold and parts, mainly pistons, won't expand to fit properly. 99 percent of engine wear takes place during the few minutes the engine is warming up. Once it's at the proper temperature very little additional wear takes place. You'll eventually spend hundreds of dollars on the engine by saving a few dollars on the thermostat.
As for that chemical test to verify a leaking head gasket, many auto parts stores rent or borrow tools so you can do it yourself. Around my town most of them will let you use the tool for free but they make you buy a bottle of the special liquid. That's because the liquid will be contaminated and rendered ineffective if any coolant gets into it. That would make it unusable to the next guy. Someone would have to replace the fluid, which costs money.
You might also look for a nearby community college with an Automotive program. The instructor will likely send a student out to do the chemical test for you. It just takes a couple of minutes. Keep in mind though you must fit into their schedule. If they're in the classroom when you show up, you'll have to wait or come back on a different day. They would stop what they're doing and perform the test as a courtesy for you, but only when it doesn't interfere with their learning activities.
Saturday, November 2nd, 2013 AT 1:50 PM