Relation between Temperature and Pressure inside a bottle in a room?

I have an empty bottle with no cap in my room, so it is supposed to be filled with air. I made the air warm inside the bottle by dropping an igniting matchstick inside the bottle. Warm air will rise and will cause the low air pressure inside the bottle.

But

What temperature inside the bottle will cause what pressure ?

Is there any equation that can tell me that how much temperature inside the bottle will cause how much low pressure as compared to the pressure of the air outside the bottle ?

"Warm air will rise and will cause the low air pressure inside the bottle." This is only a small effect if the bottle is still uncapped. The pressure of the air in the bottle is still almost atmospheric, as the bottle is open to the atmosphere. The gas is hotter in the bottle, and this, by itself would increase the pressure, but the density, $\rho$, of the air in the bottle is reduced and these two factors work in opposite directions, allowing the pressure to be almost unchanged. $$p=\frac{\rho RT}{M_{molar}}\ \ \ (T \text{= Kelvin temperature},)$$I say that the pressure in the bottle is almost unchanged, because at two points at the same level, one (B) inside the bottle and one (O) outside, the pressure outside will be greater by $h(\rho_{cold\ air}-\rho_{hot\ air}) g$ in which h is the height of the hot air column above B – a difficult thing to estimate. But, as I've said, the pressure difference will be small, in the order of 5 Pa, compared with 100 000 Pa typical atmospheric pressure.
• " how can we artificially cause much lower pressure than the surrounding air pressure in a vessel/chamber?" There's a really good way to do this! You need a tin with a tight-fitting lid. Thoroughly wash out the tin, then $with\ the\ lid\ off$, boil a few $\text{cm}^3$ of water in the tin, until there's plenty of steam, and the air is swept out of the tin. Then remove the heat and immediately fit the lid tightly. The steam will condense, leaving a vacuum. Atmospheric pressure will be enough to make a flimsy tin collapse! – Philip Wood Apr 14 '18 at 21:23