14th September 2011 - 11:13 PM
You are correct that raising the temperature will raise the pressure inside the bottle. The air pressure will rise per the ideal gas law; the vapour pressure will rise per Clausius-Clapeyron because there is more heat energy available to cause molecules of the liquid to evaporate.
When the bottle was stoppered, the gas mixture in the neck would have been at atmospheric pressure (100kPa). Dalton's Law tells you that the total pressure is the sum of the partial pressure of air and the partial pressure of vapourised solvent. If you can find the vapour pressure of the solvent at 20C (let's call this P_v) then you can work out the partial pressure of air at 20C (let's call this P_a, and it is given by 100kPa-P_v).
Now raise the temperature to 30C. The ideal gas law says that pressure is proportional to temperature. If we are happy to treat air as an ideal gas, and assume that any change in the solubility of air in the solvent is negligible, then we can write P_a'=P_a.(T'/T)=P_a.(303/293). This is a rise of about 3.5%.
According to Wikipedia
(a skim read of their source
seems sound), you can relate vapour pressure P1 at temperature T1 to vapour pressure P2 at temperature T2 by
where R is the gas constant and DH is the molar enthalpy of vapourisation, which you will have to look up for your solvent. For this particular case, then:
which is approximately
Per Dalton's Law again, the final pressure is P'=P_v'+P_a'.
As an example, if your solvent is ethanol (cheers) the vapour pressure at 20C is 5.83kPa
and the enthalpy of vapourisation of 38.6kJ/mol
. Numbers are also from Wikipedia, and health warnings abound because it quotes a vapour pressure of 5.95kPa elsewhere
That makes the values at 30C:
P_v'=P_v.exp(1.335x10^-5 x 38.6x10^3)=9.85kPa
That makes the total pressure at 30C:
This is a rise of about 7% compared to 20C. I very much doubt that that would threaten your bottle integrity, but you might want to calculate the area of your stopper and make a quick estimate of how much force is needed to pop off the top.
You will get different answers for different solvents, obviously.
If the vapour pressure at 30C exceeds atmospheric pressure (100kPa), it boiled.
The filling level does not make a difference, assuming that there's more than a tiny drop of liquid in the bottom.