What happened?
Under normal conditions the air pressure is equal to the pressure by the gas inside the container, but
when you pour out the hot water and seal the bottle, what is left is hot water vapor. The water vapor starts
to cool down and as it does so the water vapor condenses. This condensed water vapor has less pressure than
the air pressure outside the bottle. Resulting in the water bottle being compressed.
This can occur even with cold water during the winter. The cool water starts becoming cold and
it condenses making the air pressure in the water bottle smaller than the air pressure outside.
The pressure inside the container
Recall, the ideal gas law is pV=nRDT. This can be rewritten as p =
nRT
V
.
In this experiment I used a 500mL water bottle, but the ideal gas law requires the units to be in liters, and there are 1000 mL to every 1L,
therefore I have a 0.5L water bottle. Next, I need to find the amount of moles of water vapor in the water bottle. In order to do this I need
to know the conversion from liters to moles. There is 1 mole to every 22.4 liters. I have 0.5L, which means I have 0.022 moles of water vapor.
Now, let's say that the hot water is initailly at the boiling point, which is 100 °C (the water doesn't need to be that hot for this experiment
to work), right after I pour out the hot water the temperature inside the water bottle is lower than water, but it is increasing, this is because there
is water, albeit very little, still left that is giving off heat. The length of time that the temperature will increase depends on the size of the container
and how well sealed the container is. For this experiment I drilled a hole that is slightly bigger than the diameter of the thermometer, so there is some
unacounted heat loss.
For simplicity's sake, let's say that the temperature of the gas is 100 °C. The temperature needs to be in Kelvin in order to use the ideal gas law,
all you need to do is add 273 to the degrees Celcius. The initial temperature of the gas is 273 + 100 = 373 K. According to the zeroeth law of Thermodynamics,
when one system (the water vapor) is in thermal equillibrium with a second system (the water bottle), and the second system is in equillibrium with a third system
(the surroundings), then the first and third are in thermal equillibrium with each other, all this is to say that all three systems will have the same final
temperature, and in this experiment that temperature is the temperature of the room. The temperature of the room is 71 °F, and since temperature needs to be in
Kelvin I need to change 71 °F into Kelvin. In order to do this I first need to convert Fahrenheit to Celcius. The conversion from Fahrenheit (labeled as Tf)
to Celcius (labeled as Tc) is Tc= 5⁄9 * (Tf + 40) - 40. I get that 71 °F is 21.6667 °C. The final step is to add 273 to get
Kelvin. The final temperature in Kelvin is 294.6667 K.
The change in temperature is 373 - 294.6667 = 78.333 K. Now I can use the ideal-gas law to find the pressure inside the container
The pressure inside the water bottle is
0.022 * 0.0821 * 78.333
0.5
= 0.283 ATM. This is substantially smaller than the 1 ATM outside the water bottle, this is why the water bottle compresses. Then the
water bottle stops compressing when the temperature of the gas is equal to the temperature of the room.
Outside of the bounds of this experiment, but if you wanted to increase the pressure, thus bringing the water bottle back to full volume, then you would warm up the water bottle,
either by pouring hot water on the water bottle, or by leaving the water bottle in a really hot room. This causes the gas inside to seek a new equillibrium temperature, which is hotter than
21.667 °C. As the gas heats up, the gas expands, which increases the temperature and decompresses the water bottle.
Click the Differing Conditions tab to continue.
