Charles Law inverse for cooling? Is the inverse of Charles law also true? If I double the volume, the temperature will be halved? 
In the fire service we ventilate a structure by opening up the ceiling to the attic. If the volume of the attic was identical to the volume of the floor below, would the temperature be halved?
 A: Check out the description of Charles' law on wikipedia.  Charles' law relates the temperature and volume of a single body of gas at constant pressure.
If you're mixing two bodies of air at different temperatures by punching a hole in the wall between them, you're not changing the volume of a body of gas at constant pressure, you're combining two bodies of gas.  Charles' law wouldn't really apply.
When you mix two quantities fluids at different temperatures, the final temperature should be an average of the two initial temperatures, weighted by the number of particles in the initial quantities of fluid.  This is just a consequence of conservation of energy:
$$T_{final} = \frac{T_1^{initial} N_1 + T^{initial}_2 N_2}{N_1 + N_2}$$
You'll find that if the two gases have equal numbers of particles to begin with ($N_1 = N_2$) as in your hypothetical question, then the final temperature after mixing will be halfway between the two initial temperatures:
$$T_{final} = \frac{T_1^{initial} + T^{initial}_2}{2}$$
Of course, this is a simplified analysis that applies when the two gases mix completely without exchanging heat with their surroundings.  A real-life situation in a building will be more complex.
A: By just adding on another volume (the attic space), you wouldn't really be applying Charles' Law since you've increased the amount of gas.
The temperature would go down though, since you're effectively diluting the hot gas with cooler gas from the attic space.
Charles's Law is simply that for a fixed mass of gas at a constant pressure, V and T are proportional. In practical terms it is why a balloon expands and contracts as you heat and cool it. If you force the volume to change (stretching or squishing the ballon) the law doesn't apply since you are changing the pressure - then you'd be into Boyle's Law.
