How Does an Air Conditioner Alter the Volume, Temperature, and Pressure of Gas?

[Pudulax1]

I was just pondering the way an air conditioner works, and I have a basic question. When the compressor compresses the gas, how does it change the volume, temperature, and pressure? Using pv=nrt, there does not seem to be a way to figure out what the proportion that temperature and pressure will increase in relation to the new smaller volume. Does one increase more than the other? Or am I way off, is volume constant and just pressure and temperature increase? Please help me clarify because my brain may explode soon.

 

[russ_watters]

You can't use the ideal gas law for this. You use the tables for the refrigerant and find the appropriate state for a certain compression at a constant entropy. Have a look at the t-s diagram (figure 2): http://en.wikipedia.org/wiki/Refrigeration

 

[lpfr]

russ_watters is right when he says in actual heat pumps, ideal gases law is just an approximation.

But, in order to understand the principle of heat pumps, you can imagine a very poor heat pump filled with a near ideal gas as helium at atmospheric pressure. The process in the compressor is mostly adiabatic because it is fast and the gas has not time to give-up heat to the cylinder walls. Of course, there is always some heat transmitted and the compressor heats up. The gas is cooled in a heat exchanger (usually a ventilated or convection radiator) outside the zone to be cooled. Finally the cooled gas is allowed to expand adiabatically and its temperature lowers. This cool gas passes thought another radiator or heat exchanger to cool the zone or the air.
In actual heat pumps the gas used is not helium, but a gas that can be liquefied at room temperature and high pressure, as Freon, NH3, or other hydrocarbons. This allows using the heat of vaporization in the cooling cycle.

The formula for ideal gases is, written in the physical version:
$$PV=N{3\over 2}kT$$
N is the number of molecules (not moles) and k the Boltzmann constant. When the gas is formed by polyatomic molecules, the formula changes to:
$$PV=N{n\over 2}kT$$
where n is the number of degrees of freedom of the molecule.

Furthermore, when the pressure is high (far more than a few bars) the gas formula must be corrected to take into account the volume of the molecules themselves. In actual heat pumps, this is not necessary.

In an adiabatic process the quantity conserved is the product $$PV^\gamma$$ where $$\gamma$$ is the adiabatic coefficient of the gas. It is 1.67 for monatomic gases, 1.40 for diatomic gases and less for polyatomic gases.

When a mass of gas passes adiabatically from a state 1 to a state 2 we write:
$$P_1V_1^\gamma= P_2V_2^\gamma$$
$$P_1V_1= N{n\over 2}kT_1$$
$$P_2V_2= N{n\over 2}kT_2$$
If you know $$P_1$$, $$V_1$$, $$T_1$$ and $$P_2$$or $$V_2$$, you can compute $$T_2$$