An adiabatic process is one in which heat does not enter or leave the system. Though this is not strictly true in a turbocharger, it is a close approximation over a very short time period. An adiabatic temperature or pressure change can be calculated from the volume change using the following two equations (which usually are found in your physics texts):
(T2/T1) = (V1/V2)^g-1 and (P2/P1) = (V1/V2)^g,
where g is the ratio of air's heat capacity at constant pressure to air's heat capacity at constant temperature; T1, P1, and V1 are the initial temperature, pressure, and volume; and T2, P2, and V2 are the final temperature, pressure, and volume. Taking the log of both of these equations, and knowing that log ab = b log a, the following equation relating adiabatic temperature change to pressure change can be derived:
(T2/T1) = (P2/P1)^[(g-1)/g].
Physics texts will tell you that for the diatomic gasses like N2 and O2 in air, g is assigned a value of 1. 4. By substituting and rearranging we get
T2 = T1 x (P2/P1)^0. 286.
Actually, the exponent in the above equation that you usually see is 0. 283, meaning that g is actually closer to 1. 395. You can use either number, the difference it makes in the calculations is about 0. 25%. We need to note that temperature in all of these equations must be absolute temperature in either Kelvin (centigrade scale) or Rankine (Fahrenheit scale). If you want to work in C then K = C + 273. 15. If you are more comfortable with F then R = F + 459. 69. Also note that 1 degree on the Kelvin scale equals 1 degree on the Centigrade scale, and 1 one Rankine degree equals 1 Fahrenheit degree. I prefer the Fahrenheit scale and will round off 459. 69 to just 460 for the examples here. I'll assume that we are at sea level at a standard pressure of 14. 7 psi with an ambient air temperature of 70F.
The expression (P2/P1) represents the turbocharger compressor pressure ratio (PR), where P1 is atmospheric pressure on the inlet side and P2 is the absolute pressure on the outlet side. T1 is the air temperature entering the compressor and T2 is the air temperature leaving the compressor.
Many people think that 12 psi in the manifold is a moderate boost level so I'll use that as an example. If we can assume there is a 2 psi pressure loss from the turbo to the manifold, then the PR at the compressor is about 1. 95 ( (14 +14. 7)/14. 7). Let's also assume we manage to keep the intake air temperature down to 800F (or 5400R). The air temperature leaving the compressor, due to adiabatic heating only, would be 194F which is 654R = 540R x (1. 95)^0. 286. This is an increase of 114F! But the air is actually hotter than that.
Turbochargers compress the air by increasing the velocity of the molecules. When the air molecules don't move in a direction toward the intercooler (IC), they serve only to heat the air. The more the air is heated above that predicted by adiabatic compression, then the less efficient the turbocharger is. Compressor flow maps usually show at various flow amounts and pressure ratios the measured efficiency of a turbocharger. Compressors usually operate in the 55-75% efficiency range depending on engine load, pressure ratio, turbine speed, and compressor size (65-70% is average, 75% or more is where we would like to be).
When we factor in compressor efficiency (CE) the final temperature of the air leaving the compressor is
T3 = T1 + (T2-T1)/CE.
If compressor efficiency is 70% in the above example, then the final temperature is 242F, more than hot enough to boil water. If we increase the PR to 2. 36 (18 psi manifold and 2 psi IC loss), the temperature would increase to 295F. These high temperatures explain why intercoolers are so important when boost pressure is above a few psi.
Attention: TDR Forum Junkies 
) What are you using for a temp gauge? 
I'll spend spring break working on this one... 
