What is the purpose of the intercooler on a turbo powered car:in terms of PV=nRT

I have always been trying to figure this out in terms of PV=nRT.

So Cold air is denser great but why does the engine care? Why can't you just have hot compressed air going into the engine instead of cooler less compressed air? How much does the turbo increase the pressure? The only reason I could see not wanting hot air is to stop pre ignition but the turbo doesn't make that much heat or does it?

All explanation of you tube just say: cold air is denser so its the same reason you put a box around a cold air intake. Yes that makes sense you don't want hot engine air on a NA car but this is different because the amount of air is fixed by ho much the turbo pumps. Not like the engine bay where the NA car  just takes the 1 bar pressure of hot engine air or 1 bar outside engine bay air. Then it does make a difference but in a turbo its a closed system so that doesn't matter so much.

In a turbo set up: Sure cold air is denser but you have the same amount of air regardless of temp from the turbo output to the engine. Why bother to cool it down in between?

Having some real numbers to put into the equation would help.

Odd question for an electronics site.  There are some decent forums dedicated to turbocharging that you may find helpful.  I have a few books on the subject that are fairly easy to read.  If you are interested, I can provide some ISBNs.

As you compress the air, with say a tire pump, it gets hot.  Imagine taking that hot air and compressing it even further.   One problem with the heated charge is that it can auto ignite from the heat.  If you wanted to use gasoline, there are some blends that will help with this.   I use some of it to test hand held multimeters.    There is a limit what you can do with even the best gasoline.  Methanol may be a good choice.    Assuming the fuel is uses some sort of ignition source rather than heat (diesel) you may be able to change the angle where the ignition takes place to help out as well.    I imagine you would be pretty limited on what you could do with pump gas, especially without cooling the intake charge.  There used to be water/methanol injection systems that would help out.  There are other ways to cool the intake charge, for example N2O. 

Obviously, cheap fuel with low pressures could keep things safe but whats the fun in that??   

To get an idea about the more complex control systems, maybe have a look at HCCI.   

Odd question for an electronics site. There are some decent forums dedicated to turbocharging that you may find helpful. I have a few books on the subject that are fairly easy to read. If you are interested, I can provide some ISBNs.As you compress the air, with say a tire pump, it gets hot. Imagine taking that hot air and compressing it even further. One problem with the heated charge is that it can auto ignite from the heat. If you wanted to use gasoline, there are some blends that will help with this. I use some of it to test hand held multimeters. There is a limit what you can do with even the best gasoline. Methanol may be a good choice. Assuming the fuel is uses some sort of ignition source rather than heat (diesel) you may be able to change the angle where the ignition takes place to help out as well. I imagine you would be pretty limited on what you could do with pump gas, especially without cooling the intake charge. There used to be water/methanol injection systems that would help out. There are other ways to cool the intake charge, for example N2O.Obviously, cheap fuel with low pressures could keep things safe but whats the fun in that??To get an idea about the more complex control systems, maybe have a look at HCCI.