"An adiabatic process is one during which no energy enters or leaves the system by heat- that is, Q = 0... Applying the first law of thermodynamics to an adiabatic process, we see that change in internal energy is equal to work done on the gas. From this result, we see that if a gas is compressed adiabatically such that work is positive, then change in internal energy is positive and the temperature of the gas increases."
I understand all this except for that last part, the temperature increases. Where did that come from? Why does it have to be temperature, why not pressure or volume?

Question

"An adiabatic process is one during which no energy enters or leaves the system by heat- that is, Q = 0... Applying the first law of thermodynamics to an adiabatic process, we see that change in internal energy is equal to work done on the gas. From this result, we see that if a gas is compressed adiabatically such that work is positive, then change in internal energy is positive and the temperature of the gas increases."
I understand all this except for that last part, the temperature increases. Where did that come from? Why does it have to be temperature, why not pressure or volume?

anonymous · Answer

Because you already know that you are compressing the gas so that its pressure increases. This process also is accompanied by reducing the volume of the container. What left is temperature that needs to be inferred.

anonymous · Answer

And also you know that the internal energy of a typical gas is always a function of T. So it is reasonable to infer that it increases by all means.

anonymous · Answer

So your first response says it's because you know pressure is increasing and volume is decreasing. But temperature is proportional to both of those, so wouldn't it depend on whether pressure is increasing faster than volume is decreasing? So you can't say for sure that temperature is rising.

anonymous · Answer

Yes it depends, but how they affect each other we just don't know. In this case, when we are compressing the gas, they do behave in such a way that internal energy rises. Do know that the so-called form PV=NKT will fail when considering pressures greater than 40 standard atmospheres. It can be shown that, of course, in the pressure range above 40 pascal, the linear relations reduces to a more sophisticating form 

PV=A(1 + BP^2 + CP^3 + ....)

in pressures lower than 40 atmospheres, the constant A is calculated to be nRT wile other virial coefficients vanish. But as soon as the pressure range varies, i.e becomes greater than 40, then the other terms would participate and we no longer know how they would become.

anonymous · Answer

Okay, but I still don't understand how this guarantees that the temperature increases. (sorry for delay, I had to leave for a while)

aravindg · Answer

Do you know the definition of temperature?

anonymous · Answer

@AravindG I guess that would be that it's the average kinetic energy of all the molecules in the system. Also, two objects in thermal equilibrium (no heat between them) have the same temperature.