I have two questions, so if anyone could answer them it would be very helpful:
1) How can you cool a gas by increasing the volume, even though PV=nRT indicates otherwise. I thought if you increase volume, you also increase temperature?
2) High pressure and low temperature cause deviations from ideal gas behavior, so gases deviate from ideal behavior when pressure is >10atm and
temperatures near the boiling points. Shouldn't we decrease temperature? From PV=nRT if we increase temperature we can increase pressure (and decrease volume), creating intermolecular forces. But if this is true, gases deviate at extremely high temperature and extremely low temperatures, since gases come together when temperature decreases.
please help!
Both questions can be answered by telling you that the ideal gas equation can be used the way you're using it if, and only if, two parameters are variable.
Ok, so PV=nRT, so you usually think, oh, ok, i'll increase the pressure so I'll increase the temperature. the problem is, that's only true if volume doesn't change as a result of that pressure increase. if it does, then the temperature change will be a function of both pressure change and volume change.
so, for your first question, which is describing adiabatic expansion of a gas, the reason you get cooling of a gas (and trust me, I used a CO2 cylinder without a regulator, ice crystals started forming around the valve because of the cooling) must be derived:
dE=dq+dw (or you may have learned dq-dw, the sign of dw is arbitrary, just be consistent).
So, dE is defined as CvdT.
For an adiabatic process, dq is always 0.
dw=-PdV (the sign is negative because work is done by the system as volume/dV increases so the system is losing energy)
NOW you can apply PV=nRT:
Rearrange so you get P=nRT/V
Plug into dw=-PdV to get
dw=-nRT/VdV
so now you just have:
CvdT=-nRT/VdV rearrange to get T on the other side:
(Cv/T)dT=-(nR/V)dV now integrate both sides: (n and R are constants, and Cv can be taken to be a constant in many cases)
sooo
Int[(Cv/T)dT]=Int[(-nR/V)dV] take constants out:
CvInt[dT/T]=-nRInt[dV/V] we're integrating over T1 to T2 and V1 to V2
so we get:
Cv*Ln[T2/T1]=-nR*Ln[V2/V1] because the sign is negative, you flip the Ln
Cv*Ln[T2/T1]=nR*Ln[V1/V2] sooooo as you can see as you increase temperature, (so T2/T1 is greater than 1), V1/V2 is also greater than 1 which means that you're decreasing volume. the converse is also true.
For your second question, think of it like this: high temperature, if all else is constant, will tend to want to get molecules as far apart as possible. low pressure, if all else is constant, will want to get molecules as far apart as possible. if you have both high temperatures and low pressure, you'll be getting the molecules as far apart as possible. if you increase pressure, you're squeezing the molecules together. if you decrease temperature, molecules have lower avg kinetic energy so they tend to get closer together, again, if all else is constant. that's why high temp, low pressure makes it most ideal: it gets everything far apart to minimize molecular interactions.
hope this helps
