If the equation was
T×(V-b)
R/Cv=const
this sounds like an adiabatic expansion with a=0.
That is, for the adiabatic expansion of a perfect gas you'd have
T
Cv/R×V=const where Cv is the heat capacity at constant volume
Now, "b" in Van der Waal's equation represents an added volume. You can represent it as the volume taken by the molecules' electrons. It resembles (but differs slightly) the volume of the liquid substance. This "b" isn't usable for the free movement of the molecules. This distinguishes them from a perfect gas whose molecules are points. Consequently, "b" is subtracted from the gas' volume in VdW equation.
"a" represents the attraction (usually) between the molecules.
If "a" is neglected, then the imperfect gas behaves like a perfect one where you just add a chunk of iron "b" in the volume. Simply replacing V by V-b everywhere, the gas behaves as a perfect one whose molecules have no volume. That's why you get
T×(V-b)
R/Cv=const
I strongly suppose that a≠0 would not give that equation (but I won't check it, too long). Hence my question whether "a" is explicitly neglected. By the way, this would be reasonable in many technological situations like a hydraulic accumulator, where nitrogen's temperature exceeds clearly the critical point.
Now, supposing that a=0 if this is necessary, can you calculate the work? Same adiabatic expansion as a perfect gas, but with a chunk of iron "b" in the volume.
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By the way, these two are your allies for the adiabatic expansion of a perfect gas:
T
Cv/R×V=const
T
-Cp/R×P=const
because
- they're easy to memorize: CV goes with V and CP with P.
- CV/R represents half the degrees of freedom of a perfect gas
(more accurately: each translation and rotation brings R/2 to CV, each excited vibration adds R) - CP/R is 1+CV/R, since H=U+RT
- The ratio of both equations gives you PVCp/Cv=const where you recognize γ.
- It gives you quickly the temperature from P or V ratios, and the temperature tells you easily the work, the heat - in the case of a perfect gas, pretty much everything.
The previous 1.4 stands for a diatomic gas like nitrogen, oxygen and by extension air. 1.40 is even nicely accurate under usual conditions, useful to have in mind. According to the list:
- 3 transations bring 3/2×R to CV
- 2 rotations (no third rotation since N2 and O2 are linear) bring 2/2×R
- N2 and O2 don't vibrate at room temperature nor moderate heat, so nothing to add
- Hence CV=2.5×R, CP=3.5×R and γ=3.5/2.5=1.40