2.4.8 · D5States of Matter (Quantitative)
Question bank — van der Waals equation (P + a - V²)(V − b) = RT — physical meaning of a, b
The excluded-volume picture () and the attraction-tug picture () that these traps lean on:
True or false — justify
Larger always means a gas is easier to liquefy.
True — bigger means stronger attractions, which raise the critical temperature, so the gas can be squeezed into a liquid at more accessible temperatures. See Critical Constants and Liquefaction.
The constant is exactly equal to the real volume of one mole of molecules.
False — the actual molar volume, because the excluded sphere around a pair of molecules is , shared as per molecule.
At very high pressure, a real gas is harder to compress than an ideal gas.
True — molecules are jammed together and their finite size (the effect) resists further squeezing, giving .
Real pressure is always lower than the pressure an ideal gas would exert under the same conditions.
False — that holds only when attractions () dominate at low/moderate pressure; at very high pressure the term makes real pressure higher than ideal.
Both and are positive numbers for every gas.
True — molecules always occupy some space () and always attract at least weakly (); neither correction can flip sign.
A noble gas like helium can have effectively close to zero.
True in practice — helium's attractions are extremely weak, so is tiny, making He behave almost ideally over a wide range.
If we set and , the van der Waals equation becomes the ideal gas law.
True — killing both corrections leaves exactly, which is the Ideal Gas Law. That is why vdW is a repair, not a replacement.
Two gases with the same must be chemically similar.
False — only reports molecular size; two chemically different molecules of similar physical bulk can share nearly the same .
can never occur for a gas whose is zero.
True — with no attraction there is nothing to pull wall-molecules back, so pressure is never softened below ideal; only the effect () survives.
Spot the error
" corrects the volume because it appears in the term."
Wrong — the whole bracket is a pressure correction; measures attraction and patches pressure. Judge by which quantity is inside the bracket with P, not by which letter sits near a .
"To fix volume we write since molecules add extra room."
Wrong sign — molecules remove usable room, so the correction is . The available space is always less than the container volume.
"The pressure patch is because it scales with density."
Wrong power — attraction is a pair effect, so it scales with density squared: . Both the pulled molecule and the pullers scale with density.
"Add to the volume and subtract from the pressure."
Both mixed up — the term is added to P, the term is subtracted from V. Never swap which correction meets which variable.
"Since NH₃ has a big , it must also have a big ."
Non sequitur — (attraction) and (size) are independent. Small polar molecules can have large but small ; big nonpolar molecules the reverse.
" means the gas is behaving ideally plus a little extra."
Wrong — means it deviates from ideal in the repulsion/size direction; only is ideal. Any is non-ideal.
"The excluded volume for a pair of molecules is , one molecule's volume."
Wrong — two centres cannot approach closer than , so the forbidden sphere has radius and volume , eight times one molecule.
Why questions
Why must the pressure correction depend on density and not just on the total number of molecules?
Because attraction is felt locally — what matters is how crowded the neighbourhood is (molecules per volume), not how many exist in total. Density captures crowding.
Why does dividing the pair's forbidden volume by two give per molecule, not ?
The forbidden region is shared between the two molecules of the pair, so each one is bookkept for half of it to avoid double-counting when summing over the whole gas.
Why does a gas with strong attractions show at moderate pressure?
Attractions pull wall-bound molecules back, softening collisions so (hence ) falls below the ideal value , dragging under 1. Link: Intermolecular Forces.
Why do the and effects both fade at low pressure and high temperature?
Molecules are then far apart (density low) and fast-moving, so their finite size blocks little space and attractive tugs are negligible during brief, high-energy encounters. This is why the Ideal Gas Law works there.
Why is real pressure lower than ideal when attraction dominates?
A molecule heading for the wall is tugged backward by the molecules behind it, so it lands with reduced momentum and delivers a softer push — measured pressure drops below the internal ideal value.
Why can two effects ( pulling down, pushing up) both be present at once?
They arise from different physics — attraction vs finite size — and act simultaneously. Which one wins depends on pressure, and their tug-of-war traces out the full curve. See Compressibility Factor Z.
Edge cases
At the exact pressure where the and effects cancel, what is ?
, so the real gas momentarily mimics ideal behaviour even though it is not ideal — this special temperature-linked condition is the Boyle Temperature.
What happens to the term as you compress the gas toward its smallest possible volume?
approaches , so and the pressure predicted by vdW shoots toward infinity — the finite molecular volume sets a hard floor you cannot compress past.
For a monatomic ideal-like gas at extremely low density, what do and effectively contribute?
Almost nothing — with molecules far apart, and , so vdW collapses back onto from the Kinetic Theory of Gases picture.
If a hypothetical gas had molecules that were true points () but still attracted, what would and be?
(no size, no excluded volume) but (attraction remains), so only the pressure patch survives and always.
As temperature rises far above the Boyle temperature, why does stay above 1 across most pressures?
Fast molecules barely feel attraction, so the effect is swamped and the finite-size effect dominates, keeping the gas harder to compress than ideal.
What does it mean physically if a gas has a very large but a very small ?
Its molecules are physically bulky yet barely attract (large, nonpolar, floppy species), so it resists compression by size but liquefies only with great difficulty.
Connections
- Kinetic Theory of Gases — the ideal assumptions these traps exploit
- Compressibility Factor Z — the -based edge cases above
- Critical Constants and Liquefaction — why large eases liquefaction
- Boyle Temperature — the cancel-point edge case
- Intermolecular Forces — the origin of
- Ideal Gas Law — the limit