The ideal gas is only a model
The two key assumptions of an ideal gas are that the molecules themselves have no volume and that there are no forces between them. At ordinary temperatures and pressures the molecules are far apart (U3.3), both assumptions hold well, and PV = nRT is very accurate. Change the conditions, though, and the model breaks down. AP topic 3.6 asks you to explain under which conditions and for which gases the deviations are largest, and why.
Two sources of deviation
1. Attractions between molecules make the pressure lower. As a molecule is about to hit the wall, the molecules behind it pull it back, so it hits less hard; the attractions also draw molecules together so they hit the walls less often. So the real pressure is lower than the ideal value. The stronger the intermolecular forces, the larger this deviation: polar molecules (such as and ) and molecules with many electrons and strong dispersion forces (such as Xe and ) deviate most.
2. The volume of the molecules makes the pressure (or volume) higher. At very high pressure the molecules are crowded together, and the space they themselves occupy can no longer be ignored; the space they are actually free to move in is less than the container’s volume, so the real pressure is higher than the ideal value. At a given pressure, the real volume is also larger than the ideal value. The larger the molecules, the more noticeable this deviation.
Both effects are always present; what matters is which one wins:
- Moderate pressures (tens of atmospheres) and lower temperatures: the molecules are fairly close and fairly slow, attractions matter most, and the real pressure is lower than ideal.
- Very high pressures: the volume of the molecules becomes the main factor, and the real pressure is higher than ideal.
When is a gas most ideal?
At high temperature and low pressure. At low pressure the molecules are far apart, so both their volume and their attractions are negligible; at high temperature they move so fast that attractions barely change their motion. Conversely, deviations are largest at low temperature and high pressure, and at a low enough temperature the gas liquefies.
Of all gases, He and behave most ideally: their electron clouds are tiny and hardly polarizable, so their intermolecular forces are extremely weak, and the molecules themselves are very small.
Drag the temperature and the volume to compare the real pressures of several gases with the ideal value; see how the pressure changes when you account for the volume of the molecules, and then for the attractions between them:
van der Waals model (a and b from Wikipedia, “Van der Waals constants (data page)”, citing the CRC Handbook, 53rd ed.). Temperatures start at 450 K, so none of these gases liquefies here.
Enrichment (AP doesn’t ask you to calculate with it): the van der Waals equation corrects the ideal gas law with two constants:
b reflects the volume of the molecules themselves, and a the strength of the attractions between them. The a value is only 0.0346 L²·bar/mol² for He, 1.370 for , and 6.803 for (CRC): the stronger the intermolecular forces, the larger a.
Example 1: Which gas deviates more?
At the same temperature and pressure (300 K, 10 atm), which deviates more from ideal behavior, or ? Is the real pressure higher or lower than the ideal value?
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Compare the attractions between molecules.
- : a bent, polar molecule with 32 electrons; dipole–dipole forces and stronger dispersion forces.
- : a linear, nonpolar molecule with 14 electrons; only weaker dispersion forces.
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Conclusion. molecules attract one another much more strongly, so deviates more (its boiling point, −10 °C, versus −196 °C for , shows the stronger attractions).
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Direction. 10 atm is not very high, so the molecules are still fairly far apart and their own volume matters little; attractions dominate. A molecule about to hit the wall is pulled back by its neighbors and hits more gently, so the real pressure of is lower than ideal.
Example 2: Argon at extreme pressure
At several hundred atmospheres, the real volume of a given amount of Ar at a given temperature is found to be larger than PV = nRT predicts. Explain.
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Which way each effect pushes.
- Attractions between particles make the pressure (or volume) lower than ideal; they matter most at low temperature, near condensing.
- The volume of the particles makes the pressure (or volume) higher than ideal; it matters most at very high pressure.
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Which effect wins. At extreme pressure the atoms are packed tightly, and their own volume is a sizable fraction of the container volume.
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Why the volume is larger. The ideal gas model treats the atoms as having no volume, so the gas could be compressed without limit; real atoms take up space that can’t be squeezed out, so the real volume exceeds the ideal value.
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What about attractions? Ar is a monatomic noble gas with relatively few electrons, so its attractions are weak and are outweighed by the volume effect at extreme pressure.
- One gas at the same pressure: 500 K or 200 K
The higher the temperature, the smaller the effect of attractions.
- One gas at the same temperature: 1 atm or 100 atm
The lower the pressure, the farther apart the molecules.
- Same temperature and pressure: He or
He has extremely weak intermolecular forces.
- Same temperature and pressure: or
is polar and its molecules also hydrogen-bond.
- Same temperature and pressure: Xe or Ne
Xe has far more electrons, stronger dispersion forces, and larger atoms.
- Same temperature and pressure: or
has a tiny electron cloud.
- Same temperature and pressure: or
has more electrons (22 against 10), so stronger dispersion forces.
- Same temperature and pressure: or water vapor
Water molecules hydrogen-bond.
- A gas near its boiling point, or the same gas far above it
Near the boiling point, attractions are significant; the gas is close to liquefying.
- Ar at low pressure or Ar at high pressure
At high pressure the volume of the atoms themselves can't be ignored.
Common mistakes
Mistake 1: “A real gas always has a lower pressure than the ideal value.” Attractions lower the pressure, but the volume of the molecules raises it. At moderate pressures attractions usually win; at extreme pressures the volume effect wins and the real pressure is higher than ideal.
Mistake 2: “He is the most ideal gas because its atoms move fastest.” The key is that He atoms are tiny and have only 2 electrons, so they are hardly polarizable and attract each other extremely weakly. At the same temperature all gases have the same average kinetic energy; speed isn’t the reason.
Mistake 3: “At high pressure, gas molecules get squeezed bigger or flatter.” The molecules don’t change size. What changes at high pressure is the fraction of the total volume the molecules themselves occupy: the more crowded they are, the less that fraction can be ignored.
Quick check
Under which conditions does a real gas come closest to ideal behavior?
At low pressure the molecules are far apart, so their volume and their attractions are negligible; at high temperature they move fast, so attractions have little effect.
At the same temperature and pressure, which gas shows the largest deviation from ideal behavior?
is polar and has many electrons, so its molecules attract each other most strongly. He, , and Ne all have very weak intermolecular forces.
At a moderate pressure, the measured pressure of is noticeably lower than PV = nRT predicts. Which is the best explanation?
Attractions make the real pressure lower. The effect in A makes the pressure higher, the wrong direction; B and C are false.
At 500 atm, a sample of Ar occupies a larger volume than PV = nRT predicts. Which is the best explanation?
At extreme pressure the atoms themselves occupy a larger fraction of the total volume, so the gas can't be compressed as far as the ideal value. The effect in A would make the volume smaller; B is false; at constant temperature the speeds don't change (D).
and He are the two gases that behave most ideally. Which is the best explanation?
The two causes of deviation are attractions between molecules and the volume of the molecules, and both are smallest for and He. The fact in B is true but isn't the reason; C and D are false.
A sample of in a rigid container is heated from 300 K to 600 K. How does its behavior change?
The strength of the attractions barely changes, but the average kinetic energy doubles, so attractions affect the motion relatively less and the gas behaves more ideally.
AP-style free response
1.00 mol of is in a 1.00 L rigid container at 300 K.
(a) Calculate the pressure in the container, treating the gas as ideal.
(b) The measured pressure is lower than your answer to (a). Explain at the particle level.
(c) Would 1.00 mol of He under the same conditions come closer to the answer to (a)? Justify your answer.
(d) Describe conditions under which the real pressure of would be higher than the ideal value, and explain why.
Model answer and scoring
(a) P = nRT/V = 1.00 × 0.08206 × 300 ÷ 1.00 ≈ 24.6 atm.
(b) molecules attract each other through dispersion forces (22 electrons, fairly polarizable). At this pressure they are fairly close together, so a molecule heading for the wall is pulled back by its neighbors and hits less hard, making the real pressure lower than ideal.
(c) Yes, closer. He atoms have only 2 electrons and are hardly polarizable, so their attractions are extremely weak, and the atoms are very small, so both deviations are small.
(d) At very high pressure (very small volume). The molecules are crowded, the volume they themselves occupy becomes a large fraction of the total, and the space they are free to move in is less than the container’s volume, so they hit the walls more often and the real pressure is higher than ideal.
Scoring: (a) about 24.6 atm; (b) states that attractions between molecules reduce how hard they hit the walls; (c) both the conclusion and the reason (few electrons, weak forces, small atoms); (d) very high pressure plus the volume of the molecules.
Summary
- An ideal gas has molecules with no volume and no forces between them; real gases are most ideal at high temperature and low pressure, and deviate most at low temperature and high pressure.
- Attractions between molecules make the real pressure lower; the volume of the molecules makes the real pressure (or volume) higher, and it wins at extreme pressure.
- The stronger the intermolecular forces (polar molecules, molecules with many electrons), the larger the deviation; He and are the most ideal gases.
AP tip: To explain a deviation from ideal behavior, first decide which effect is at work: a lower pressure means attractions, a higher pressure means the volume of the molecules. Then compare gases by the types and strengths of their intermolecular forces. “Faster molecules” and “lighter molecules” are not reasons.