States of Matter — Gas Laws, Kinetic Theory and Real Gases

The gaseous state is the one place in chemistry where a handful of molecular assumptions produce an equation you can trust to three figures, and it is worth understanding why. The ideal gas law works because a gas at ordinary pressure is mostly empty space: the molecules occupy a negligible fraction of the container and hardly notice one another. Every deviation from ideality traces back to one of those two assumptions failing — molecules taking up room, or molecules attracting each other. Once you can say which assumption has broken in a given situation, the van der Waals equation stops being a formula to memorise and becomes a correction you could have written yourself.

The Gas Laws and the Ideal Equation

The individual gas laws and what each holds constant
LawStatementHeld constant
BoylePV = constant, so P ∝ 1/VT and n
CharlesV/T = constant, with T in kelvinP and n
Gay-LussacP/T = constantV and n
AvogadroV ∝ n at fixed T and PT and P
DaltonP_total = Σp_i, and p_i = x_i·P_totalV and T

Combining them gives PV = nRT, with R = 0.0821 L·atm·mol⁻¹K⁻¹ = 8.314 J·mol⁻¹K⁻¹ = 2 cal·mol⁻¹K⁻¹. Which value of R you use is decided entirely by the units of the other quantities, and mixing them is the commonest arithmetic loss in this chapter — litre-atmospheres demand 0.0821, joules demand 8.314. Two rearrangements are worth having ready: d = PM/RT for the density of a gas, and M = dRT/P for finding a molar mass from a measured density.

⚠️ Charles's law needs kelvin, and it is not a small difference
Heating a gas from 27 °C to 54 °C does not double its volume. In kelvin the change is 300 K to 327 K, a rise of just 9 %. The temptation is real because the celsius numbers do double, and the option corresponding to doubled volume is always offered. The same rule applies to every ratio law: the moment T appears in a division, it must be absolute. Only ΔT is the same in celsius and kelvin, which is why heat-capacity questions can quote either.

Dalton's law of partial pressures explains a practical point that appears in numericals: a gas collected over water is moist, so the pressure of the dry gas is the total minus the aqueous tension, p_gas = P_total − p_water. Forgetting to subtract the water vapour pressure inflates the answer by a few per cent, which is enough to move you to a wrong option.

Kinetic Theory and Molecular Speeds

The kinetic theory derives PV = nRT from mechanics by assuming molecules are point masses in random motion with perfectly elastic collisions and no intermolecular forces. Its central result is PV = ⅓ mNu², from which the average kinetic energy per molecule comes out as (3/2)kT — depending on temperature alone and on nothing about the gas. That single sentence explains why hydrogen and carbon dioxide at the same temperature have identical average kinetic energy while hydrogen moves far faster.

The three molecular speeds
SpeedExpressionMeaning
Most probable, u_mp√(2RT/M)The peak of the distribution curve
Average, u_av√(8RT/πM)The arithmetic mean of all speeds
Root mean square, u_rms√(3RT/M)The one tied to kinetic energy and pressure
Their ratio1 : 1.128 : 1.224u_mp < u_av < u_rms, always in that order
🎯 Why the Maxwell distribution flattens and shifts right on heating
The curve plots the fraction of molecules against speed, so the area under it is fixed at 1 — every molecule has some speed. Heating gives molecules more energy, so the peak moves to a higher speed; but because the total area cannot grow, the peak must simultaneously get shorter and the curve broader. That is the whole shape change, and it explains the chemically important consequence: the fraction of molecules above any given threshold speed rises sharply with temperature, which is exactly why reaction rates are so temperature-sensitive. A heavier gas at the same temperature has its peak further left and narrower, for the same reason — same energy spread over more massive molecules.

Graham's law follows directly: the rate of diffusion or effusion is inversely proportional to the square root of molar mass, r₁/r₂ = √(M₂/M₁). This is why hydrogen escapes a punctured balloon about four times faster than oxygen, and it is the principle behind the separation of ²³⁵UF₆ from ²³⁸UF₆ — a mass difference of under 1 % gives an enrichment factor of only 1.0043 per stage, which is why thousands of stages are needed.

Real Gases, van der Waals and Liquefaction

Real gases deviate because molecules do occupy volume and do attract one another. Van der Waals corrected both: (P + an²/V²)(V − nb) = nRT, where a measures the strength of intermolecular attraction and b the effective volume of the molecules themselves. Attraction reduces the pressure the gas exerts on the wall, so a is added to the measured P; the molecules' own bulk reduces the space available, so nb is subtracted from V. Reading the correction in that order makes the signs impossible to misremember.

Reading the compressibility factor Z = PV/nRT
Value of ZWhat dominatesTypical condition
Z = 1Neither correction mattersLow pressure, high temperature
Z < 1Attraction — the gas is easier to compress than idealModerate pressure; a-dominated
Z > 1Molecular volume — harder to compress than idealVery high pressure; b-dominated
Z > 1 at all pressuresAlmost no attraction to correct forH₂ and He at room temperature
🧠 Order gases by a for liquefaction and by b for molecular size
A large a means strong attraction, so the gas liquefies readily and has a high critical temperature — which is why NH₃ and CO₂ have large a and are sold as liquids in cylinders, while H₂ and He have tiny a and resist liquefaction to within a few kelvin of absolute zero. A large b simply means big molecules. So when a question ranks four gases for "ease of liquefaction", it is asking you to rank a; when it asks which is "most easily compressed" at high pressure, it is asking you to rank b. The critical temperature itself follows as T_c = 8a/27Rb, which is why a and T_c always move together.
⚠️ Above the critical temperature no pressure can liquefy a gas
The critical temperature T_c is not a threshold that pressure can overcome — it is the temperature above which the distinction between liquid and gas disappears entirely. CO₂ has T_c = 31 °C, so on a warm day no amount of squeezing will produce liquid CO₂; the cylinder in a fire extinguisher works because it is below 31 °C inside. A question that offers "apply higher pressure" as the way to liquefy a gas above its critical temperature is offering something physically impossible. The correct first step is always cool below T_c, then compress.

Two further points close the chapter. The Boyle temperature is the temperature at which a real gas obeys the ideal law over a wide pressure range, because the a and b corrections happen to cancel; below it Z dips under 1 first, above it Z rises above 1 immediately. And liquids are characterised by two properties the paper tests qualitatively: surface tension, which falls with temperature and explains capillary rise and spherical droplets, and viscosity, which also falls with temperature in liquids but rises with temperature in gases — an inversion worth remembering, because in a gas viscosity comes from momentum transfer by faster-moving molecules rather than from cohesion.

Key takeaways

  • PV = nRT with R = 0.0821 L·atm or 8.314 J — the units of the other quantities pick the value.
  • Any T in a ratio must be in kelvin: 27 °C to 54 °C is a 9 % rise, not a doubling.
  • Gas collected over water is moist — subtract the aqueous tension to get the dry-gas pressure.
  • Average kinetic energy is (3/2)kT — it depends on temperature alone, never on the identity of the gas.
  • u_mp < u_av < u_rms in the fixed ratio 1 : 1.128 : 1.224; Graham gives r ∝ 1/√M.
  • Heating shifts the Maxwell peak right and flattens it, since the area under the curve is fixed at 1.
  • Z < 1 means attraction dominates, Z > 1 means molecular volume dominates. H₂ and He have Z > 1 throughout.
  • Large a means easy liquefaction and high T_c; above T_c no pressure whatsoever will liquefy a gas.

Practice questions (8)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. A gas at 27 °C is heated to 54 °C at constant pressure. Its volume becomes:

    1. Double the original
    2. 1.09 times the original
    3. Half the original
    4. Unchanged
    Show answer

    Answer: B — 1.09 times the original

    Charles's law uses absolute temperature, so the ratio is 327/300 = 1.09, a rise of 9 %. Doubling is the trap, and it is a strong one because the celsius readings do double — but 0 °C is not the zero of temperature, so celsius ratios are meaningless. Only differences in temperature are the same in the two scales.

  2. At the same temperature, H₂ and O₂ have equal:

    1. Root mean square speed
    2. Average kinetic energy per molecule
    3. Density
    4. Rate of diffusion
    Show answer

    Answer: B — Average kinetic energy per molecule

    Average kinetic energy per molecule is (3/2)kT, which contains no reference to mass, so it is identical for every gas at a given temperature. Everything else in the list depends on molar mass: u_rms = √(3RT/M) makes hydrogen four times faster, its density is sixteen times smaller, and by Graham's law it diffuses four times faster.

  3. For a real gas at moderate pressure, Z = PV/nRT is less than 1. This indicates that:

    1. Molecular volume dominates over attraction
    2. Intermolecular attraction dominates, making the gas more compressible than ideal
    3. The gas is behaving ideally
    4. The temperature is above the critical temperature
    Show answer

    Answer: B — Intermolecular attraction dominates, making the gas more compressible than ideal

    Z < 1 means the observed volume is smaller than the ideal law predicts, so attractive forces are pulling the molecules together and the gas compresses more readily than an ideal one would. Z > 1 is the opposite case, where the finite size of the molecules resists further compression — that is what dominates at very high pressures and for H₂ and He at room temperature.

  4. CO₂ has a critical temperature of 31 °C. At 40 °C, liquid CO₂ can be obtained by:

    1. Applying a pressure of 100 atm
    2. Applying any sufficiently high pressure
    3. No pressure at all — it must first be cooled below 31 °C
    4. Reducing the pressure below 1 atm
    Show answer

    Answer: C — No pressure at all — it must first be cooled below 31 °C

    Above the critical temperature the liquid and gaseous states cease to be distinguishable, so liquefaction is impossible no matter how hard the gas is squeezed — compressing it simply gives a denser supercritical fluid. The only route to liquid CO₂ at 40 °C is to cool it below 31 °C first and then apply pressure.

  5. The correct order of molecular speeds for a gas at a given temperature is:

    1. u_rms < u_av < u_mp
    2. u_mp < u_av < u_rms
    3. u_av < u_mp < u_rms
    4. All three are equal
    Show answer

    Answer: B — u_mp < u_av < u_rms

    The fixed ratio is u_mp : u_av : u_rms = 1 : 1.128 : 1.224, so the most probable speed is always the smallest and the root mean square the largest. The ordering never changes with temperature or with the gas — only the absolute values do, since all three scale as √(T/M).

  6. Which gas would be expected to have the largest van der Waals constant a?

    1. He
    2. H₂
    3. NH₃
    4. N₂
    Show answer

    Answer: C — NH₃

    The constant a measures intermolecular attraction, and ammonia is strongly hydrogen bonded, so its a is much the largest of the four. Helium and hydrogen have only feeble dispersion forces, which is why both have tiny a values, resist liquefaction almost to absolute zero, and show Z > 1 at ordinary temperatures.

  7. On raising the temperature, the Maxwell speed distribution curve:

    1. Shifts right and its peak becomes taller
    2. Shifts right and its peak becomes shorter and broader
    3. Shifts left and becomes narrower
    4. Does not change shape, only area
    Show answer

    Answer: B — Shifts right and its peak becomes shorter and broader

    Heating raises the most probable speed so the peak moves right, but the area under the curve is fixed at 1 because it represents all the molecules — so a wider spread forces a lower peak. The last option is impossible for exactly that reason: the area can never change. The broadening is what makes the fraction of molecules above any activation threshold rise so steeply with temperature.

  8. Viscosity of a liquid decreases with rising temperature, while viscosity of a gas:

    1. Also decreases
    2. Increases
    3. Remains constant
    4. Becomes zero
    Show answer

    Answer: B — Increases

    In a liquid, viscosity comes from cohesive forces, which heating weakens, so it falls. In a gas there is almost no cohesion; viscosity arises from molecules carrying momentum between layers, and heating makes them move faster, so the transfer becomes more effective and viscosity rises. This inversion is the whole point of the question — the two states behave oppositely for genuinely different reasons.