Real gases — deviations from ideality, compressibility factor Z: Visual walkthrough

Step 1 — What "ideal" secretly assumes

WHAT. The ideal gas law says $P V = n R T$ . Here $P$ is the push of the gas on the walls (pressure), $V$ is the volume of the container, $n$ is how many moles of molecules there are, $R$ is a fixed number (the gas constant), and $T$ is the temperature. This law hides two pretend-facts.

WHY look at them. If we want to know how a real gas breaks the law, we must first name exactly what the law assumes. You can only fix a lie you can see.

PICTURE. On the left, the ideal cartoon: molecules are shrunk to dots with no size, and no strings pull between them. On the right, reality: molecules are little balls that take up room, and faint springs (attractions) tug neighbours together.

Step 2 — Naming the "how wrong" number, $Z$

WHAT. We define one single score: $Z = \frac{P V _{m}}{R T}$ where $V_{m} = V / n$ is the molar volume (the volume that one mole occupies).

Let us read the fraction term by term:

$P V_{m}$ (top) = what we actually measure: real pressure times real molar volume.
$R T$ (bottom) = what the ideal law predicts $P V_{m}$ should equal.

So $Z$ is literally $\frac{reality}{ideal prediction}$ .

WHY this ratio and not a difference? A ratio is dimensionless — it has no units — so the same number compares any gas at any scale. And it has a built-in reference point: if reality matches the prediction, top equals bottom, and $Z = 1$ exactly. Any distance from $1$ is a pure fingerprint of non-ideality.

PICTURE. A number line with $Z = 1$ marked in the middle. Left of it ( $Z < 1$ ): "reality is smaller than ideal → attractions squeeze the gas." Right of it ( $Z > 1$ ): "reality is bigger than ideal → the gas resists squeezing."

Recall Why does

$Z < 1$ mean "more compressible"? If attractions pull molecules inward, the gas collapses to a smaller $V_{m}$ than ideal for the same $P$ . Smaller measured $V_{m}$ makes the top of the fraction smaller, so $Z$ drops below $1$ . "Easier to squeeze" = "more compressible". ::: Because attraction shrinks the real volume below the ideal prediction, pulling $P V_{m}$ under $R T$ .

Step 3 — Fixing the volume lie (the constant $b$ )

WHAT. Molecules are not dots; each is a hard ball. Two balls cannot sit in the same place, so a fixed chunk of space is permanently off-limits to the centres of other molecules. Call the off-limits volume per mole the excluded volume $b$ (units $L\cdotmol^{- 1}$ ). The room a molecule can actually roam in is therefore not $V_{m}$ but $V_{m} - b .$

Reading it: $V_{m}$ is the whole container per mole, and we subtract $b$ because that much is blocked by the bodies of the molecules themselves.

WHY subtract, not add? The molecules steal free space from each other, so the usable volume shrinks. Less usable room means the gas is harder to squeeze further — this pushes $Z$ up.

PICTURE. Two hard spheres touching. Around one, a dashed sphere of twice the radius marks the forbidden zone its neighbour's centre can never enter. That forbidden bubble, summed over a mole, is $b$ .

Step 4 — Fixing the force lie (the constant $a$ )

WHAT. A molecule flying toward the wall gets pulled back by the crowd of molecules behind it. So it strikes the wall softer than an ideal molecule would. The real pressure is therefore less than the "no-attraction" pressure. We add a correction $\frac{a}{V _{m}^{2}}$ back to $P$ to recover the pressure that would exist without attraction: $P + \frac{a}{V _{m}^{2}} .$

Reading it:

$a$ = the strength of attraction (units $atm\cdotL^{2} mol^{- 2}$ ). Stickier molecules → bigger $a$ .
$\frac{1}{V _{m}^{2}}$ = the squared density. One factor of density $(\frac{1}{V _{m}})$ counts how many molecules are hitting the wall; the second factor counts how many are doing the pulling. A pull needs two partners, so density appears twice.

WHY add here (but subtract in Step 3)? The measured $P$ is too low because of attraction, so we add the amount that was lost. In Step 3 the volume was too high (we ignored molecular bulk), so we subtracted. Opposite lies, opposite signs.

PICTURE. A molecule near the wall with a red arrow (its outward flight) shortened by orange spring-arrows tugging it inward. The wall registers a gentler tap. Contrast a bulk molecule surrounded on all sides — its tugs cancel, so only near the wall does attraction bite.

Step 5 — Turning van der Waals into $Z$

WHAT. Solve the van der Waals equation for $P$ , then feed it into $Z = P V_{m} / R T$ . Isolating $P$ : $P = \frac{R T}{V _{m} - b} - \frac{a}{V _{m}^{2}} .$

Multiply by $V_{m}$ and divide by $R T$ (that is exactly the recipe for $Z$ ): $Z = \frac{P V _{m}}{R T} = size term > 1 \frac{V _{m}}{V _{m} - b} - attraction term > 0 \frac{a}{R T V _{m}} .$

Reading the two pieces:

$\frac{V _{m}}{V _{m} - b}$ : numerator bigger than denominator (we subtracted $b$ ), so this is always above 1 — it raises $Z$ . This is the size/repulsion effect.
$- \frac{a}{R T V _{m}}$ : a negative piece — it lowers $Z$ . This is the attraction effect. Notice $T$ sits on the bottom: heat the gas and attraction matters less (fast molecules ignore the tug).

WHY split like this? Because now the whole real-gas story is a tug of war you can read off directly: whichever term is larger decides whether $Z$ sits above or below $1$ .

PICTURE. A see-saw: the blue "size" term pushes $Z$ up on one side, the orange "attraction" term pulls it down on the other, balanced over the $Z = 1$ pivot.

Step 6 — The low-pressure limit and where the slope starts

WHAT. At low pressure molecules are far apart, so $V_{m}$ is large and $b / V_{m}$ is tiny. Use the approximation $\frac{1}{1 - x} \approx 1 + x$ for small $x$ (here $x = b / V_{m}$ ): $\frac{V _{m}}{V _{m} - b} = \frac{1}{1 - b / V _{m}} \approx 1 + \frac{b}{V _{m}} .$ Substituting into the $Z$ expression: $Z \approx 1 + \frac{1}{V _{m}} (b - \frac{a}{R T}) .$

Reading it: the whole deviation from $1$ is packed into the bracket $(b - \frac{a}{R T})$ :

If $\frac{a}{R T} > b$ → bracket is negative → $Z$ starts below 1 (attraction wins).
If $b > \frac{a}{R T}$ → bracket is positive → $Z$ starts above 1 (size wins, e.g. $H_{2}$ , $He$ at room $T$ ).

WHY this approximation and not the full formula? Near $P \to 0$ we only want the initial slope of the $Z$ -vs- $P$ curve. Since $1/ V_{m}$ grows in proportion to $P$ at low pressure, the sign of the bracket is the sign of that starting slope. The exact formula would hide this simple decision behind clutter.

PICTURE. Three starting curves leaving the point $(0, 1)$ : one sloping down (attraction wins, bracket $< 0$ ), one flat (bracket $= 0$ ), one sloping up (size wins, bracket $> 0$ ).

Step 7 — Every case, including the extremes

WHAT. We must cover the whole curve and the degenerate inputs, so the reader never meets an unshown scenario.

$T > T_{B}$ (hot): bracket positive from the start → $Z$ climbs above $1$ immediately, no dip.
$T < T_{B}$ (cool): attraction wins first → $Z$ dips below $1$ , reaches a minimum, then climbs.
$T = T_{B}$ (exactly Boyle): flat start, hugs $Z = 1$ over a range of low $P$ .
Very high $P$ (any gas): $V_{m} \to b$ from above, so $\frac{V _{m}}{V _{m} - b} \to \infty$ . The size term dominates and every real gas ends up with $Z > 1$ . The low- $P$ dip is only ever an intermediate feature.
Degenerate case $a = 0, b = 0$ : $Z = 1$ at all $P, T$ — we recover the perfect ideal gas.
Degenerate case $b = 0$ but $a \neq = 0$ ("sticky points"): $Z = 1 - \frac{a}{R T V _{m}} < 1$ always — a gas that only attracts is always more compressible than ideal.

WHY spell these out? Because the single worked curve most textbooks draw ( $N_{2}$ -like, $T < T_{B}$ ) hides the hot case, the exact-Boyle case, and the guaranteed high- $P$ rise. Missing any one leaves the reader stranded when they meet $H_{2}$ or a squeezed gas.

PICTURE. One graph with $P$ across and $Z$ up. Four curves: hot (rises straight up), Boyle (flat then up), cool (dips then up), and a marker showing all curves converging upward at very high $P$ .

Step 8 — Numbers on the picture (worked checks)

WHAT. Anchor the abstractions with the parent's three numbers, each read off the diagrams above.

(These match the parent's Critical Constants and Liquefaction and Kinetic Theory of Gases framing: $a$ and $b$ are the same molecular facts, seen from a different door.)

The one-picture summary

Everything above, compressed: the two lies (Step 1) become the two corrections $a$ and $b$ (Steps 3–4), which assemble into van der Waals (Step 4), which rearranges into a two-term tug of war for $Z$ (Step 5), whose low- $P$ bracket decides the starting slope and defines $T_{B}$ (Step 6), while high $P$ forces every gas above $1$ (Step 7).

Recall Feynman: the whole walkthrough in plain words

The pretend gas is a crowd of invisible dots that never touch and never hold hands. Two things are wrong. First, real people have bodies — they take up room, so the crowd can't pack as tight as dots; we shrink the "free room" from $V_{m}$ to $V_{m} - b$ , and this makes the gas resist squeezing, nudging the score $Z$ up. Second, real people hold hands — attractions pull a person heading for the wall back into the crowd, so they bump the wall softer and the pressure reads low; we add $a / V_{m}^{2}$ back to fix it, and this drags $Z$ down. Put both fixes into $P V = R T$ and you get van der Waals. Rearrange it and $Z$ becomes a see-saw: a size term always above $1$ versus an attraction term always below. At low pressure hand-holding usually wins, so $Z$ dips under $1$ — unless the molecules barely attract (like $H_{2}$ , $He$ ), where bodies win and $Z$ starts high. The exact temperature where the two effects tie at the start is the Boyle temperature $T_{B} = a / R b$ . And no matter what, squeeze hard enough and bodies always win — every real gas ends up with $Z$ above $1$ .