Critical constants Tc, Pc, Vc; law of corresponding states: Foundations

0. How to read this page — and one standing convention

The parent note Critical constants & corresponding states uses a lot of notation. Below we earn each symbol from nothing, in order, so that nothing is used before it is defined. Read top to bottom.

1. The three things we measure about a gas

Imagine gas trapped in a cylinder (holding one mole) with a piston you can push. Three numbers describe its state.

Figure — Critical constants Tc, Pc, Vc; law of corresponding states

Figure 1 — A one-mole cylinder. The red arrows are molecules banging the wall (that is pressure $P$ ); the box width is the molar volume $V$ ; the blue balls' jiggle speed is temperature $T$ . This anchors the three lab knobs every later equation relates.

Why these three? They are the knobs you can actually turn on real gas in a lab. Everything in the parent note is a relationship between these three.

2. Counting molecules: the mole and $R$

Why $R$ appears everywhere: any time we relate $P V$ (an energy) to $T$ (a jiggle), $R$ is the bridge. See Ideal Gas Equation.

3. The starting model: the ideal gas

Now that $R$ and $n$ are defined, we can write the simplest relationship between $P$ , $V$ , $T$ .

See Ideal Gas Equation and Real Gases and Compressibility Factor Z.

4. Fixing the model: intermolecular forces and molecular size

Real molecules do two non-ideal things.

Figure 2 — Left: the ideal picture (size-less dots that never stick). Right: the real picture — molecules have a hard size (yellow, that is $b$ ) and pull on neighbours (red dashed springs, that is $a$ ). This figure's job is to make the two van der Waals corrections physical before we write them algebraically.

Intuition WHY the attraction correction is

$a / V^{2}$ (derivation sketch) The pressure drop from stickiness isn't a random $1/ V^{2}$ — you can reason it out.

A molecule near the wall is pulled back inward by its neighbours, so it strikes the wall softer. The size of this "pull-back" is proportional to how many neighbours are nearby — i.e. to the local density of molecules, $ρ = n / V$ .
But the number of wall-hitting molecules being slowed is also proportional to that same density $ρ = n / V$ .
Pressure-drop $\propto (how many get pulled) \times (how hard each is pulled) \propto \frac{n}{V} \cdot \frac{n}{V} = \frac{n ^{2}}{V ^{2}}$ .

So the correction goes as $\frac{n ^{2}}{V ^{2}}$ ; folding the proportionality constant into $a$ gives $\frac{a n ^{2}}{V ^{2}}$ , and with our standing $n = 1$ this becomes exactly $\frac{a}{V ^{2}}$ . Two factors of density, two powers of $V$ in the denominator — that is the whole reason the exponent is 2.

See Van der Waals Equation of State.

5. The behaviour picture: isotherms and the plateau

Figure 3 — Three isotherms (fixed- $T$ curves). Blue ( $T < T_{c}$ ) has a wiggly plateau region where liquid and gas coexist; yellow ( $T = T_{c}$ ) touches its flat spot at exactly one point (red dot, the critical point); green ( $T > T_{c}$ ) falls smoothly with no plateau. Note every curve lives to the right of $V = b$ . The figure shows how the plateau shrinks to a point as $T$ rises to $T_{c}$ .

6. Naming the special point: $T_{c}$ , $P_{c}$ , $V_{c}$

Now that we have the critical point as a picture, we give its three coordinates names — the symbols the parent note uses constantly.

Definition The critical constants

$T_{c}$ , $P_{c}$ , $V_{c}$ The critical point is one specific state of the gas — the single dot where the plateau has just collapsed. Its three coordinates are:

== $T_{c}$ (critical temperature) = the temperature of that special isotherm== — the exact $T$ at which the plateau shrinks to a point. Above $T_{c}$ no liquefaction is possible.
== $P_{c}$ (critical pressure) = the pressure at the critical point== — the height of the red dot on the $P$ – $V$ plot.
== $V_{c}$ (critical molar volume) = the molar volume at the critical point== — the horizontal position of the red dot (per mole, by our standing convention).

The little subscript " $c$ " everywhere just means "measured at the critical point." So whenever you meet $P_{c}$ , $V_{c}$ or $T_{c}$ later — including inside $Z_{c} = P_{c} V_{c} / (R T_{c})$ — read it as "the pressure / volume / temperature of that one critical dot."

7. The calculus tool: what a derivative sees

The parent note pins the critical point using two conditions on derivatives. Here's what those symbols mean, from zero.

Figure 4 — Blue curve: a shape with an inflection where the tangent (yellow ruler) is flat AND the curve stops bending — this is the critical-point geometry. Green dashed curve: an ordinary peak that is flat but STILL bends, to contrast. The figure exists to show why we need TWO conditions, not one.

8. The scoreboard number: compressibility factor $Z$

9. Reduced variables: measuring in "fractions of critical"

Equipment checklist

Test yourself — you are ready for the parent note when you can answer each without peeking.

On this page, what does a lone

V

always mean, and why?

Molar volume (space per one mole), because we fixed

n = 1

throughout.

What does

P

physically represent?

How hard gas molecules push on the walls per unit area (atm or Pa).

Why must

T

be in Kelvin?

So temperature is never negative; it starts at absolute zero and measures average molecular jiggle.

What is

R

and why does it appear?

The gas constant,

0.0821 L atm mol^{- 1} K^{- 1}

; it bridges the units of

P V

(energy) and

T

What does the ideal gas law assume, and why can't it liquefy?

Point-sized, non-sticky molecules; with no size and no attraction there is nothing to hold a liquid together.

What does the van der Waals constant

a

measure, and its units?

Strength of intermolecular attraction (stickiness); units atm·L²·mol⁻².

Why is the attraction correction proportional to

1/ V^{2}

Both the number of molecules pulled back and the strength of each pull scale with density

n / V

; their product gives

n^{2} / V^{2}

, i.e.

a / V^{2}

n = 1

What does the van der Waals constant

b

measure, and its units?

Excluded volume per mole — the space the molecules themselves take up; units L·mol⁻¹.

What is the physical domain of the vdW equation and what happens at the edge?

V > b

; as

V \to b^{+}

the term

R T / (V - b)

diverges to

+ \infty

(can't compress past the molecules' own bulk).

What is an isotherm?

A curve of

P

V

at one fixed temperature.

What does the plateau on an isotherm mean?

Gas and liquid coexisting; squeezing at constant pressure converts gas to liquid.

What do the subscripts in

T_{c}

P_{c}

V_{c}

mean?

"At the critical point" — the temperature, pressure and molar volume of that single dot where the plateau collapses.

What does

(\partial P / \partial V)_{T} = 0

mean geometrically?

The isotherm is momentarily flat — a horizontal tangent (with

T

held fixed).

What does

(\partial^{2} P / \partial V^{2})_{T} = 0

mean?

Zero curvature — the curve isn't bending at that instant (inflection).

Why do we need BOTH derivatives zero at the critical point?

The critical point is flat AND un-bending — an inflection with horizontal tangent — which a single condition can't pin down.

What does

\partial

(vs

d

) signify here?

Vary

V

only while holding

T

constant.

Define the compressibility factor

Z

, and give

Z_{c}

for vdW.

Z = P V / R T

; at the critical point

Z_{c} = P_{c} V_{c} / R T_{c} = 3/8 = 0.375

for every gas.

What is a reduced variable?

A property divided by its critical value, e.g.

T_{r} = T / T_{c}

— a unitless "fraction of critical" that cancels gas identity.