CHM-02

van der Waals equation

(p + a/Vm²)(Vm − b) = R T. Attraction and covolume.

Reading speed

Equation of statevan der Waals 1873

Governing equation

\left(p+\dfrac{a}{V_m^2}\right)(V_m-b)=RT

$V_m \in \left[0.05, 50\right]$

where

V_m: Molar volume (L/mol)
T: Temperature (K)
a: Attraction a (L²·bar/mol²)
b: Covolume b (L/mol)
p: Pressure (bar)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

00:0 / 00:08

①

Governing law — red

\left(p+\dfrac{a}{V_m^2}\right)(V_m-b)=RT

$V_m \in \left[0.05, 50\right]$

$V_m = 0.05$ $p = 1719.6$

Along the curve

②

Domain of the variable — blue

V_m_{\min} = 0.05,\quad V_m_{\max} = 50

$V_m \in \left[0.05, 50\right]$

$V_m = 0.05$ $V_m = 0.05$

Along the curve

③

Response curve — green

\left(p+\dfrac{a}{V_m^2}\right)(V_m-b)=RT

$V_m \in \left[0.05, 50\right]$

$V_m = 0.05$ $p = 1719.6$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}p}{\mathrm{d}V_m}

$V_m \in \left[0.05, 50\right]$

$V_m = 0.364$ $p^{\prime} = -2679.5$

Along the curve

Lecture brief

Historical brief

Ideal-gas law, van ’t Hoff, Nernst, Michaelis–Menten and Clausius–Clapeyron are physical chemistry’s working equations of equilibrium and rate. The lab is pressure, potential and kinetics. This sheet (CHM-02 — van der Waals equation) is the form associated with van der Waals 1873. Working symbols: $V_m$ , $T$ , $a$ , $b$ $\rightarrow$ $p$ . The first cubic EOS. a corrects for attractions, b for finite molecular volume.

Purpose

Purpose: compute

p

from

V_m

T

a

b

in Physical chemistry via

\left(p+\dfrac{a}{V_m^2}\right)(V_m-b)=RT

(p + a/Vm²)(Vm − b) = R T. Attraction and covolume. Use it when a real physical chemistry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

V_m = 2.000\,\mathrm{L/mol}

T = 300.000\,\mathrm{K}

a = 1.370\,\mathrm{L^{2}·bar/mol^{2}}

b = 0.039\,\mathrm{L/mol}

, the governing relation

\left(p+\dfrac{a}{V_m^2}\right)(V_m-b)=RT

yields

p = 12.377\,\mathrm{bar}

. One mole. a in L²·bar/mol², b in L/mol, p in bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Molar volume V_m2 L/molTemperature T300 KAttraction a a1.37 L²·bar/mol²Covolume b b0.039 L/mol

Outputs

Pressure p12.377 bar

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

One mole. a in L²·bar/mol², b in L/mol, p in bar.

The first cubic EOS. a corrects for attractions, b for finite molecular volume.

Reading speed

Watch on YouTube