**James Richard Fromm**

The **state** of any amount of substance is something which chemists often
find it necessary to specify clearly. Such a specification for even the simplest substance
must include at least *n*, the amount of substance present; *p*, the pressure,
and *T*, the absolute temperature. In most cases it is found convenient to specify a **standard
state** of a system as well, so that different chemical systems can be compared
under the same conditions. As a standard, the choice of *n* = 1 is universal;
however, the pressure is commonly chosen as either *p* = 100 kPa (1 bar) or as *p*
= 101.325 kPa (1 atm). The temperature is commonly chosen as either *T* = 298.15 K
(25^{o}C) or as *T* = 273.15 K (0^{o}C). **Throughout all of
the sections of this work the values of p = 100 kPa and T = 298.15 K will be
used as standard.** For gases, other works may employ "STP" (standard
temperature and pressure) as

The simplest known equation of state, which is an equation linking at least the three
properties of temperature, pressure, and volume of a chemical system, is the ideal gas
law, *pV* = *nRT*. This equation accurately describes an ideal gas, but it
describes a real gas such as oxygen or carbon dioxide accurately only at pressures below
atmospheric. As the pressure increases or the temperature decreases, real gases are found
to deviate significantly from the behavior expected of ideal gases.

Many more complex equations of state have been proposed to describe the behavior of gases, liquids, and solids but their physical interpretation is often not obvious. We will consider only one of them, the van der Waals equation.

In 1873, Johannes van der Waals, a physics professor at the University of Amsterdam, developed an equation to account more accurately for the behavior of real gases. It was considered a sufficiently important development to justify the award of the Nobel Prize in 1910.

The **van der Waals equation** is the second most simple equation of
state; only the ideal gas law is simpler. It is used to describe the behavior of gases
when pressures are higher, or temperatures are lower, than those at which the ideal gas
law is sufficiently accurate. The van der Waals equation describes the relationship
between the physical quantities of pressure, temperature, and volume more accurately than
does the ideal gas law. It does so, however, at the cost of a more complex equation and
the use of a unique set of two van der Waals coefficients for each different gas. The
usual form of the van der Waals equation is:

(*p* + (*n*^{2}a/*V*^{2}))(*V* - *n*b)
= *nRT*

In the van der Waals equation, the term *n*^{2}a/*V*^{2}
reflects the fact that the attractive forces between molecules are not zero. The measured
pressure is thus less than it should be because the attractive forces act to reduce it.
The term *n*b reflects the fact that the volume of the molecules of a real gas is not
zero, and so the volume in which the molecules may move is less than the total measured
volume. A table of the van der Waals coefficients for several common gases is given below.

Gas |
atm dm^{6}/mol |
dm^{3}/mol |

Ideal | 0.0 | 0.0 |

He | 0.034 | 0.0237 |

Ar | 1.345 | 0.0322 |

O_{2} |
1.360 | 0.0318 |

N_{2} |
1.390 | 0.0391 |

CO_{2} |
3.592 | 0.0427 |

CH_{4} |
2.253 | 0.0428 |

H_{2} |
0.244 | 0.0266 |

Example. The volume of one mole of oxygen **molecules** is 31.8 mL
according to the van der Waals coefficient values tabulated. If the molecular diameter is
taken as 0.370 nm, an approximate molar volume would be *N*_{A}*d*^{3},
or 30.5 mL. The volume actually occupied by one mole of oxygen **gas** at 25^{o}C,
according to the ideal gas law, is 24465 mL. The molecules of the gas actually occupy only
0.13% of the total volume occupied by the gas at 25^{o}C. The vast majority of the
volume of the gas is empty space through which the molecules move.

Copyright 1997 James R. Fromm