**Nanomedicine,
Volume I: Basic Capabilities**

**©
1999 Robert A. Freitas Jr. All Rights
Reserved.**

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

**10.3.2 The Van der Waals
Equation**

The compressibility of gases is most simply described by the well-known ideal gas law:

_{}
{Eqn. 10.17}

where p_{gas}, V_{gas} and n are gas pressure,
volume, and number of moles, respectively. However, all real gases deviate to
some extent from the ideal gas law, even at STP. Most real gases obey the ideal
gas law to within a few percent at low densities -- that is, at low pressures
(e.g., <~1 atm) and at temperatures well above their condensation points.
For real gases at high pressure, finite molecular volumes and intermolecular
attractions cause significant deviation from the ideal gas law. In 1873, J.D.
van der Waals deduced an empirical equation of state (subsequently derived from
statistical mechanics using suitable approximations^{1031})
that reproduces the observed behavior of real gases with moderate accuracy:

_{}
{Eqn. 10.18}

where A is a measure of intermolecular attraction and B is
a measure of finite molecular volumes. The van der Waals gas "constants," given
in Table
10.1 for various gases, are known to vary slightly with temperature. Nevertheless,
while Eqn. 10.18 is only one of several expressions
commonly employed to represent real gas behavior, it is the simplest to use
and to interpret. The van der Waals equation remains approximately valid even
at temperatures and molar volumes so low that the gas has become a liquid.^{1031}
Note that the critical temperature T_{crit}, the highest temperature
at which gas and liquid may exist as separate phases at any pressure, is approximated
by T_{crit} = 8 c_{1} A / 27 B R_{gas}, where c_{1}
= 1.01 x 10^{5} J/m^{3}-atm and R_{gas} = 8.31 J/mole-K;
critical pressure p_{crit} = A / 27 B^{2}.^{390}
For example, in the case of water, at temperatures and pressures above T_{crit}
= 647.3 K and p_{crit} = 218.3 atm,^{763}
the vapor and liquid phases become indistinguishable.

The boiling point of a pure liquid as a function of pressure
may be approximated by:^{2036}

_{}
{Eqn. 10.19}

where T_{1} and T_{2} are the boiling points
(in K) at pressures p_{1} and p_{2}, respectively, DH_{vap}
is the molar heat of vaporization for the liquid (Table
10.8), and R_{gas} = 8.31 J/mole-K. (This formula assumes that DH_{vap}
is constant over the temperature range from T_{1} to T_{2}.)
Taking T_{1} = 373.16 K, DH_{vap}
= 40,690 J/mole, and p_{1} = 1 atm for water, then at p_{2}
= 6.4 atm the boiling point has risen to T_{2} = 435 K. Boiling point
also is altered by the presence of solute (Section 10.5.3).

Table
10.2 shows the molecular number density achieved inside gas storage vessels
maintained at various pressures at T_{gas} = 310 K. Note that there
are significant (near-linear) gains in gas molecule packing density up to 1000
atm, minor (non-linear) gains up to 10,000 atm, and no significant gain at >100,000
atm as density approaches a limiting value for the liquid or solid state. Table
10.2 also shows that pressures calculated using the ideal gas law vary from
pressures calculated using the van der Waals equation by ~5% at 100 atm and
~50% at 1000 atm.*

* For comparison, pressures of ~1000 atm occur
naturally in the deepest part of the ocean (the Marianas Trench), ~10,000 atm
at the crust-mantle interface (the Mohorovic discontinuity), 3.64 million atm
at the Earth's core, and ~10^{9} atm at the center of the Sun.

One early experiment provided evidence of stably trapped (Section
10.3.4) room-temperature gases at ~1300 atm pressure inside carbon nanotubes.^{1169}
Single-walled carbon nanotubes have been experimentally charged with trapped
hydrogen gas up to 5-10% by weight,^{2024}
and with argon gas at ~600 atm.^{2034}

Last updated on 24 February 2003