What equations describe the behavior of real gases beyond PV=nRT?

[newton1]

the ideal gas law PV=nRT only can used on the ideal gas, right?
if the we want consider the real gas...
what equation should we used??

 

[dg]

The most famous 'real' gas equation of state is the so-called Van der Waals equation

(P+a*(n/V)^2)(V-nb)=nRT

that can be derived by ideal gas equation substituting

V->V-nb
P->P+a*(n/V)^2

The first substitution compensate for the volume occupied by each molecule (we can think of b as the volume occupied by a mole of gas at 0 Kelvin)
The second substitution compensate for the internal energy density due to intermolecular interaction

a and b are considered constants dependent on the gas only.

You can note that the equation depends on P, T and V/n only, once a and b are fixed.

Try and search on the web for Van der Waals equation...

 

[enigma]

There are other equations also which are more precise than Van der Waals.

The problem is, they are get more and more complex as the precision gets higher. For many applications, Van der Waals is sufficient.

Examples:

Beattie-Bridgeman

P=R_uT/v²*(1-c/(vT³))*(v=B)-A/v²

Benedict-Webb-Rubin

P=R_uT/v + (B₀R_uT - A₀ - C₀/T²)*1/v² + (bR_uT - a)/v³ + a* α / v⁶ + c/(v³T²)*( 1 + γ / v²)* e^{- γ / v^2}

(Don't ask me how to apply those... I don't even claim to know...)

 

[Andersen]

in fact, the ideal gas law is usually sufficient too - especially at high temps and low pressures (if you can arrange both) - just allow for extra degrees of freedom in the specific heat if its polyatomic.

to add more detail, there are various levels of thermodynamics through to statistical mechanics that you can apply, if needed - you can model for the "exact" interaction your gas has (in prinicple - these things are hard to solve sometimes, and you'll probably have to use perturbation theory)

for "maximum realness", you'll need Quantum Stat-Mech, but that is probably serious overkill.


Joe

 

[chroot]

Yep, in general most situations where the pressure < 150 bar and the temperature > 200K are very accurately modeled by the ideal gas law.

- Warren

 

[Mike H]

More a historical curiosity than anything else:

van der Waals never really wanted a and b to be taken as constants, in fact, if you look at his later work in the area, he sought to see how they varied with changing parameters. However, it tends to be something that is not overly productive and has long since fallen by the wayside.

Back on topic...

A good bit of determining equations of state for real fluids is done computationally/numerically, with the algebraic expression extracted after fitting the data. While you not unexpectedly see this in chemical engineering, you also see this quite a bit in condensed matter/chemical physics where we still can't seem to model water accurately all the time. A good bit of the interest in formulating better quality models of fluids is due to the interest in biological systems, where figuring out solvation can be a non-trivial exercise.