# What equation of state is needed for liquid states?

I'm familiar with the ideal gas law $$PV=nRT$$ but I don't think it applies to liquids like water. If I'm wrong, please correct me! If I'm right, then what equation of state applies to liquids such as water?

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For all intents and purposes, you can use an incompressible equation of state:

$$V = constant$$

That's it. No matter what pressure and temperature, you have the same volume. It's not completely true, but in relation to gasses it is true enough to make it that pressure work is negligible in liquids compared to gasses, and for liquids, you can just deal with the heat content without considering any work done in the expansions and contractions required to change temperature.

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what about change in volume with temprature? –  Prathyush Oct 13 '12 at 12:26
That makes a lot of sense. In this case, does the equation of state remain as the ideal gas law? –  Paul Oct 13 '12 at 13:06
@Prathyush: no change in volume with either temperature or pressure--- this is the approximation which works for 99% of heat engines. You only have non-negligible volume changes in gasses. The reason is that liquids have touching atoms, and don't compress or decompress well. They do compress a little bit, but this is not a significant amount of work during heating and cooling cycles, or pressure increase/decrease cycles. It's not like in gasses, where the work and the heat are always comparable. –  Ron Maimon Oct 13 '12 at 15:28
@Paul: The equation of state is what I said V=constant, no dependence of V on P or T. So for this description, you need a Gibbs ensemble, P and T are independently varied and V is constant. If you use another ensemble and want to take the constant V limit, you have to give V a tiny dependence on pressure/temperature. –  Ron Maimon Oct 13 '12 at 15:30
@RonMaimon: Ok... I think I understand now... If the fluid is incompressible, then pressure and temperature do not affect the volume occupied by a constant quanitity of mass. Compressibility necessarily implies that pressure and temperature affect the volume occupied by a constant quanitity of mass. Am I stating this correctly? –  Paul Oct 13 '12 at 16:04

Supercritical fluids are well described by real and ideal gas laws.

A common equation of state for both liquids and solids is

$$V_m = C_1 + C_2 T + C_3 T^2 - C_4 p - C_5 p T$$

where $V_m$ is molar volume, $T$ is temperature, $p$ is pressure, and $C_1$, $C_2$, $C_3$, $C_4$, and $C_5$ are empirical constants, all positive and specific to each substance.

The Peng & Robinson equation of state has been found to be useful for both liquids and real gasses:

$$p = \frac{RT}{V_m - b} - \frac{a(T)}{V_m (V_m + b) + b (V_m - b)}$$

where $a$ and $b$ are empirical constants and $R$ the ideal gas constant.

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Is there a name for your first equation for $V_m$? –  Paul Oct 13 '12 at 13:10
Sorry, I don't know. –  juanrga Oct 13 '12 at 13:33
$V_m$ stands for molar volume. That is, the actual volume divided by the number of moles. –  Paul J. Gans Oct 14 '12 at 1:34

To add just a bit to Ron Maimon's answer, the fact is that we do not have anything like a general equation of state for liquids. They are simply too complex for present day techniques.

We can do a bit better for solids since solids have an idealized crystalline form. But even there solids are very complex.

But don't fret. The situation isn't too much better for real gases, especially around the critical point.

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Accurate equations of states (EOS) for real gases, liquids, or solids are (in contrast to nice theoretical models such as an ideal gas or debye solid) quite complex, and must be fitted to experimental data.

For example, a very accurate equation of state for water and steam can be found in
Wagner and Pruss, The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use, 1995
http://www.teos-10.org/pubs/Wagner_and_Pruss_2002.pdf

But the water EOS is not very intelligible, except for a computer.

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