Is there a generalized steady state state equation(s) that considers all states of matter?

Question

The Ideal Gas Law (aka State Equation), $PV=nRT$ relates the properties of pressure, amount of matter, volume and temperature for a gaseous state of matter under steady state conditions, but is there a more generalized theory, equation or set of equations that relates these steady state thermodynamic properties for any state of matter? One that results in the state equation for a gas, but also relationships for liquids and solids?

Answer 1

The answer is negative and we can already see this with gases. The idea gas law, which I will write in the form

$$\frac{P}{kT}=\frac{N}{V},$$

where $N$ is the number of molecules and $k$ the Boltzmann's constant, is only valid for diluted gases, i.e. for small a density $\frac{N}{V}$. Real gases deviate from it. It is not difficult to understand why: the ideal gas law can be derived by assuming that (i) atoms/molecules have a zero volume, and (ii) there is no long distance interaction between then: they only feel each other when they collide, elastically, at one single point because of the zero volume (hard ball model). Both hypotheses are violated as the density $\frac{N}{V}$ increases.

First, obviously, as there are more atoms/molecules per unit of volume, their volume is not negligible. Moreover, they are not as far and apart as one can neglect the electrostatic forces between them. The existence of such forces could come as a surprise considering atoms and molecules are neutral. But for many of them the barycentre of the positive nuclei and the negative electrons are distinct, thus creating an electric dipole. Even for monoatomic gases for which isolated atoms do not have such a dipole, when two atoms approach each other, they can induce a dipole in the other one. Then the strength of the force between two dipoles separated by a distance $r$ scale as $\frac{1}{r^6}$ (van der Waals force) as opposed to the $\frac{1}{r^2}$ for electric charges: thus dipole-dipole interaction is weak, and that's why the ideal gas law works, if only approximately. Taking into account the non-zero volume of atoms/molecules and their van der Waals interaction, one can work out corrections to the ideal gas law. The first correction is proportional to the square of the density, the second one to the cube of the density, etc. This should not come as a surprise, since this is a larger $\frac{N}{V}$ that made the effects I have just discussed relevant. The coefficients of those powers of the density depends on the only independent variable let, i.e. the temperature. Mathematically, it reads

$$\frac{P}{k} = \frac{N}{V} + B_2(T)\left(\frac{N}{V}\right)^2 + B_3(T)\left(\frac{N}{V}\right)^3 + \cdots$$

This is called the virial expansion and $B_2(T)$, $B_3(T)$, etc are called the virial coefficients.

It it tempting to conclude that we have a universal gas law but the virial coefficients will depends on the type of atoms/molecules of the gas. This should be clear from my explanation of their origin: different atoms/molecules have different size and different permanent or induced electric dipoles, and this translate to different virial coefficients. There is an easier manner to visualise this: there exists a couple of heuristic equations which are somehow an approximate summation of the virial expansion. Popular ones are the van der Waals equation and the Redlich–Kwong equation. The former can be written in the following form, for example,

$$\frac{P}{kT}=\frac{N}{V - N b'} - a'\left(\frac{N}{V}\right)^2.$$

The constants $a'$ and $b'$ are different for different gases: generally, they have to be measured.

The morale is that, even for gases, we have only universal patterns of equations, patterns in which we can plug different numbers for each gas.

Since I introduced the subject by looking at what happens with an increasing density $\frac{N}{V}$ above, let me consider liquids. They indeed come naturally into this discussion since by increasing the density, at some point, the gas will condense into a liquid. But then, as you surely know, liquids are nearly incompressible. Thus the equations worked out so far for gases, which clearly let $V$ decrease when $P$ increases are immediately all wrong after the phase transition to a liquid. So for liquids, even the patterns which we made emerge for gases fall apart. What would be an equation of state for a liquid then? At this point, I should dwelve into the isotherms in the $P$ versus $V$ diagram and show how at the phase transition between gas and liquid one has to patch up the van der Waals isotherms to impose the observed behaviour (a quick googling found this page which does a good job to explain the issue) but that discussion is already too long and instead I will finish with another phenomenon which was irrelevant for gases.

You have noticed that for gases, I have not considered the effect of gravity, or of any other external force acting on the whole volume. Thus for the equivalent for a liquid, we shall consider for example a small volume of water in the weightless conditions of the International Space Station. You already know what it looks like, I am sure, i.e. a small ball of water, and here we have a completely new phenomenon: surface tension. The molecule of water at the surface of the ball attract each others, and in order to minimise the potential energy, a ball is formed. The physics model is completely different than that for a gas.