Chemistry - What is a rigorous definition of gas volume, and how is the Van der Waals equation derived?
Solution 1:
I edited the first van der Waals equations in your question, because it was incorrect.
First, the volume available to the gas is pretty much was you think: it's the space left for it to occupy, i.e. the volume delimited by the container. If you think of a gas tank, it's the interior volume of the tank. For systems of macroscopic dimensions, there is no real ambiguity as to its definition, which can be purely geometrical. For nanoscale systems, such as accessible volume in porous media, the definition is much harder: it is typically done by constructing a Gibbs dividing surface between the two phases considered (here, a solid and a gas).
Second, importance of both terms in a real-life example. Let's write the van der Waals equation in terms of density, rather than volume and quantity:
$$ \left(P+a\rho^2\right)\left(\frac{1}{\rho}-b\right)=RT $$
Let's take a practical example of a gas with a large quadrupolar moment, CO2. We have $a$ and $b$ parameters for it from this source, and we can plot $\rho(P)$ at $T=300$ K:
(black: ideal gas; red: van der Waals gas; green: only $b$; blue: only $a$). We see that excluded-volume correction by itself is not much of a gain, but it's actually crucial once you include the attraction, because otherwise the phase transition (the cusp in the blue curve) occurs at much too low pressure.
Finally, regarding the derivation of the van der Waals equation, this is typically a topic of statistical thermodynamics and there are many such demonstrations. My favorite is the classical one by Hill, but it may not use the notations in modern textbooks. I'm also partial to that in McQuarrie’s book. All statistical physics derivation introduce the excluded volume in a very natural fashion, but they show more variation on the way they arrive at the $a$ term (so of them find the exact analytical form of the equation, while in some derivations you find a series expansion whose first term is the van der Waals equation).
Solution 2:
The van der Waals equation can't be derived from first principles. It is an ad-hoc formula. There is a "derivation" in statistical mechanics from a partition function that is engineered to give the right answer. It also cannot be derived from first principles.
A gas is a collection of molecules that do not cohere strongly enough to form a liquid or a solid. The volume of a gas is, as long as the molecules fit into it, the volume of the container holding the gas. In other words, the volume of a gas is NOT a property of the gas directly, but a property of the container.