$3-1$ Pad$$ approximation applied to cubic equations of state. The theoretically$-$substantiated

virial equation of state is a series expansion in the molar density, or alternatively, in the inverse of

the molar volume: $z=1+\frac{B}{v}+\frac{C}{v^2}+\frac{D}{v^3}+...$

The coefficients $B(T),C(T),D(T),$dots, are the second, third, fourth, etc. virial coefficients. They

are functions of temperature only. They reflect, respectively, the interactions between two, three

or more molecules in the system. Unfortunately, this EOS converges very slowly. This means

that many terms must be considered, each with its own temperature$-$dependent coefficients.

Therefore, approximations such as the cubic equations of state are used instead.

Functions are commonly approximated using polynomial expansions. Often, it is better to use a

Pad$$ approximant instead. These are rational functions defined by the ratio of two polynomials.

For example, a $P_{2,2}(x)$ approximant for a function in the variable $x$ is defined by the ratio of two

quadratic polynomials: $P_{2,2}(x)=\frac{a{x}^2+bx+c}{x^2+dx+e}$

$($c$)$ The van der Waals equation of state $($EOS$)$ is defined by: $(P+\frac{a}{v^2})(v-b)=RT.$ Show

that the van der Waals EOS is in fact a $P_{2,2}(v)$ Pad$$ approximation for the compressibility

function $z=P\frac{v}{R}T.$

$($d$)$ Assume that you were unaware of the existence of the van der Waals EOS. Use the virial

EOS and construct a $P_{2,2}(v)$ Pad$$ approximation with exactly the same functional form as

the one determined for the van der Waals EOS in $($c$).$ Start your analysis considering the

limiting value of $z$ as the molar volume becomes large. Hint: The problem is only

tractable when the denominator polynomial does not feature a constant term, i$.$e$.$ consider

as starting point $z=\frac{v^2+av-d}{v^2-bv+c}$ and let $c=0$