Question

At low densities, the vander waals equation, $(p+\frac{a}{V_m^2})(V_m-b)=RT$ and the Dieterici's equation $P(V_m-b)=RTexp(-a/RT{V}_m)$ give essentially the same value of p.
Enter 1 if the statement is true else 0.

Answer 1

We know density $=\frac{m}{V}\propto \frac{n}{V}$

In the given equations $n=1$, thus for low density Volume would be high.

The Dieterici's equation is $P(V_m-b)=RTexp(-a/RT{V}_m)$

Suppose $x=a/RT{V}_m$. The exponntial can be written as $e^{-x}$

Expand it as power series (assuming that $x<<1$).

$e^{-x}=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+....$
$P=\frac{RT}{V_m-b}(1-\frac{a}{RT{V}_m}+\frac{a^2}{2R^2{T}^2{V}_m^2}.....)$

The higher powers can be neglected.
As $P=\frac{RT}{V_m-b}-\frac{a}{V_m^2}$ ($b<V_m\because$ volume is high)

$(p+\frac{a}{V_m^2})(V_m-b)=RT$

This is the vander waals equation.
Thus, at low densities, the van der Waals equation and the Dieterici's equation give essential the same value of p.