Page 54 - Essentials of physical chemistry

16 Essentials of Physical Chemistry

Although the van der Waals gas equation is generally more accurate than the ideal gas law it has a
problem in that it is not easy to solve a cubic equation to obtain the volume. It is easy to get the
pressure:

2
nRT n a
P ¼ 2
(V nb) V
but we need a trick to get the volume. By rearranging the equation to a polynomial in V set to zero,
we can use the Newton–Raphson root finding method. Thomas [3] shows this method will converge
to the nearest root from some initial guess using an iterative process as

f (x guess )
:
x new ¼ x guess
df (x guess )=dx
This method requires that the function of the polynomial be set to zero and a good initial guess for
the root. Ultimately you may have to sketch the curve to see where the nearest root is but if the
temperature is relatively high and the pressure is relatively low (with respect to the critical point) we
can make an estimate using the ideal gas equation. First we need the derivative with respect to V.
2
2
3PV 2n(bP þ RT)V þ n a ¼ 0:
Thus, we have a method to obtain V from the van der Waals equation provided we can make a good
initial guess, possibly from the ideal gas equation or from a sketch of the polynomial in V as to
where the curve crosses the horizontal axis. That is why we set the equation to zero so that the curve
will go through zero at the roots of the equation:

" #
2
3
3
2
PV n(bP þ RT)V þ n aV g n ab
g g
:
V new ¼ V guess 2 2
3PV 2n(bP þ RT)V g þ n a
g
V guess
Obviously this is a very complicated procedure but it could be coded as an equation in BASIC on a
PC to run in a millisecond. You need to set up an iterative loop so that each new value is used for the
next guess and stop iterating when the desired number of decimal places in the answer is reached.
Because this method is so tedious it is very satisfying to see the answer converge and usually five or
less iterations are adequate if the initial guess is good. One of the homework problems will suffice to
carry out this procedure to see that it can converge and that an automated way is needed to make it
practical.
In Figure 1.7 the schematic diagram plots V ¼ (V=V-critical) and P ¼ (P=P-critical) to yield an
equation independent of the (a, b) parameters as we will soon show. The main point is that the
isotherms at higher temperature appear as one branch of a hyperbola as expected but at the point
marked as the ‘‘critical point’’ the curve has an inflection point and then at still lower temperatures
the curve shows its behavior as a cubic curve in V. Careful measurements within the conditions at
temperatures lower than the critical point actually confirm the presence of liquid droplets. This leads
to a definition of the critical temperature as that temperature above which a gas cannot be
compressed into a liquid phase at any pressure. At temperatures lower than the critical temperature
of a material the gas can be compressed (squeezed) into the liquid form by applying higher pressure.
The inflection point of the van der Waals equation at the critical point is very helpful in a
mathematical sense since not only is the first-derivative zero at that point but the first-derivative
(slope) changes sign on either side of the critical point so the second derivative (curvature) must also
go through zero. We now embark on a series of mathematical manipulations to determine the (a, b)
parameters of a given gas from the experimental critical point parameters (P c , V c , T c ).