Before you read this, I suggest you read post 18.25.
In post 18.25, we saw that for most gases, under most conditions, the pressure, p, and volume, V, of n moles (post 17.48) a gas were related to its temperature, T (on the Kelvin scale, post 16.34) by the equation
pV = nRT
where R is a constant called the ideal gas constant. So, if we double T, pV doubles, and if we triple T, pV triples. This means that, for an ideal gas, a graph of pV plotted against T is a straight line, as shown in the picture above.
However, at very high pressures, this ideal gas equation does not work so well. The gas is then called a non-ideal gas; I have called this post “Real gases” because, our explanation of the behaviour of ideal gases is a simple model for the behaviour of a gas that may not describe real gases exactly. In post 18.26, we saw that the perfect crystal is a model, not an exact description, of a real crystalline solid.
When the ideal gas equation ceases to describe the behaviour of a gas, a graph of pV plotted against T is no longer a straight line. One attempt to describe the behaviour of non-ideal gases was to modify the ideal gas equation, for example, by writing it as
pV = nRT(1 + a’T)
or even as
pV = nRT(1 + a’T + a’’T2)
where a’ and a’’ are called virial coefficients and their values are chosen to make a graph, of pV plotted against T, go through the experimental data points.
Some books give the impression that this approach provides a better understanding of non-ideal gases. I think that it’s just a trick to make a curve go through a set of points and doesn’t lead to much advance in understanding (see appendix).
A real advance in understanding non-ideal gases was made by the Dutch physicist Johannes van der Waals (1837-1923).
He realised that the ideal gas equation works if the volume of the molecules in a gas is negligible. In a real gas, each molecule has a volume and excludes other molecules form the space it occupies. When we think about all the molecules in a sample of gas, this exclusion volume will be proportional to the number of moles. So, van der Waals replaced V in the ideal gas equation by V – nb, where b represents the volume excluded by a mole of the gas in the sample. The modified equation becomes
p(V – nb) = nRT.
The other condition for the ideal gas equation to work is that the gas exerts a pressure only because its molecules collide with the walls of its container – collisions between molecules are negligible. However, van der Waals realised that, as the molecules in a gas are pushed closer, collisions between molecules are increasingly likely. Then the change in momentum (post 16.13), when molecules collide (post 17.30), is not all transferred to the walls of the container. So the force exerted by these collisions (post 16.13) and, therefore, the pressure (post 17.5) in the gas is greater than the value for the pressure that is used in the ideal gas equation. The probability of collision between molecules will be proportional to the number of moles of gas divided by the volume they occupy, n/V. Also, when a molecule collides with the wall, its momentum (and, hence, the force it exerts in a collision) is reduced by the number of its previous collisions which, as we have seen, is proportional to n/V. So the pressure in the gas is greater by an amount that is proportional to (n/V) multiplied by (n/V) which equals n2/V2; the amount by which the pressure is greater can then be written as an2/V2, where a is a constant for the gas being considered.
So, van der Waals replaced p, as well as V, in the ideal gas equation to get the result
(p +an2/V2)(V – nb) = nRT.
This equation is called the van der Waals equation and describes the behaviour of most gases at high pressures reasonably well.
In this appendix, I want to show that using the virial coefficients, to describe the behaviour of a non-ideal gas, is just a trick to make a curve go through a bunch of points.
In this appendix, the value of y depends only on the value of x and the values of one or more constants, a0, a1, a2, a3… We call x an independent variable and y a dependent variable.
The graph above shows some red data points. I have fitted the best straight line between them, shown in blue. To fit this line, I have found the values of a0 and a1 that minimise the sum of the squares of the differences between the y values of the data points and those calculated using the equation y = a0 + a1x.
To find each calculated value of y, I have put the x value for each data point into the equation. Why have I minimised the sum of the squares of the differences and not the sum of the differences? The reason is that differences can be positive or negative, but squares of differences are always positive. By minimising the sum of the squares, I minimise the true discrepancy between the line and the data points.
The line straight line does not fit the data points very well!
In the graph above, I have fitted the curve y = a0 + a1x + a2x2 through the data points. The fit between the curve and the data points is much better.
In the graph above, you can see that fit for the curve y = a0 + a1x + a2x2 + a3x3 is better still.
So the use of virial coefficients to “explain” the properties on non-ideal gases is simply an exercise in curve-fitting. If you want to fit curves like this – just use the polynomial option in the Trendline function of “Excel”; the straight line is a first-order polynomial, the next curve is a second-order polynomial and the final curve is a third-order polynomial.
Fitting straight lines and curves to data points can be useful but needs to be done with care – see post 16.8.