# 0.1 The kinetic molecular theory  (Page 2/7)

 Page 2 / 7

## Observation 1: limitations of the validity of the ideal gas law

To design a systematic test for the validity of the Ideal Gas Law , we note that the value of $\frac{PV}{nRT}$ , calculated from the observed values of $P$ , $V$ , $n$ , and $T$ , should always be equal to 1, exactly. Deviation of $\frac{PV}{nRT}$ from 1 indicates a violation of the Ideal Gas Law . We thus measure the pressure for several gases under a variety of conditions by varying $n$ , $V$ , and $T$ , and we calculate the ratio $\frac{PV}{nRT}$ for these conditions.

Here , the value of this ratio is plotted for several gases as a function of the "particledensity" of the gas in moles, $\frac{n}{V}$ . To make the analysis of this plot more convenient, the particle density is given in termsof the particle density of an ideal gas at room temperature and atmospheric pressure ( i.e. the density of air), which is $0.04087\frac{\mathrm{mol}}{L}$ . In this figure , a particle density of 10 means that the particle density of the gas is 10 times the particle density of air at roomtemperature. The x-axis in the figure is thus unitless.

Note that $\frac{PV}{nRT}$ on the y-axis is also unitless and has value exactly 1 for an ideal gas. We observe inthe data in this figure that $\frac{PV}{nRT}$ is extremely close to 1 for particle densities which are close to that of normal air. Therefore, deviations fromthe Ideal Gas Law are not expected under "normal" conditions. This is not surprising, since Boyle's Law , Charles' Law , and the Law of Combining Volumes were all observed under normal conditions. This figure also shows that, as the particle density increases above the normalrange, the value of $\frac{PV}{nRT}$ starts to vary from 1, and the variation depends on the type of gas we are analyzing. However, even forparticle densities 10 times greater than that of air at atmospheric pressure, the Ideal Gas Law is accurate to a few percent.

Thus, to observe any significant deviations from $PV=nRT$ , we need to push the gas conditions to somewhat more extreme values. The results for such extreme conditions areshown here . Note that the densities considered are large numbers corresponding to very high pressures. Under theseconditions, we find substantial deviations from the Ideal Gas Law . In addition, we see that the pressure of the gas (and thus $\frac{PV}{nRT}$ ) does depend strongly on which type of gas we are examining.Finally, this figure shows that deviations from the Ideal Gas Law can generate pressures either greater than or less than that predictedby the Ideal Gas Law .

## Observation 2: density and compressibility of gas

For low densities for which the Ideal Gas Law is valid, the pressure of a gas is independent of the nature of the gas, and is therefore independent of the characteristics of theparticles of that gas. We can build on this observation by considering the significance of a low particle density. Even at thehigh particle densities considered in this figure , all gases have low density in comparison to the densities of liquids. To illustrate,we note that 1 gram of liquid water at its boiling point has a volume very close to 1 milliliter. In comparison, this same 1 gram of water, onceevaporated into steam, has a volume of over 1700 milliliters. How does this expansion by a factor of 1700 occur? It is not credible that theindividual water molecules suddenly increase in size by this factor. The only plausible conclusion is that the distance betweengas molecules has increased dramatically.

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