Grokking The Kinetic Theory of Gases Part I


©Fernando Caracena2014k0

Matter is Quantized

The idea that matter comes in chunks—discrete units of matter that make up all material things—is a very old idea. Lucippus (5th century BCE) and his student, Democritus ( c. 460 – c. 370 BC) advocated this idea among the ancient Greek philosophers. Lucretius (c. 99 BC – c. 55 BC) presented this idea in a long poem called "De Rerum Natura" , which translates to "On the nature of things". This idea about the basic structure of matter was known as atomism. What may have held back the development of this idea in the natural sciences was the logical contradiction in the basic idea as presented by Democritus and Luccipus: that in really there is nothing but atoms and the void, atoms being hard, irreducible bits of the various known substances that moved around in nothingness, the void. The logical contradiction comes from the concept that the void, or nothing, exists. The idea that nothing has real existence is a logical contradiction. Modern physicists, of course, still talk about the vacuum, but the present idea about it is that the physical vacuum is not really empty. It may just be empty of particles of matter, but it is full of activity, which physicists now describe as virtual particles. Perhaps, the mistake of these ancient philosophers is that they were too specific in their details about how the peices were supposed to work and fit together. Anyway, let us go on.

The quantization of matter was developed before that of the quantum theory, as a result of trying to understand the laws of chemistry and the behaviour of gases. Here the French scientist, Antoine Lavoisier played a prominent role. Basically, he found that mass is conserved during chemical processes. The equating of total mass of all the components entering a chemical reaction, with that of the products resulting from that reaction, made quantitative chemical analysis possible. Further, chemical analysis could reduce chemical compounds to irreducible elements that defied further chemical analysis. This fit the idea that the elements were made up of different types of atoms in large numbers and compounds were collections of composite structures of atoms glued together in what what were called molecules. For example. pure gold consisted of a collection of a very large number of the smallest bits, or atoms of gold. What atoms actually were was not as important as the idea that they existed and that they could go into various associations to form molecules that were the smallest units of chemical compounds.


Ideal Gas

Antoine Lavoisier worked with volumes of gases finding that elemental gases, such as oxygen, combined chemically with other elemental gases, such as hydrogen, to form compounds, such as water in gaseous form (or water vapor), according to various proportions by volume. Use some measured volume for the purpose of this discussion. Because the volume of a gas depends on both the pressure and temperature that it is subject to, we also have to specify that the measure of volume is taken, always at the same temperature and pressure. In that case, two volume measures of hydrogen gas combine (boom) with one volumetric measure of oxygen gas to form water vapor. The complication here is that the reaction of hydrogen and pure oxygen produces an explosion that releases a lot of heat. There are other complications in that water vapor is difficult to cool and at the same time keep in its gaseous form without allowing some of it to condense into drops of water.

Chemists found that equal measures of elemental gases do not weigh the same. A volume measure of oxygen weighs roughly sixteen times as much as one of hydrogen. The physical properties of gases therefore loomed large in understanding the atomic and molecular structure of matter, and in relating that structure to the laws of chemistry. Remember that by "laws" we refer here to empirical relations discovered by experimentation and not some metaphysical connections.


Other Results from Chemistry

Another contribution to our understanding of atoms and molecules came from quantitative studies in chemistry of Joseph Louis Proust (1754 – 1826), who provided an  important insight in this regard in his law of definite proportions of chemical combinations, stated as follows in the hyperlinked Wikipedia article on Proust, cited above:

"In chemistry, the law of definite proportions, sometimes called Proust's Law, states that a chemical compound always contains exactly the same proportion of elements by mass. An equivalent statement is the law of constant composition, which states that all samples of a given chemical compound have the same elemental composition by mass. For example, oxygen makes up about 8/9 of the mass of any sample of pure water, while hydrogen makes up the remaining 1/9 of the mass. Along with the law of multiple proportions, the law of definite proportions forms the basis of stoichiometry."

If we combine what is stated above in this blog, "a volume measure of oxygen weighs roughly sixteen times as much as one of hydrogen" with another, "two volume measures of hydrogen gas combine (boom) with one volumetric measure of oxygen gas to form water vapor", then ee can calculate the above example numbers given in the Wikipedia reference.

Use one volumetric unit of oxygen as one unit of mass.

In that case, two volumetric units of hydrogen combining with one of oxygen has a total mass of

2*(1/16)+1=9/8 , units.

The proportion of the mass of hydrogen in the reaction to the total mass is


"    =1/9;

and that of oxygen is

1/(9/8) = 8/9  .

This result comes out logically from the ideas that chemical reactions involve certain multiples of volumetric measure, and that different elemental gases have different masses for the same volumetric measure.


Prelude to Kinetic Theory

The ideal gas law applies to the properties of real gases diluted to low enough densities. We have discussed these in a previous blog called, "Grokking the Thermodynamic Theory of Gases". For any contained gas sample that follows the ideal gas law, the gas laws, independent of units are summarized in the simple equation,

P V = K T,                                 (1a)

where K is an arbitrary constant that is determined by the initial conditions of the gas sample.

In this case, (1a) sums up the behavior of very dilute gases, which is the form we shall use here to grok atomic and molecular structures of gases. As stated in (1a),  K is and arbitrary constant that applies to a particular sample of a gas and nothing else. There is nothing fundamental about it at all. It just sums up the gas laws for sufficiently dilute gases.

First, we shall show that K does not depend on the type of gas selected; but to eliminate that possibility in this discussion, suppose we take a number of samples of the same type of gas  at the same temperature and pressure but differing only in volume. What we see in (1a) by looking at it carefully is that K scales in direct proportion to the initial volumetric measure, and nothing else, except perhaps the type of gas selected,

K=(P0/T0) V,                              (1b)

since the initial pressure and Temperature were the same for all samples.

Now, let us think about it deeply. If there is some smallest unit of a gas, a molecule, then statistically, there is a smallest volume of a gas sample (V0) that can be selected at a particular temperature and pressure. All other samples of that gas must be integer multiples  of that volume. In that case we can writ (1b) as

K=n (P/T) V0,                              (1c)

where n is an integer. Or,

K=n k0,                                      (1d)

where k0 is the constant of proportionality of a gas consisting of only one molecule, which of course, is true only in some statistical sense.

Using (1d) we can rewrite (1a) as,

P V = n k0T.                                                                     (2)

In the form (2) the ideal gas laws are written in a much more fundamental form that (1a) because n relates to the number of molecules in an initial volume of a gas sample.  Unfortunately, at this stage we are unable to specify the number of molecules within any volumetric measure of gas; and so, (2) is just equivalent logically to (1a). However, in the form (2), the ideal gas law suggests an approach to exploring the atomic and molecular structure of gases.

We can point out something else from (2) by using a dimensional analysis. Let us enquire into the units of the term on the left hand side of (2)

[P V]=[F]/[A] * [L]3




[P V]=[F]* [L]

[P V]=[Energy].

The product of pressure times the volume of the gas contained is equivalent to units of energy. In that case, pressure represents an energy density. Further, (2) suggests that  k0T being the energy of the gas divided by the total number of molecules is just the average energy per molecule,

k0T = PV/n   .                                                                               (3)

This suggests that temperature is simply a measure of the average energy of each molecule of a gas.


The Remaining Questions

So far, we have gotten a lot of milage out of a bit of simple algebra and the empirical gas laws for dilute gases.

Is (2) all there is about the ideal gas law. If so, what remains to be explained about the constituents of a gas? For example, does helium have a different constant per molecular volume than hydrogen and oxygen? Do we have to specify various values of k for various kinds of gases? One answer that pops out here, almost automatically, is that since k0T is a measure of the average energy of each molecule of a gas, the mass of the molecule is already included implicitly in this term, and therefore must be a universal term. This means that k0 is simply a scaling term between temperature and the energy that it represents. But, we shall show that this is true in a future blog, where the mechanism of pressure is analysed in terms of molecular dynamics, viz. kinetic theory of gases.





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