The Molecular Theory of Physics.
From a consideration of the chemical properties of the elements we shall now turn to an examination of the physical characteristics, although in a certain sense chemistry itself is but one special phase of physics.
If matter is really constructed of independently existing particles—atoms and molecules—the interplay of the individual parts must determine not only the chemical activities, but also the other properties of matter. Since most of these properties are different for different substances, or in other words are “molecular properties,” it is reasonable to suppose that in many cases explanations can be more readily given by considering the molecules as the fundamental parts. It is natural that the first attempts to develop a molecular theory concerned gases, for their physical properties are much simpler than those of liquids or solids. This simplicity is indeed easily understood on the molecular theory. When a liquid by evaporation is transformed into a gas, the same weight of the element has a volume several hundred times greater than before. The molecules, packed together tightly in the liquid, in the gas are separated from each other and can move freely without influencing each other appreciably. When two of them come very close to each other, mutually repulsive forces will arise to prevent collision. Since it must be assumed that in such a “collision” the individual molecules do not change, they can then to a certain extent be considered as elastic bodies, spheres for instance.
From considerations of this nature the kinetic theory of gases developed. According to this a mass of gas consists of an immense number of very small molecules. Each molecule travels with great velocity in a straight line until it meets an obstruction, such as another molecule or the wall of the containing vessel; after such an encounter the molecule travels in a second direction until it collides again, and so on. The pressure of the gas on the wall of the container is the result of the very many collisions which each little piece of wall receives in a short interval of time. The magnitude of the pressure depends upon the number, mass and velocity of the molecules. The velocity will be different for the individual molecules in a gas, even if all the molecules are of the same kind, but at a given temperature an average velocity can be determined and used. If the temperature is increased, this average molecular velocity will be increased, and if at the same time the volume is kept constant, the pressure of the gas on the walls will be increased. If the temperature and the average velocity remain constant while the volume is halved, there will be twice as many molecules per cubic centimetre as before. Therefore, on each square centimetre of the containing wall there will be twice as many collisions, and consequently the pressure will be doubled. Boyle’s Law, that the pressure of a gas at a given temperature is inversely proportional to its volume, is thus an immediate result of the molecular theory.
The molecular theory also throws new light upon the correspondence between heat and mechanical work and upon the law of the conservation of energy, which about the middle of the nineteenth century was enunciated by the Englishman, Joule, the Germans, Mayer and Helmholtz, and the Dane, Colding. A brief discussion of heat and energy will be given here, since some conception of these phenomena is necessary in understanding what follows.
To lift a stone of 5 pounds through a distance of 10 feet demands an expenditure of work amounting to 5 × 10 = 50 foot-pounds; but the stone is now enabled to perform an equally large amount of work in falling back these 10 feet. The stone, by its height above the earth and by the attraction of the earth, now has in its elevated position what is called “potential” energy to the amount of 50 foot-pounds. If the stone as it falls lifts another weight by some such device as a block and tackle, the potential energy lost by the falling stone will be transferred to the lifted one. If the apparatus is frictionless, the falling stone can lift 5 pounds 10 feet or 10 pounds 5 feet, etc., so that all the 50 foot-pounds of potential energy will be stored in the second stone. If instead of being used to lift the second stone, the original stone is allowed to fall freely or to roll down an inclined plane without friction, the velocity will increase as the stone falls, and, as the potential energy is lost, another form of energy, known as energy of motion or kinetic energy, is gained. Conversely, a body when it loses its velocity can do work, such as stretching a spring or setting another body in motion. Let us suppose that the stone is fastened to a cord and is swinging like a pendulum in a vacuum where there is no resistance to its motion. The pendulum will alternately sink and rise again to the same height. As the pendulum sinks, the potential energy will be changed into kinetic energy, but as it rises again the kinetic will be exchanged for potential. Thus there is no loss of energy, but merely a continuous exchange between the two forms.
If a moving body meets resistance, or if its free fall is halted by a fixed body, it might seem as if, at last, the energy were lost. This, however, is not the case, for another transformation occurs. Every one knows that heat is developed by friction, and that heat can produce work, as in a steam-engine. Careful investigations have shown that a given amount of mechanical work will always produce a certain definite amount of heat, that is, 400 foot-pounds of work, if converted into heat, will always produce 1 B.T.U. of heat, which is the amount necessary to raise the temperature of 1 pound of water 1° F. Conversely, when heat is converted into work, 1 B.T.U. of heat “vanishes” every time 400 foot-pounds of work are produced. Heat then is just a special form of energy, and the development of heat by friction or collision is merely a transformation of energy from one form to another.
With the assistance of the molecular theory it becomes possible to interpret as purely mechanical the transformation of mechanical work into heat energy. Let us suppose that a falling body strikes a piston at the top of a gas-filled cylinder, closed at the bottom. If the piston is driven down, the gas will be compressed and therefore heated, for the speed of the molecules will be increased by collisions with the piston in its downward motion. In this example the kinetic energy given to the piston by the exterior falling body is used to increase the kinetic energy of the molecules of the gas. When the molecules contain more than one atom, attention must also be given to the rotations of the atoms in a molecule about each other. A part of any added kinetic energy in the gas will be used to increase the energy of the atomic rotations.
The next step is to assume that, in solids and liquids, heat is purely a molecular motion. Here, too, the development of heat after collision with a moving body should be treated as a transformation of the kinetic energy of an individual, visible body into an inner kinetic energy, divided among the innumerable invisible molecules of the heated solid or liquid. In considering the internal conduct of gases it is unnecessary (at least in the main) to consider any inner forces except the repulsions in the collisions of the molecules. In solids and liquids, however, the attractions of the tightly packed molecules for each other must not be neglected. Indeed the situation is too complicated to be explained by any simple molecular theory. Not all energy transformations can be considered as purely mechanical. For instance, heat can be produced in a body by rays from the sun or from a hot fire, and, conversely, a hot body can lose its heat by radiation. Here, also, we are concerned with transformations of energy; therefore the law for the conservation of energy still holds, i.e. the total amount of energy can neither be increased nor decreased by transformations from one form to another. For the production of 1 B.T.U. of heat a definite amount of radiation energy is required; conversely, the same amount of radiation energy is produced when 1 B.T.U. of heat is transformed into radiation. This change cannot, however, be explained as the result of mechanical interplay between bodies in motion.
The mechanical theory of heat is very useful when we restrict ourselves to the transfer of heat from one body to another, which is in contact with it. When applied to gases the theory leads directly to Avogadro’s Law. If two masses of gas have the same temperature, i.e., if no exchange of heat between them takes place even if they are in contact with each other, then the average value of the kinetic energy of the molecules must be the same in both gases. If one gas is hydrogen and the other oxygen, the lighter hydrogen molecules must have a greater velocity than the heavier oxygen molecules; otherwise they cannot have the same kinetic energy (the kinetic energy of a body is one-half the product of the mass and the square of the velocity). Since the pressure of a gas depends upon the kinetic energy of the molecules and upon their number per cubic centimetre, at the same temperature and pressure equal volumes must contain equal numbers of oxygen and of hydrogen molecules. As Joule showed in 1851, from the mass of a gas per cubic centimetre and from its pressure per square centimetre, the average velocity of the molecules can be calculated. For hydrogen at 0° C. and atmospheric pressure the average velocity is about 5500 feet per second; for oxygen under the same conditions it is something over 1300 feet per second.
All these results of the atomic and molecular theory, however, gave no information about the absolute weight of the individual atoms and molecules, nor about their magnitude nor the number of molecules in a cubic centimetre at a given temperature and pressure. As long as such questions were unsolved there was a suggestion of unreality in the theory. The suspicion was easily aroused that the theory was merely a convenient scheme for picturing a series of observations, and that atoms and molecules were merely creations of the imagination. The theory would seem more plausible if its supporters could say how large and how heavy the atoms and molecules were. The molecular theory of gases showed how to solve these problems which chemistry had been powerless to solve.
Let us assume that the temperature of a mass of gas is 100° C. at a certain altitude, and 0° C. one metre lower, i.e., the molecules have different average velocities in the two places. The difference between the velocities will gradually decrease and disappear on account of molecular collisions. We might expect this “levelling out” process or equilibration to proceed very rapidly because of the great velocity of the molecules, but we must consider the fact that the molecules are not entirely free in their movements. In reality they will travel but very short distances before meeting other molecules, and consequently their directions of motion will change. It is easy to understand that the difference between the velocities of the molecules of the gas will not disappear so quickly when the molecules move in zigzag lines with very short straight stretches. The greater velocity in one part of the gas will then influence the velocity in the other part only through many intermediate steps. Gases are therefore poor conductors of heat. When the molecular velocity of a gas and its conductivity of heat are known, the average length of the small straight pieces of the zigzag lines can be calculated—in other words, the length of the mean free path. This length is very short; for oxygen at standard temperature and pressure it is about one ten-thousandth of a millimetre, or 0·1 μ, where μ is 0·001 millimetre or one micron.
In addition to the velocity of the molecules, the length of the mean free path depends upon the average distance between the centres of two neighbouring molecules (in other words, upon the number of molecules per cubic centimetre) and upon their size. There is difficulty in defining the size of molecules because, as a rule, each contains at least two atoms; but it is helpful to consider the molecules, temporarily, as elastic spheres. Even with this assumption we cannot yet determine their dimensions from the mean free path, since there are two unknowns, the dimensions of the molecules and their number per cubic centimetre. Upon these two quantities depends, however, also the volume which will contain this number of molecules, if they are packed closely together. If we assume that we meet such a packing when the substance is condensed in liquid form, this volume can be calculated from a knowledge of the ratio between the volume in liquid form and the volume of the same mass in gaseous form (at 0° C. and atmospheric pressure). Then from this result and the length of the mean free path the two unknowns can be determined. Although the assumptions are imperfect, they serve to give an idea about the dimensions of the molecules; the results found in this way are of the same order of magnitude as those derived later by more perfect methods of an electrical nature.
The radius of a molecule, considered as a sphere, is of the order of magnitude 0·1 μμ, where μμ means 10⁻⁶ millimetre or 0·001 micron. Even if a molecule is by no means a rigid sphere, the value given shows that the molecule is almost unbelievably small, or, in other words, that it can produce appreciable attraction and repulsion in only a very small region in space.
The number of molecules in a cubic centimetre of gas at 0° C. and atmospheric pressure has been calculated with fair accuracy as approximately 27 × 10¹⁸. From this number and from the weight of a cubic centimetre of a given gas the weight of one molecule can be found. One hydrogen molecule weighs about 1·65 × 10⁻²⁴ grams, and one gram of hydrogen contains about 6 × 10²³ atoms and 3 × 10²³ molecules. The weight of the atoms of the other elements can be found by multiplying the weight of the hydrogen atom by the relative atomic weight of the element in question—16 for oxygen, 14 for nitrogen, etc. If the pressure on the gas is reduced as much as possible (to about one ten-millionth of an atmosphere) there will still be 3 × 10¹² molecules in a cubic centimetre, and the average distance between molecules will be about one micron. The mean free path between two collisions will be considerable, about two metres, for instance, in the case of hydrogen.
The values found for the number, weight and dimensions of molecules are either so very large or so extremely small that many people, instead of having more faith in the atomic and molecular theory, perhaps may be more than ever inclined to suppose the atoms and molecules to be mere creations of the imagination. In fact, it is only two or three decades ago that some physicists and chemists—led by the celebrated German scientist, Wilhelm Ostwald—denied the existence of atoms and molecules, and even went so far as to try to remove the atomic theory from science. When these sceptics, in defence of their views, said that the atoms and molecules were, and for ever would be, completely inaccessible to observation, it had to be admitted at that time that they were seemingly sure of their argument, in this one objection at any rate.
A series of remarkable discoveries at the close of the nineteenth century so increased our knowledge of the atoms and improved the methods of studying them that all doubts about their existence had to be silenced. However incredible it may sound, we are now in a position to examine many of the activities of a single atom, and even to count atoms, one by one, and to photograph the path of an individual atom. All these discoveries depend upon the behaviour of atoms as electrically charged, moving under the influence of electrical forces. This subject will be developed in another section after a discussion of some phenomena of light, an understanding of which is necessary for the appreciation of the theory of atomic structure proposed by Niels Bohr.
In the molecular theory of gases, where we have to do with neutral molecules, much progress has in the last years been made by the Dane, Martin Knudsen, in his experiments at a very low pressure, when the molecules can travel relatively far without colliding with other molecules. While his researches give information on many interesting and important details, his work gives at the same time evidence of a very direct nature concerning the existence of atoms and molecules.