620 MOLECULE exerting an attractive force -4 and a repulsive force --7, we might [L [1? define the molecule to be a sphere of radius a, such that ^j = "^7. In like manner, regarding a molecule as a centre of force, repelling according to the law of the inverse fifth power of the distance, we might define the magnitude of a molecule as a sphere of radius equal to the least distance to which two molecules, whose relative velocity is equal to the mean velocity of the centres of force, approach each other in a direct encounter. If on any hypothesis concerning the nature of a molecule, or the law of force which acts during encounters, we can calculate the co efficient of viscosity or diffusion analytically, a comparison of the analytical result with results obtained by experiment may afford the means of determining the absolute numerical value of the con stants used in the analysis. For example, if we consider the mole cules as elastic spheres, and if we consider for a moment Meyer s results as correct, or approximately correct, the coefficient of Nm viscosity for any single gas can be put m the form -y- ul u> where N is the number of molecules in unit_pf volume, m the mass of a molecule. Now, for every value of wZ w , the mean free path of a molecule with that velocity is equal to /h ]y^2, where s is twice the radius of a molecule multiplied by a numerical factor which can be determined to any required degree of accuracy. ^Also at given temperature and pressure the numerical value of Jh is known. It follows that we can calculate the numerical value of the coefficient of viscosity by analytical methods in terms of NTTS* to any required degree of accuracy. Let it be w~^ If by experi ments on viscosity we can determine the numerical value of the same coefficient in the form 0-,, when C-, is a mere numerical quan- C C tity, we have immediately the equation C = *r^5 , or Nirs 2 = -~ . This gives in absolute numerical measure the value of NITS-, or four times the sum of the great circle areas of all the molecules in unit of volume, supposing them to be spherical. If we attempt to use the coefficient of diffusion instead of viscosity in this method, we are met by the difficulty that the analytical result contains now two unknown quantities instead of one namely, the radii of the respective molecules of the two gases in question. If this difficulty be got over by a comparison of results obtained in different ex periments, the greater certainty attending the observations on dif fusion might perhaps compensate for the additional mathematical difficulty, and render diffusion at least equally trustworthy with viscosity as a method for estimating molecular dimensions. Again, on the hypothesis of repulsion between molecules according to the law of the inverse fifth power of the distance, we can calculate analytically the rate of diffusion between two reservoirs connected by a tube as above described, the result containing only one unknown constant, viz., /*, the constant of absolute force. Com paring the analytical result with the results of experiments on diffusion through such a tube as above described, if we find them capable of being harmonized by attributing any numerical value to /j., we should have good reason for concluding that the law of force assumed is to a certain extent at least the true law, and that the particular value of fi is that which harmonizes the analytical with the experimental results. And the determination of p, the absolute force, corresponds to, or indeed is, the determination of the size of the molecule. Until all the mathematical hypotheses have been fully developed, no very great reliance can be placed on the results of such com parisons, even assuming that the experimental results themselves are to be depended upon. However valuable the experiments may be for other purposes, they are not valuable for the purpose of determining molecular dimensions until our mathematical analysis is sufficiently advanced to enable us to interpret the experiment. At present it is perhaps impossible to deduce from the experiments any other result bearing on this question than that the coefficients of diffusion and viscosity increase with increasing temperature, and probably contain an important term proportional to the square root of the absolute temperature. If, indeed, it can be shown that that is the only term, and if it can be also shown that the density of one of two diffusing gases in a tube through which steady diffusion is going on tends to vary in geometrical progression, then the analysis will lead us to the conclusion that molecules of gases behave in their physical relations to each other as if they were elastic spheres. The following method has also been suggested for estimating the magnitude of molecules of mercury. Mercury is regarded by most chemists as monatomic. Let us assume that its molecules are con ducting spheres ; on that assumption we may calculate the specific inductive capacity of mercury vapour on Faraday s hypothesis to be 1 4- 2X yiT > wnerc ^ ls the ratio which the aggregate volume of all the spherical molecules in unit volume bears to unit volume. If now K, the specific inductive capacity of mercury vapour, can be deter- 1 + 2X mined experimentally, the equation K yi^ an " or ds a ground for estimating the value of X, that is, the aggregate volume of the molecules. Another method, originally proposed by Van der "Waals, is founded on the small deviations from Boyle s law observed in all gases. Suppose a vessel of volume V containing a number & of elastic spheres, each of mass m, moving with a certain average kinetic energy. Let p be the pressure. Let a second class of elastic spheres, in number N 2 , each of the same mass m as the former class and having the same average kinetic energy, be introduced into the vessel. If the second class of spheres could freely penetrate the first, and vice versa, so that there should be no restrictions on a sphere of the first class and a sphere of the second being in the same place at the same time, then the pressure on the walls of the vessel would be increased in the exact proportion n 2 . Boyle s law would be exactly fulfilled. But if the spheres cannot pene trate each other, the volume occupied by the second class of spheres 4 is not V, but V- y N^-in 3 , if r be the radius of a sphere of the first class. Consequently, the pressure due to the second class of spheres is rather greater than it should be, and there is a small deviation from Boyle s law. Van der Waals treats the pressure as proportional to the number of encounters, and therefore inversely proportional to the mean free path, which is evidently diminished by any increase in the magnitude of the spheres, and diminished more than in proportion by any increase in the number. (H. W. W. S. H. B.) CHEMICAL ASPECT. The word Molecule is used by chemists to express the unit of a pure substance, that quantity of it which its formula ought to represent. What this quantity is, in any particular case, must be ascertained by studying the chemical actions by which the substance is produced and the chemical changes which it undergoes. We may give one or two illustrations to show how this can be done, as well as to indicate the limits within which these methods can be applied. The formula usually assigned to acetic acid is C 2 H 4 O 2 . This agrees with almost all the chemical actions in which it takes part. Thus, one quarter of the hydrogen is replaceable by other metals, as in C 2 H 3 KO 2 , &c. ; and one, two, or three quarters of the hydrogen can be replaced by chlorine. There must, therefore, be four (or a multiple of four) atoms of hydrogen in the molecule. Similarly, half of the oxygen can be replaced by sulphur, and one-half of the oxygen along with one-quarter of the hydrogen can be replaced by chlorine. There must, therefore, be two (or a multiple of two) atoms of oxygen in the molecule. Again, the formation of marsh gas and carbonate of soda, when acetate of soda is heated with caustic soda, and the formation of aceto-nitrile from cyanide of potassium and iodide of methyl, show that the carbon in acetic acid is divisible by two, or that the molecule contains two (or a multiple of two) atoms of carbon. C 2 H 4 O 2 is the simplest formula which fulfils these conditions, but the existence of an acid acetate of potash and an acid acetate of ammonia, the formulae of which are usually written C 2 H 3 KO 2 , C 2 H 4 O 2 and C 2 H 3 (NH 4 )O 2 , C 2 H 4 O 2 , as if these were com pounds derived from two molecules of acetic acid, might lead us to C 4 H 8 O 4 , as this shows that the hydrogen is divisible by eight. In the same way, we can easily satisfy ourselves that C 6 H 10 O 5 , or some multiple of it, is the formula of starch ; that C S H-NO, or some multiple of it, is the formula of indigo blue, and so on. But it is not easy to determine by purely chemical methods whether these formula? themselves, or multiples of them, really represent the molecule. A simple formula may suffice for a great many of the reactions of a substance, and may enable us to represent a great many of its derivatives, and yet reactions and derivatives may be discovered which require a multiple of that simple formula. This has already been
indicated in reference to acetic acid, and a very striking