Since the process is reversible, (3) gives the work which may be done during the mixing of a volume vo of two gases under the partial pressures Q, and Q, with a large quantity of the same gases under an equal total pressure, but with partial pressures P, and P. The quantity denoted by W can never be negative. To verify this from (3), write it in the form y = a'log + y log = log { (~)*(4)′′}, (4) Q1 + Q2' Q1 + Q2 so that x+y=x' + y' =]. Now (Todhunter's 'Algebra,' p. 392) if a, b, c,... be any positive quantities, Suppose that a, b, c, ... consist of p equal quantities a and q equal quantities B; then >anß. and therefore, since x+y=1, that W is always positive, unless a=B, in which case the composition of the two mixtures is the same, and W vanishes. 1 We have now to show how the formula for the mixture of two pure gases may be derived from (3). Let v, be the volume of the first gas and v2 of the second, at the constant pressure P1+ P2. The value of the interdiffusion of v, and v must be the same as that of their diffusion into a large quantity of a mixture whose composition is identical with that of the mixture of v, and v. For, on this supposition, the separation of the two mixtures spoken of would have no mechanical value. Now by (3) the value of W for the diffusion of a quantity vo of pure gas into a large quantity of a mixture whose partial pressures are P1 and P is (since Q, and Q, log Q, vanish) and hence the value of W for the interdiffusion of the quantities v, and v2 is W=v, (P1+ P2) log This equation agrees with the rule enunciated at the beginning of this paper, inasmuch as (P,+P)v1 log 1⁄2 +v1⁄2 represents the work gained in the expansion of the first gas from volume Vi to volume v12, and (P,+P2)v, log01 +2, sponding quantity for the second gas. represents the corre The significance of equation (5) may perhaps be more fully brought out by the following investigation of it. Whatever the relative proportions of the two gases in the reservoir may be, it will always be possible by going high enough to obtain a small quantity of the lighter gas in any required degree of purity. The removal of this at the top of the tube, its condensation to the pressure in the reservoir, the fall to the level of the reservoir, and the introduction into the reservoir would, on the whole, réquire no work to be done if this kind of gas had alone been present. The only effect of the heavier gas is to render necessary a greater condensation in the third operation; and thus W is the work that is required to condense the gas from the partial pressure P to the total pressure in the reservoir P1+P2, whence equation (5) follows at once. If it is desired to isolate a small quantity of the heavier gas, the tube must be taken downwards. It is to be observed that the work required to force a given quantity of gas into a large reservoir containing gas at the same pressure is independent of this pressure, since, according to Boyle's law, v is diminished in the same proportion that p is increased. The principle of dissipation may be employed to prove that the pressure in a vertical column of mixed gases is greater 318 On the Work that may be gained during the Mixing of Gases, when there is free diffusion than when the gases are uniformly mixed; for if the gases be allowed to rise from the reservoir tolerably quickly (or if a series of movable pistons be interpolated), the composition in the tube will be the same as in the reservoir. If free diffusion be now allowed, there must be dissipation. The original state of things will be restored if the mixture be slowly forced back into the reservoir; and accordingly the work consumed in condensation must be greater than that gained in the expansion. In fact it may be proved algebraically by a process somewhat similar to that applied to equation (3), that the pressure of the gases under free diffusion P, where is greater than the pressure of a uniform mixture p', where (7) (8) It is, however, possible to imagine other distributions which shall give a pressure greater than (7). The mechanical equilibrium gives one equation involving the two quantities p、 and Po and the subsidiary conditions are that p1=P1, P2=P,, when z=0. Hence we may take as the most general solution, where X is an arbitrary function of x. Thus the total pressure P1+P2=P1e-12 + Pâ€ ̃μg≈ For free diffusion X=0; but it could always be taken so as to make the integral either positive or negative, as might be desired. The work required to decompose a mixture of gases is in general small, and could scarcely be of much importance from an industrial point of view. When, however, the proportion of one ingredient is very insignificant, more work is required. Thus the separation of the carbonic anhydride from the atmosphere would require, relatively to the quantity obtained, a much larger expenditure of work than the separation of the oxygen. This consideration shows that extreme purity in any gas will always be attained and maintained with difficulty. Even when the necessary work is small, as in the separation of oxygen from the atmosphere, it is well to bear in mind that some work is absolutely essential. The reversible absorption of the oxygen of air may be effected by a substance like baryta ; but we must not expect to recover the pure oxygen at the same temperature and under a pressure equal to the total pressure at which it was absorbed. Either the temperature must be raised, or the gas must be exhausted at a pressure less than that under which it existed in the mixture during the absorption. It is just possible that this point might be found to be of practical importance in the solution of the problem of extracting oxygen from the air. XXXV. Notices respecting New Books. An Elementary Treatise on the Integral Calculus, containing Applications to Plane Curves and Surfaces, with numerous Examples. By BENJAMIN WILLIAMSON, A.M., Fellow and Tutor, Trinity College, Dublin. London: Longmans, Green, and Co. 1875 (crown 8vo, pp. 267). WE E have already noticed Mr. Williamson's 'Elementary Treatise on the Differential Calculus' (vol. xliii. p. 307), and have now to perform the same office for his companion volume on the Integral Calculus,—a work characterized by the same excellences as those which marked the previous volume. It is written with clearness and accuracy, and is illustrated with an abundance of examples. Its contents may be briefly described as consisting of three parts. In the first are given the ordinary methods of integration, viz. by reduction to known forms, of rational fractions, by successive reduction, and by rationalization. In this part of the work, which occupies the first five chapters, integration is treated simply as the inverse of the process of differentiation. In the next part, the sixth chapter, a sufficient account is given of Definite Integrals; and here the student is introduced to the notion of integration as a process of summation. In the latter part of this chapter some account is given of the Eulerian Integrals, particularly of the second; and at the end there is a table of values of the gamma function (log (p)+10) from p=1.000 to p=1.999 to six places of decimals; but the method of constructing this table is omitted as "too complicated for insertion in an elementary treatise." The third part of the book, comprised in chapters seventh, eighth, and ninth, explains the application of the Integral Calculus to finding areas and lengths of curves and volumes of solids. In this part will be found many useful and interesting properties of curves; e. g. in addition to those which we might expect to find almost as a matter of course, we may mention Lambert's theorem on the area of an elliptic sector, areas of roulettes, the theory of Amsle meter, Landen's theorem on the length of a hyperbolic ner's theorem on the rectification of roulettes, and some o It will be seen, from this brief account of the contents work is of a strictly elementary character. Such subjec expansion of functions in trigonometrical series, elliptic f and the Calculus of Variations are simply omitted; while t formation of the independent variable and double integr but briefly noticed. Within the limits which the author has to himself, however, the treatment is very full and satisfact the work is well adapted to the wants of those for whom i ten, viz. students in the Universities, very few of whom suppose) will find it necessary to enter into the subject be contents of the present volume. XXXVI. Proceedings of Learned Societies. ROYAL SOCIETY. [Continued from p. 237.] June 11, 1874.-Joseph Dalton Hooker, C.B., President, Chair. HE following communications were read :— THE "Spectroscopic Notes.-No. III. On the Molecular St of Vapours in connexion with their Densities." By J. Lockyer, F.R.S. 1. I have recently attempted to bring the spectroscope upon the question whether vapours of elements below the temperatures are truly homogeneous, and whether the of different chemical elements, at any one temperature, are similar molecular condition. In the present note, I beg before the Royal Society the preliminary results of my resea 2. We start with the following facts : I. All elements driven into vapour by the induced give line-spectra. II. Most elements driven into vapour by the voltaic a us the same. III. Many metalloids when greatly heated, some at o temperatures, give us channelled-space spectra. IV. Elements in the solid state give us continuous spe 3. If we grant that the spectra represent to us the vibr of different molecular aggregations (this question is discus Note II.), spectroscopic observations should furnish us with of some importance to the inquiry. 4. To take the lowest ground. If, in the absence of all ledge on the subject, it could be shown that all vapours at all of temperature had spectra absolutely similar in character, t would be more likely that all vapours were truly homogeneou |