Parsonage / Robinson / Irving The Gaseous State

The Commonwealth and International Library: Chemistry Division
1. Auflage 2013
ISBN: 978-1-4831-8103-5
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 184 Seiten

The Gaseous State provides a comprehensive discussion on the various areas of concerns in gases. The main concern of the title is the interpretation of the properties of bulk gases in terms of the characteristics of the constituent molecules. The text first details the perfect gas equation, and then proceeds to tackling various gaseous properties. The coverage of the selection includes gas imperfection, collisions, viscosity, thermal conductivity, and diffusion, and energy transfer. The title also covers the Brownian movement and the determination of Avogadro's number. The book will be most useful to undergraduate students of chemistry.

Autoren/Hrsg.

Parsonage, N. G.

Robinson, Robert

Irving, H. M. N. H.

Staveley, L. A. K.

Leseproben

CHAPTER 1

THE PERFECT GAS EQUATION

Publisher Summary

This chapter discusses properties of the perfect gas equation. For calculation of the pressure exerted by the molecules on the walls of box, it is assumed that the molecules are infinitesimal, there are no forces between them, collisions with the walls are elastic, and the walls are perfectly smooth. It follows from the last two assumptions that reflections of molecules from the wall are specular, that is, the angle of incidence is equal to the angle of reflection. By considering the effect of collisions on a system in which initially all the speeds are equal, it is seen that the molecules of a gas do not all have the same speed. Only in the rare cases where the velocities of the molecules in any given pair are equally inclined to the line of centers at impact would the speeds of the two molecules remain equal. In a mixture of gases, the total bombardment pressure exerted on the walls is equal to the sum of the pressures exerted by each of the components. The pressure caused by the molecules of any one component is equal to the pressure that the same molecules would exert if they alone occupied the entire volume.

The idea that the pressure exerted by a gas on its container is due to the bombardment of the walls by the molecules of the gas is an old one. Bernoulli (1738)(1) had this idea, although since this was before Dalton’s atomic theory he speaks of particles or corpuscles rather than of molecules. He did not, apparently, think that the particles in a body of gas might have unequal speeds, and he did not, therefore, find it necessary to perform an averaging over all the speeds of the particles. He deduced correctly that the pressure should be proportional to the square of the speed, since both the frequency and the “intensity” of the impacts were proportional to the speed. He also arrived at the correct dependence of the pressure on the volume (Boyle’s or Mariotte’s law): = constant. The reasoning which led him to this result was, however, in part incorrect. The Boyle’s law equation, as will be seen later, is only valid for infinitesimal particles. In his treatment, Bernoulli initially considered the particles to have finite size. He then derived—incorrectly—the equation corresponding to this model. Nevertheless, if the size of the particles was put equal to zero in Bernoulli’s equation the Boyle’s law result was obtained.

The first published paper in which the Perfect Gas Equation was derived correctly was one by Clausius in 1857,(2) in which the ideas were also clearly set out. Apart from the results presented in the paper, the clarification of the fundamental principles of the kinetic theory of gases acted as a spur to others, and notably to Maxwell, who in the following 5 years was able to extend the theory to the consideration of collisions between molecules, the distribution of molecular speeds, and the viscosity of gases.

DERIVATION OF THE EQUATION

In this calculation of the pressure exerted by the molecules on the walls of a box containing them, it is assumed that the molecules are infinitesimal, that there are no forces between them, that collisions with the walls are elastic, and that the walls are perfectly smooth. It follows from the last two assumptions that reflections of molecules from the wall are specular, that is, the angle of incidence is equal to the angle of reflection. It is not, however, assumed that the molecules necessarily have the same speed. Indeed, that the molecules of a gas do not all have the same speed is readily seen by considering the effect of collisions on a system in which initially all the speeds are equal. Only in the rare cases where the velocities of the molecules in any given pair are equally inclined to the line of centres at impact would the speeds of the two molecules remain equal.

Let the gas be confined to a cubic container of side (Fig. 1.1).

FIG. 1.1

The total force on a face perpendicular to the -axis is, by Newton’s third law, equal and opposite to the force exerted on the gas molecules by the wall. Further, by Newton’s second law, this latter force is equal to the rate of change of momentum of the gas molecules at the wall:

force exerted

force exerted

rate of change of the

on the = —

on the

= — momentum of the

wall

molecules

molecules at the wall.

Hence the problem becomes that of calculating the rate of change of the momentum of the molecules at the walls. Consider a single molecule approaching the wall and having velocity components in the -, - and -directions of , and , respectively. Since the walls are smooth, the velocity components and are unchanged on collision with the wall under consideration, which is perpendicular to the -axis. Immediately before impact with the wall the velocity component in the -direction is and, after impact, since the collision is assumed to be elastic, it is - (Fig. 1.2). There is thus a change of momentum of 2 where is the mass of the molecule. To determine the rate of change of momentum due to such molecules it is necessary to know the frequency of such collisions with the wall. Clearly, all those molecules having the velocity considered which are within a perpendicular distance of the wall will collide with it within one second. This will be a fraction / of the total number of molecules in the box having this velocity. If is the number of molecules in the box, and ()d is the fraction of all these molecules which have -components of the velocity in the range to + d, then the number of molecules having this velocity component in this range is () d (see Appendix 2). Hence the total change of momentum per second due to molecules within the given velocity range is

FIG. 1.2 Specular reflection.

(1.1)

The total change of momentum per second due to molecules of all velocities is then

(1.2)

The integral extends only over positive values of because only these lead to collisions with the wall. By symmetry, () = (–), since the choice of the positive direction for the -axis is arbitrary. Hence the integrand is symmetrical about = 0, and

The final integral is, of course, the mean value of (see Appendix 2), and subsequently the notation will be used for this.

The expression for the total change of momentum per second at the wall then becomes

Hence the pressure on the wall is given by

(1.3)

or, since

But, by symmetry,

Therefore

(1.4)

The quantity <2> is known as the mean-square speed of the molecules.

It would be expected that <2>, and consequently also the total molecular translational kinetic energy , would be greater the higher temperature. Evidence that <2> is not also a function of pressure came from the experiments of Joule (1845). He allowed air to expand into a vacuum adiabatically and found that there was no temperature change. The argument which was then used is applicable, strictly, only to monatomic gases, which were still unknown at that time: it is that if the gas is perfect then the expansion does not involve any work against the forces between the molecules, and, since no external work is done either, the total translational kinetic energy of the molecules must remain constant. (If the molecules can have rotational or vibrational energy then this conclusion cannot be drawn with certainty.) If this translational kinetic energy were a function both of and , a change in temperature would be expected on changing the pressure such that

(1.5)

Since no change in temperature was observed, it was concluded that the kinetic energy was not a function of pressure. In fact, it is now known that had Joule’s experiments been more sensitive he would have detected a small temperature change resulting from the work done in the expansion against the attractive forces between the molecules of the imperfect gas which he used. It was fortunate for the advancement of the kinetic theory that Joule’s experiments produced the result which is correct only for a perfect gas.

With the negative result of Joule’s experiment, which may be written

eqn. (1.4) becomes

(1.6)

where the function ? may differ from one gas to another. (1.6) is a statement of Boyle’s experimental law, that for a given amount of gas at a given temperature =...