Page:EB1911 - Volume 18.djvu/688

systems for which the initial values of p₁, q₁, . . . p_n, q_n lie within a range such that

p₁ is between p₁ and p₁ +dp₁

q₁ is between q₁ and q₁ +dq₁ and so on.

Let the product dp₁ dq₁ . . . dp_n, dq_n be spoken of as the “extension” of this range of values.

After a time dt the value of p₁ will have increased to p₁+p₁dt, where p₁ is given by equations (1), and there will be similar changes in q₁, p₂, q₂, . . . q_n. Thus after a time dt the values of the co-ordinates and momenta of the small group of systems under consideration will lie within a range such that

p₁ is between p₁ + ṗ₁dt and p₁+dp₁ +(ṗ₁+∂ṗ₁/∂p₁dp₁)dt

q₁ is between q₁ + q̇₁dt and q₁+dq₁ +(q̇₁+∂q̇₁/∂q₁dq₁)dt

and so on. Thus the extension of the range after the interval dt is

dp₁ (1+∂ṗ₁/∂p₁dt)dq₁(q̇₁+∂q̇₁/∂q₁dt) . . .

or, expanding as far as first powers of dt,

dp₁dq₁ . . . dp_ndq_n ${\Big \{}$1+ Σ_1ⁿ${\Big (}$∂ṗ₁/∂p₁ + ∂q̇₁/∂q₁${\Big )}$dt${\Big \}}$.

From equations (1), we find that

∂ṗ₁/∂p₁ + ∂q̇₁/∂q₁＝0.

so that the extension of the new range is seen to be dp₁dq₁ . . . dp_ndq_n, and therefore equal to the initial extension. Since the values of the co-ordinates and momenta at any instant during the motion may be treated as “initial” values, it is clear that the “extension” of the range must remain constant throughout the whole motion.

This result at once disposes of the possibility of all the systems acquiring any common characteristic in the course of their motion through a tendency for their co-ordinates or momenta to concentrate about any particular set, or series of sets, of values. But the result goes further than this. Let us imagine that the systems had the initial values of their co-ordinates and momenta so arranged that the number of systems for which the co-ordinates and momenta were within a given range was proportional simply to the extension of the range. Then the result proves that the values of the coordinates and momenta remain distributed in this way throughout the whole motion of the systems. Thus, if there is any characteristic which is common to all the systems after the motion has been in progress for any interval of time, this same characteristic must equally have been common to all the systems initially. It must, in fact, be a characteristic of all possible states of the systems.

It is accordingly clear that there can be no property common to all systems, but it can be shown that when the system contains a gas (or any other aggregation of similar molecules) as part of it there are properties which are common to all possible states, except for a number which form an insignificant fraction of the whole. These properties are found to account for the physical properties of gases.

Let the whole energy E of the system be supposed equal to E₁+E₂, where E₂ is of the form

E2＝1/2$\sum _{s}$(mu2+mv2+mw2+α1θ12+ . . . +αnθn2)+1/2$\sum _{s}'$(m′u′+m′v′2+mw′2+β1φ12+β2φ22+ . . . +βn′φn′2)

(2)

where θ₁,θ₂, . . . θ_n and similarly φ₁,φ₂, . . . φ_n′ are any momenta or functions of the co-ordinates and momenta or co-ordinates alone which are subject only to the condition that they do not enter into the coefficients α₁, α₂, &c.

In this expression the first line may be supposed to represent the energy (or part of the energy) of s similar molecules of a kind which we shall call the first kind, the terms 1/2(mu²+mv²+mw²) being the kinetic energy of translation, and the remaining terms arising from energy of rotation or of internal motion, or from the energy, kinetic and potential, of small vibrations. The second line in E₂ will represent the energy (or part of the energy) of s′ similar molecules of the second kind, and so on. It is not at present necessary to suppose that the molecules are those of substances in the gaseous state. Considering only those states of the system which have a given Value of E₂, it can be proved, as a theorem in pure mathematics^[1] that when s, s′, . are very large, then, for all states except an infinitesimal fraction of the whole number, the values of u, 1/, w lie within ranges such that

(i) the values of u (and similarly of v, w) are distributed among the s molecules of the first kind according to the law of trial and error; and similarly of course for the molecules of other kinds:

＝$\sum _{s}$1/2 m′u′2/s′＝$\sum _{s}$1/2 β1φ12/s′＝. . .

(3)

A state of the system in which these two properties are true will be called a “normal state”; other states will be spoken of as “abnormal.” Let all possible states of the system be divided into small ranges of equal extension, and of these let a number P correspond to normal, and a numberThe Normal State. p to abnormal, states. What is proved is that, as s, s′, . . . become very great, the ratio P/p becomes infinite. Considering only systems starting in the p abnormal ranges, it is clear, from the fact that the extensions of the ranges do not change with the motion, that after a sufficient time most of these systems must have passed into the P normal ranges. Speaking loosely, we may say that there is a probability P/(P+p), amounting to certainty in the limit, that one of these systems, selected at random, will be in the normal state after a sufficient time has' elapsed. Again, considering the systems which start from the P normal ranges, We see that there is a probability p/(P+p) which vanishes in the limit, that a system selected at random from these will be in an abnormal state after a sufficient time. Thus, subject to a probability of error which is infinitesimal in the limit, we may state as general laws that—

A system starting from an abnormal state tends to assume the normal state; while

A system starting from the normal state will remain in the normal state.

It will now be found that the various properties of gases follow from the supposition that the gas is in the normal state.

If each of the fractions (3) is put equal to 1/4h, it is readily found, from the first property of the normal state, that, of the s molecules of the first kind, a numberLaw of Distribution of Velocities.

$s{\sqrt {(h^{3}m^{3}{\mbox{/}}\pi ^{3})}}e^{-hm(u^{2}+v^{2}+w^{2})}dudvdw$

(4)

have velocities of which the components lie between u and u+du, v and v+dv, w and w+dw, while the corresponding number of molecules of the second kind is, similarly,

$s'{\sqrt {(h^{3}m'^{3}{\mbox{/}}\pi ^{3})}}e^{-hm'(u^{2}+v^{2}+w^{2})}dudvdw$

(5)

If c is the resultant velocity of a molecule, so that c²=u²+v²+w², it is readily found from formula (4) that the number of molecules of the first kind of which the resultant velocity lies between c and c+dc is

$4\pi s{\sqrt {(h^{3}m^{3}{\mbox{/}}\pi ^{3})}}e^{-hmc^{2})}c^{2}dc$

(6)

These formulae express the “law of distribution of velocities” in the normal state: the law is often called Maxwell’s Law of Distribution.

If ${\tfrac {1}{2}}{\overline {mu^{2}}}$ denote the mean value of ${\tfrac {1}{2}}mu^{2}$ averaged over the s molecules of the first kind, equations (3) may be written in the form

${\tfrac {1}{2}}{\overline {mu^{2}}}={\tfrac {1}{2}}{\overline {mv^{2}}}={\tfrac {1}{2}}{\overline {mw^{2}}}={\tfrac {1}{2}}{\overline {\alpha _{1}\theta _{1}^{2}}}\cdots =1/4h$

(7)

Equipartition
of Energy. showing that the mean energy represented by each term in E₂ (formula 2) is the same. These equations express the “law equipartition of energy,” commonly spoken of as the Maxwell-Boltzmann Law.

The law of equipartition shows that the various mean energies of different kinds are all equal, each being measured by the quantity 1/4h. We have already seen that the mean energy increases with the temperature: it will now be supposed that the mean energy is exactly proportional to theTemperature. temperature. The complete justification for this supposition will appear later: a partial justification is obtained as soon as it is seen how many physical laws can be explained by it. We accordingly put 1/2h=RT, where T denotes the temperature on the absolute scale, and then have equations (7) in the form

${\overline {mu^{2}}}={\overline {mv^{2}}}{\mbox{RT.}}$

(8)

When a system is composed of a mixture of different kinds of molecules, the fact that h is the same for each constituent [cf. formulae. (5) and (6)] shows that in the normal state the different substances are all at the same temperature. For instance, if the system is composed of a gas and a solid boundary, some of the terms in expression (2) may be supposed to represent the kinetic energy of the molecules of the boundary, so that equations (7) show that in the normal state the gas has the same temperature as the boundary. The process of equalization of temperature is now seen to be a special form of the process of motion towards the normal state: the general laws which have been stated above in connexion with the normal state are seen to include as special cases the following laws:—

Matter originally at non-uniform temperature tends to assume a uniform temperature; while

Matter at uniform temperature will remain at uniform temperature.

It will at once be apparent that the kinetic theory of matter enables us to place the second law of thermodynamics upon a purely dynamical basis. So far it has not been necessary to suppose the matter to be in the gaseous state. We now pass to the consideration of laws and properties which are peculiar to the gaseous state.

A simple approximate calculation of the pressure exerted by a gas on its containing vessel can be made by supposing that the molecules are so small in comparison with their distances apart that they may be treated as of infinitesimal size. Let a mixture of gases contain per unit volume v moleculesPressure of
a Gas. of the first kind, v′ of the second kind, and so on. Let

us fix our attention on a small area dS of the boundary of the

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1. See Jeans, Dynamical Theory of Gases (1904), ch. v.