The compressibilities and expansion coefficients of …rspa.royalsocietypublishing.org/content/royprsa/173/953/201.full.pdf · The compressibilities and expansion coefficients of

The compressibilities and expansion coefficients of gases at low pressures, and their relation to

molecular volume

B y J. B. M. Coppock

Chemistry Department, Battersea Polytechnic

(Communicated by J . Kenyon, F.R.S— Received 29 March 1939.— Revised 20 June 1939)

Introduction

The determination of molecular and atomic weights by the limiting density and limiting pressure methods has furnished considerable data on the behaviour of gases in the neighbourhood of atmospheric pressure. I t is customary in this work to express the deviations from the ideal gas laws

U Vin terms of either the compressibility 1 + A = or the compressibility

Pivxcoefficient A, where 1PoV0

, p 0v0 is the limiting value of pv at in

finitely low pressure, p 1v1 the value at 1 atm., and the temperature 0° C. The present paper describes a simple method of expressing compressibilities and related low pressure data for non-polar gases in terms of their effective molecular volume.

In previous papers (Coppock 1933, 1935 a) it has been shown how 1 + A values and expansion coefficients at low pressures may be calculated with considerable accuracy from the Beattie-Bridgeman (1928, 1930) equation of state, in which the constants have been derived from high-pressure data. This equation may be written in the virial form as follows:

where

pV

P

S

m + y + y 2 + p g , (1)

Ticr t b . - a , - ^ ,

— RTB0b + A 0a RB0bc

RB0c

l0, a, B0, b and c are characteristic constants for each gas.[ 201 ]

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202 J. B. M. Coppock

For condensable gases such as carbon dioxide the major contribution in representing the deviation from the ideal laws at N.T.P. is contained in the fi/V term, the other terms affect by about 1 part in 100,000.

Keesom has pointed out that many equations of state are of the type

(2)

so that at 1 atm. pressure we may identify the /?/F of the Beattie equation with B.

The virial coefficient B in equations of type (2) is related to the van der Waals constants a and b by the simple relation

thus at constant temperature B is proportional to b — ka, where k is a constant. In the case of non-polar gases at low pressures the intermolecular forces would be at a minimum, and a itself may not be of great importance, hence B should be a function of the molecular volume and deviations of these gases from the ideal laws expressed in terms of their compressibility would be a linear function of their actual volume. In this connexion it should be noted that Cawood and Patterson (1933) concluded from a critical study of the van der Waals and Dieterici equations of state that the limiting value of the compressibility of a gas at the critical temperature is directly proportional to the critical volume, i.e. it depends only on the volume occupied by the molecules.

Keesom (1921) suggested for non-polar gases that B — § 3, where Nis the Avogadro number and d the molecular diameter. For polar gases

B — - Nud^i 1 — - _____ — _____}* 3 \ 3 d*R*T*]’

where p is the dipole moment.If B for non-polar gases were completely represented by the effective

molecular volume, then it would be impossible to account for values of 1 + A for hydrogen, helium and neon less than unity, for here the contribution of a/RT must exceed that of b. Nevertheless with increasing molecular size the term including a will become less important and Keesom’s method of representing B by molecular volume of greater importance. The expression given for B is based on the argument that since molecules which approach one another cannot have their centres within the hemisphere of radius d, then B = %Nnd3, i.e. the effective volume B is four times the actual volume. Values of d are difficult to assign, because owing to the



mutual repulsion of the electronic shells the radius of the hemisphere will be greater than the molecular diameter as calculated herein from the lengths of the links and size of atoms involved. I t will be shown on the basis of the relations subsequently developed that the actual molecular diameter appears to be approximately *J2 times the calculated diameter.

N ew data for several gases

Recently Beattie and his co-workers (1935, 1937? 1939) have evaluated the five constants for their equation of state for ethane, propane and normal butane, and the previous work of the author (1933, 1935a) has been extended to these gases, 1 4- A values having been calculated using the convergence method. The expansion coefficients of methane have been recalculated, being previously in error. The results are given in table 1 and for comparison the values for ethylene and acetylene are included, in the latter case the calculation of the expansion coefficients is described

Compressibilities of gases at low pressures 203

subsequently.Table 1

Gas 10«a„ 106a„ 1 + A Ach4 3684 3680 1*00233 0*00232

36830bs. 3680Obs.1 — —c2h 6 3740 3721 1*00964 0*00955c3h 8 3813 3772 1*01893 0*01857wC4H10 3928 3861 1*03427 0*03314c2h 4 3724 3711 1*00741 0*00736

— — l*00732obs. 0*00727Obs.;c2h 2 3735 3718 l*00892Obs. 0*00884obs.

(Throughout th e p ap er values of th e coefficients of expansion a t co n stan t pressure a v an d co n s tan t volum e ocv are over th e range 0-100° C unless o the r tem p era tu re s are specifically s ta ted . The coefficients ocv are a t, an d ocv in itia lly a t, 1 a tm . pressure. R eferences in tab le are (1) K eyes and co-workers (1922, 1927), (2) Cawood an d P a tte rso n (1933).)

The results obtained for the paraffins are as expected for an homologous series, the values for n-butane probably represent the maximum deviations from the ideal laws for a non-polar gas at N.T.P. (b.p. 1° C). The diminution in 1 + A and a values for ethylene compared with ethane, and the subsequent approach of the acetylene values towards those of ethane are discussed later in terms of the spatial structure of these molecules. The reliability of the 1 + A values for these gases may be demonstrated by calculating the atomic weight of carbon in each case, taking the normal densities as cited in the International Critical Tables, i.e. from C2H6, C = 12*03 ; C2H4, C = 12*02, and C2H2, C = 12*005.




The expansion coefficients of acetylene, nitric oxide and hydrogen chloride have been calculated from a single 14- A value at 0° C by the following method. I t has been shown (Coppock 1933) that the compressibility of a given gas at any temperature T° abs. may be expressed in terms of the virial coefficient /? at that temperature by the equation

(1 + A)r R 2T 2 + /3‘ (3)

Hence A(1+A)

R2T 2, (4)

where R = 0*08206 1. atm.An inspection of the difference ( A)between the virial coefficients /?0

and /?100 shows that it is approximately proportional to /?0 values provided the gas is not very compressible, e.g. /?0 for C2H6 = —4*7632, = +1*23and /?0 for C2H4 = —3*6770, A = + 1*11. Thus /?0 calculated for acetylene from its 1+A value (Howarth and Burt 1925) is —4*441 and assuming A = +1*20, /?100 = —3*24. These values have been used to calculate the expansion coefficients as previously described giving 106av = 3735 and 10 6ocp =3718. In the case of nitric oxide 1 + A = 1*00119 (Addingley) and /?0 — 0*5971, assuming A = 0*43 as for oxygen 106oq, = 3675 and lO6̂ = 3674. For hydrogen chloride Whytlaw-Gray and Burt (1909) give 1 + A = 1*00748, fi0 = — 3*730, which is almost identical with that for carbon dioxide for which A = + 1*20, hence 106o+ = 3726 and 106ap = 3713. The error involved is probably not greater than 1 part in 2000.

In the discussion of the relation between expansion coefficient and molecular volume some of the experimental data required were not made over the temperature interval 0-100° C. The necessary corrections have been applied based on the fact that the (+ ) difference between two coefficients, say a0_60 and a0_100 is a linear function of the divergence of the coefficient from the ideal value of 0*003661, the greater the divergence the larger the difference. The method of evaluating the difference has been described by Coppock and Whytlaw-Gray (1934) and the values of the volume coefficients* obtained by these authors for five gases have been corrected in this way giving 10Gav for NO = 3677: SFe = 3785: dimethyl ether = 3888: CO = 3673 and N20 = 3727.

* In th e eq u a tio n given to co rrec t for ad so rp tio n (p. 503) t2 should read T —12, i.e . th e te m p era tu re in te rval.




The relation to molecular volume

I t has been suggested that a linear relation between 1 + A values and effective molecular volume B might be expected. Recently, Lewis (1938) has shown for similar groups of substances that critical temperatures, boiling points, viscosities and molar heats of vaporization are linear functions of the molecular volume as represented by the parachor.

The molecular volumes used in this paper are obtained from a summation of atomic radii and bond distances based on the Sidgwick-Pauling additivity rule. The data used are taken from the summaries given by Sidgwick (1933), Clark (1934), Glasstone (1936) and Brockway (1937). Sidgwick (1932) has emphasized the different interpretations of atomic radii depending on the mode of measurement and an attempt has been made to compute molecular radii r from comparable data. In many cases these have been obtained from electron diffraction measurements on the gaseous molecules. The values of r are given in table 2 together with the effective volume of one molecule V, and B the volume per gram-molecule as calculated from the van der Waals-Keesom formula F = § and B = N V, corresponding values of 1 + A and a are also tabulated.

This method of calculating B assumes the molecules are spinning rapidly enough to present themselves as spheres of revolution in the gaseous state. This is justifiable for diatomic molecules, and in the case of “ long” molecules Mack and Melaven (1931) believe end-over-end spins are to be expected and consequently r is taken as the maximum radius for the hydrocarbons and other similar molecules. This probably leads to some error in B for propane and butane, the true volume lying between the spheroidal and spherical condition considerably displaced towards the latter. The normal tetrahedral angle for the carbon valencies is assumed. The value of r for C0 2 is based on Wierl’s (see Glasstone 1936a) determination of the carbon-oxygen link; for N20 the extreme oxygen and nitrogen atoms are 2-38 A apart and r is the mean value for the resonance structures N ~=N + = 0 and N=N+—0 ~; r for dimethyl ether, a long molecule, assumes the oxygen intervalency angle of 111°, and for S0 2 the sulphur valency angle is taken as 124°. In the case of SF6, rF = 0-64, the observed S—F distance is 1*58 A hence r — 2*22 A (additivity rule 2-32 A).

Comparable values of r for the inert gases are difficult to assign, they appear to be best represented by the radii of the adjacent alkali metal ions. The values of Zachariasen and Wasastjerna obtained by independent methods for these ions are of the same order as the mean of Grimm’s values for the inert gases (for complete data see Clark (1934)), and the




former are those assumed. Viscosity measurements would give slightly larger values not fully comparable with the other data.

I t will be observed from figures 1 and 2 that for non-polar gases the compressibilities and coefficients of expansion at constant pressure are linear functions of B, the molecular volume. The discrepancies which occur

Table 2B in

Gas r in A V in A3 c.c./g. m ol. 106a„ 1 + AH e 0-68 5-27 3-19 3659 0-99948h 2 0-74 6-79 4-12 3660 0-99939d 2 0-741 6-79 4-12 36592 —

N e 0-98 15-77 9-56 3660 0-99951n 2 1 1 0 22-30 13-52 3670-5 1-00044NO 115 25-49 15-45 3677 1-00119o 2 1-20 28-96 17-55 3674 1-00094CO 1-20 28-96 17-55 3673 1-000483A 1-33 39-43 23-89 3673 1-00099HC1 1-37 43-09 26-11 3726 1-00748n h 3 1-45 51-09 30-96 3847 1-0151c h 4 1-46 52-16 31-60 3683 1-0024K r 1-48 54-33 32-92 3688 1-00279X 1-75 89-8 54-4 3721 1-00699c o 2 1-78 94-51 57-27 3723 1-006844c 2H4 1-80 97-72 59-23 3724 1-007325n 20 1*81 99-38 60-21 3727 1-007354s o 2 1-86 107-8 65-34 3900 1-02394C2H 6 1-93 120-5 73-11 3740 1-00964c 2h 2 2-03 140-2 84-96 3735 1-00892sf 6 2-22 183-3 111-1 3785 —

(c h 3)2o 2-42 237-5 143-9 3888 1-02815C3H 8 2-50 261-8 158-7 3810 1-01893nC4H 10 3 1 4 518-8 314-4 3928 1-03427

(D a ta n o t cited, in th is or p revious papers or ca lcu la ted herein bear th e ap p ro p ria te reference: (1) C lark (1939), (2) Coppock (19356), (3) W oodhead an d W hy tlaw -G ray (1933), (4) B atuecas (1934), (5) Cawood an d P a tte rso n (1933).)

are not large, particularly as B involves the cube of the molecular diameter and determinations of 1 + A and ocv are frequently affected by adsorption errors. The volume coefficient was chosen to illustrate the relation owing to its greater sensitivity to adsorption errors in its measurement than the corresponding pressure coefficient. I t was not expected that such a simple relation would hold so well in the case of expansion coefficients where T, and hence the attraction term a/BT, is variable. That it does hold supports the view of the minor importance of a and its subsequent evaluation in terms of B in the calculation of 1 + A for non-polar gases.



1 + A

The deviations from the linear relations shown by the polar gases is well marked, and expressed in terms of B the divergence appears to be proportional to the dipole moment, the values lying on a smooth curve as may be anticipated.


1-024

1-022

polar gases

-CO NO,

10000-998

B (c.c./g. mol.)

F igure 1. P lo t of 1 + A against B.

The interesting result has been pointed out that 1+A and a values for acetylene are apparently abnormal, falling between those for ethane and ethylene. I t is clear that shortening of the carbon-carbon distance due to the double bond will diminish r as we pass from C2H6 to C2H4, but in the case of acetylene the further diminution caused by the triple bond is




more than counterbalanced by the “ end-on” positions of the hydrogen atoms. Thus r and B are greater than for ethylene, actually they are larger than for ethane which is not in accordance with the 1+A values. Two explanations are possible: (1) Indeterminancy in the value of rH in the

3 8 6 0 -

B (c.c./g. mol.)F ig u r e 2. P lo t o f a v aga in st B.

C—H link which affects only slightly rmaxlmum for C2H6 and C2H4 owing to the spatial distribution of the hydrogen atoms, but is of major importance for C2H 2. (2) C2H 2 is a thinner molecule than either C2H6 or C2H4 and B calculated from the spherical formula is probably too large. Nevertheless, in spite of these defects the anomalous position of acetylene is explained simply and without the need of postulating an abnormal structure for the molecule.

No experimental value of 1 + A for isobutane is known, but compared with w-butane (B = 184-7 and 314-4 respectively) the former should deviate




less from a perfect gas (estimated 14- A = 1*022) as may be seen by a comparison of boiling points (— 17 and 1° C). In this connexion it is suggestive that in the case of buatadiene which has been ascribed the resonance structure CH2̂ i CH,̂ i CH„^:CH2, the boiling point is 3*6° lower than tha t of w-butane ( — 2*6 and 1° C). This is 3° short of the value anticipated by comparing B for the formula CH2= C H —CH =CH 2 (268*7) with the values for w-butane and isobutane where the difference in boiling points is 18° C. The resonance formula would suggest a slightly longer molecule (i.e. greater B value) than the older mode of writing the structure.

The position of sulphur hexafluoride in figure 2 is more normal when r is based on the measured S—F bond distance rather than on the additivity rule. There is also evidence from both curves, particularly in the expansion coefficient case, of the more polar tendencies of N—O than for C—O attributed to the additional electron centred on the nitrogen atom.

In view of the linear relations obtained values of B were inserted in the Keesom equation (2) and 1 + A values calculated therefrom, but the results were in considerable error. Obviously the true “ effective ” volume L must be of the same magnitude as the fijV term of the Beattie equation, and the corresponding term of the Keesom equation likewise. By evaluating 1 + A for definite values of L and determining from the slope of the curve in figure 1 the corresponding B values it is found that B must be multiplied by 2*85 to give L. This means that the actual diameter is very nearly .^2 times the calculated diameter, and suggests that the nearest distance of approach of non-polar molecules at low pressures is conditioned by their size and is not a uniform amount as represented by an envelope of say 0*5 A thick. I t also explains the apparent lessening of the van der Waals a effect with increasing molecular volume. I t was suggested earlier that the a/BT term is important in the cases of hydrogen, helium and neon, and as it is found that 1 + A and <x are linear functions of B, the correction for a has been evaluated in terms of B. Thus for non-polar gases we may write

P ivi = p 0v0+ (0-027— 2-85B)a t 0° C,

v and B are in litres per g. mol., and 22*4131 1. atm., the constantsare of course tentative. Values of 1+A calculated from this equation are given in table 3 and as errors in the molecular diameter are greatly magnified in calculating B the agreement is regarded as satisfactory.

Thus by utilizing this simple relation it should be possible to predict for non-polar molecules accurate values of the physical constants discussed in this paper from a knowledge of their molecular radii.




Table 3Gas 1 + ^calc. 1 + A>bs.

h 2 0-99932 0-99939n 2 1-00052 1-00044o 2 1-00103 1-00094c h 4 1-00291 1-0024K r 1-00317 1-00279co2 1-00617 1-00684C3H 8 1-0193 1-0189

The author wishes to express his indebtedness to Dr J. Kenyon, F.R.S., Professor R. Whytlaw-Gray, F.R.S., and Dr J. W. Smith for helpful criticism and advice.

Summary

A linear relation is shown to exist for non-polar gases at low pressures between their compressibilities and expansion coefficients and the corresponding molecular volumes.

I t is suggested that the nearest distance of approach of such gaseous molecules a t low pressures is dependent on their size and is not a uniform amount as represented by an envelope of about 0*5 A thick.

Additional data have been calculated for certain hydrocarbons and the behaviour of some of them discussed in the light of their spatial structure.

R eferences

A ddingley, C. G. P riv a te com m unication .B atuecas, T. 1934 J . Chim. Phys. 31, 65.B eattie , J . B. an d B ridgem an, O. C. 1928 Amer. Chem. 50, 3151.-------- 1930 Z . Phys. 62, 95.

B eattie , J . B . an d co-w orkers. 1935 J . Chem. Phys. 3, 93.— 1937 J . Amer. Chem. Soc. 59, 1589.— 1939 J- Amer. Chem. Soc. 61, 26.

B rockw ay, L. O. 1937 A nn . Rep. Chem. Soc. p . 196.Cawood, W . an d P a tte rso n , H . S. 1933 J . Chem. Soc. p. 619.Clark, C. H . D. 1934 Electronic structure and properties of matter, 1. L ondon :

C hapm an an d H all.— 1939 Phil. Mag. 27, 389.

Coppock, J . B . M. 1933 J . Phys. Chem. 37, 995.— 1935 a Phil. Mag. 19, 446.— 19356 Trans. Faraday Soc. 31, 913.

Coppock, J . B. M. and W hytlaw -G ray , R . 1934 Proc. Roy. Soc. A, 143, 487. G lasstone, S. 1936 A nn . Rep. Chem. Soc. p . 65.

— 1936a Recent advances in general chemistry, p . 209. L ondon : Churchill. H ow arth , J . T. an d B u rt, F . P . 1925 Trans. Faraday Soc. 20, 544.




K eesom , W . H . 1921 Phys. Z . 22, 129.K eyes, F . G. an d co-w orkers 1922 J . Math. Phys. Mass. Inst. Tech. 1, 191.

— 1927 J . Amer. Chem. Soc. 49, 1403.Lewis, D . T . 1938 J . Chem. Soc. pp . 261, 1056, 1061.M ack, E . an d M elaven, R . M. 1931 J . Amer. Chem. Soc. 54, 888.Sidgwick, N . V. 1932 A nn . Rep. Chem. Soc. p . 67.

— 1933 The covalent link. Cornell U niv . Press.W hytlaw -G ray , R . an d B u rt, F . P . 1909 J . Chem. Soc. 95, 1633. W oodhead, M. an d W hytlaw -G ray , R . 1933 J . Chem. Soc. p . 846.

On relativistic wave equations for particles of arbitrary spin in an electromagnetic field

By M. Fierz and W. Pauli Physikalisches Institut der E Technischen

Hochschule, Zurich

(Communicated by P.A. M. Dirac, F.R.S.—Received 31 May 1939)

1. Introduction

The investigations of Dirac (1936) on relativistic wave equations for particles with arbitrary spin have recently been followed up by one of us (Fierz, 1939, referred to as (A)) I t was there found possible to set up a scheme of second quantization in the absence of an external field, and to derive expressions for the current vector and the energy-momentum tensor. These considerations will be extended in the present paper to the case when there is an external electromagnetic field, but we shall in the first instance disregard the second quantization and confine ourselves to a c-number theory.

The difficulty of this problem is illustrated by the fact that the most immediate method of taking into account the effect of the electromagnetic field, proposed by Dirac (1936), leads to inconsistent equations as soon as the spin is greater than 1. To make this clear we consider Dirac’s equations for a particle of spin 3/2, which in the force-free case run as follows:

Vol. 173. A.

K b = p^PaPp = p0Pa“p, Kat/3 = V * p b f = PppbPi,

14

( 1 )



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The compressibilities and expansion coefficients of …rspa.royalsocietypublishing.org/content/royprsa/173/953/201.full.pdf · The compressibilities and expansion coefficients of