Equipartition of energy, Avogadro law and ratio of specific heats

Oliveira, Mário J. de

doi:10.1590/1806-9126-RBEF-2018-0307

Mário J. de Oliveira ^**Correspondence email address: oliveira@if.usp.br.

Universidade de São Paulo, Instituto de Física, São Paulo, SP, Brasil

https://orcid.org/0000-0002-4139-1445

^*Correspondence email address: oliveira@if.usp.br.

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Figure 1
Pressure p, in inches of mercury, versus volume v of atmospheric air from experimental data published by Boyle in 1662 ^[12][12] R. Boyle, A Defence of the Doctrine touching the Spring and Weight of the Air (Thomas Robinson, London, 1662).. The unit of volume is that in which v = 1 at the atmospheric pressure. The inset shows the same data plotted in the diagram p versus 1/v.

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Figure 2
Apparatus used by Gay-Lussac in the experiment on thermal expansion of gases reported in 1802 ^[14][14] J.L. Gay-Lussac, Annales de Chimie 43, 137 (1802).. Gas coming from the bell jar M is introduced into the flask B. With the tap R opened, the flask is heated and part of the gas escapes from the flask. When the temperature reaches 100 °C, the tap R is closed and then the flask is submerged in water with ice. When it reaches the temperature of 0 °C, the tap R is opened and a certain amount of water flushes inside the flask. The volume of water that entered the flask equals the increase in volume of the gas when it is heated from 0 to 100 °C. The volume of water is measured by weighting the flask B with water. Figure from reference ^[14][14] J.L. Gay-Lussac, Annales de Chimie 43, 137 (1802)..

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Figure 3
Representation in the p−v diagram of a sudden compression of a gas inside a vessel with heat conducting walls. The gas is compressed adiabatically from state A to B, causing an increase in its temperature. From B to C, the gas delivers heat at constant volume and its temperature decreases. At states A and C the gas has the same temperature, that is, the dashed line is an isotherm.

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Figure 4
Apparatus used by Clément and Desormes in their experiment reported in 1819 ^[45][45] C.B. Desormes and N. Clément, Journal de Physique 89, 321 (1819).. AB is a glass flask with a capacity of 28.4

ℓ

with a tap M having an aperture of about 1.4 cm in diameter. The glass flask communicates to a pneumatic machine through a tube NB. This tube also communicated with two other tubes GF and CE, the first of which was immersed in mercury and the second in water. The height of the water in tube CE gives a measure of the pressure inside the flask as the tap L is open with the taps B and M closed. Figure from reference ^[45][45] C.B. Desormes and N. Clément, Journal de Physique 89, 321 (1819)..

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Figure 5
Representation in the p−v diagram of a processes followed by the gases contained in two connected vessels. One gas follows the process ABC and the other the process

A^{'} B^{'} C

. The processes AB and

A^{'} B^{'}

are adiabatic. The processes BC and

B^{'} C

are processes at constant volume. At states A, C, and

A^{'}

the gases have the same temperature, that is, the dashed line is an isotherm.

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Figure 6
Apparatus of Joule to measure the heat evolved by compression of the air ^[56][56] J.P. Joule, Philosophical Magazine 26, 369 (1845).. The pump C and the cooper receiver R are immersed in water contained in a vessel impermeable to heat, in which there is a sensible thermometer. The pump is operated several times, compressing dry air, which is admitted through the pipe A, until the air inside the receiver reaches about twenty two atmospheres, generating a certain quantity of heat which warms the water. Figure from reference ^[56][56] J.P. Joule, Philosophical Magazine 26, 369 (1845)..

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Figure 7
Joule free expansion apparatus ^[56][56] J.P. Joule, Philosophical Magazine 26, 369 (1845).. R and E are two cooper receivers with capacities of 136.5 and 134 cubic inches. The two receivers are connect by a pipe with a stopcock and immersed in water contained in a vessel impermeable to heat. Figure from reference ^[56][56] J.P. Joule, Philosophical Magazine 26, 369 (1845)..

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Figure 8
Volume diagrams used by Gaudin ^[69][69] M.A. Gaudin, Annales de Chimie et de Physique 52, 113 (1833).. Each one of the identical volumes (represented by squares) contains the same number of molecules (just one molecule), in accordance with Avogadro law. Left: Chlorine and hydrogen giving hydrogen chloride,

{Cl}_{2} + H_{2} \to 2 HCl

. Center: Oxigen and hydrogen given water vapor,

O_{2} + 2 H_{2} \to 2 H_{2} O

. Right: Nitrogen and hydrogen giving ammonia,

N_{2} + 3 H_{2} \to 2 {NH}_{3}

. Gaudin used Dalton symbols instead of chemical symbols.

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Table 1
Column 2: Thermal expansion

α_{0}

per °C according to Regnault ^[22][22] V. Regnault, Annales de Chimie et de Physique 4, 5; 5, 52 (1842).. Column 3: Temperature

θ_{0} = 1 / α_{0}

in °C, obtained from column 2. Column 4 and 5: Density

ρ

in g/

ℓ

at 0° C and 1 atm according to Gay-Lussac ^[26][26] J.L. Gay-Lussac, Mémoires de physique et de chimie de la Societé d'Arcueil 2, 235 (1809). and Biot and Arago ^[25][25] J.B. Biot and F. Arago, Mémoires de la Classe des Sciences Mathématiques et Physiques de l'Institut National de France 7, 301 (1806).; and according to Regnault [²⁷[27] V. Regnault, Relation des Expériences pour Déterminer les Principales Lois et les Données Numèrique qui Entrent dans le Calcul des Machines à Vapeur (Firmin Didot, Paris, 1847).,²⁸[28] V. Regnault, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 36, 676 (1853).].

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Table 2
Specific heat

c_{p}

and volume specific heat

{\bar{c}}_{p}

of gases obtained experimentally by Delaroche and Bérard (column 2 and 4) ^[32][32] F. Delaroche and J.E. Bérard, Annales de Chimie 85, 72 (1813)., by Delarive and Mercet ^[37][37] A. Delarive and F. Marcet, Annales de Chimie et de Physique 75, 113 (1840). (column 5), and by Regnault (columns 3 and 6) ^[38][38] V. Regnault, Relation des Expériences pour Déterminer les Lois et les Données Physiques Nécessaires au Calcul des Machines à Feu (Firmin Didot, Paris, 1862), v. 2.. Specific heat

c_{p}

in cal/g°C. Volume specific heat

{\bar{c}}_{p}

in units in which that of air is unity.

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Table 3
Experimental values of the ratio γ of specific heats, and the velocity of sound

v_{s}

in m/s of the atmospheric air at the temperature

θ

in degrees centigrade. CD, Clément and Desoremes; GLW, Gay-Lussac and Welter; MVB, Moll and Van Beek; BM, Bravais and Martins. The second column gives the year of publication of γ.

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Table 4
Ratio γ of specific heats for several gases determined by Dulong ^[36][36] P.L. Dulong, Annales de Chimie et de Physique 41, 113 (1829)., by Rankine ^[59][59] W.J.M. Rankine, Philosophical Magazine 5, 483 (1853)., by Masson ^[47][47] A. Masson, Annales de Chimie et de Physique 53, 257 (1858)., and by Cazin ^[49][49] A. Cazin, Annales de Chimie et de Physique 66, 206 (1862)..

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Table 5
Column 2: Density

ρ

of gases in units where that of air is taken as unity, according to Gay-Lussac ^[26][26] J.L. Gay-Lussac, Mémoires de physique et de chimie de la Societé d'Arcueil 2, 235 (1809).. Column 3: Molecular weight

μ^{*}

of gases with respect to molecular weight of hydrogen taken as unity, according to Avogadro ^[66][66] A. Avogadro, Journal de Physique, de Chimie, d'Histoire naturelle et des Arts 73, 58 (1811).. Column 4 and 5: Molecular weight

μ

of gases with respect to molecular weight of hydrogen taken as equal to 2, and chemical formulas (cf), according to Cannizzaro ^[64][64] S. Cannizzaro, Nuovo Cimento 7, 321 (1858)..

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

gas	100 $α_{0}$	$θ_{0}$	$ρ$	$ρ$
air	0.3665	272.85	1.29230	1.29318
oxygen			1.42617	1.42980
nitrogen	0.36682	272.61	1.25241	1.25616
hydrogen	0.36678	272.64	0.09461	0.08957
carbon monoxide	0.36667	272.72	1.25072	1.2510
nitric oxide			1.33929	1.3436
hydrogen chloride	0.36812	271.65	1.651	1.6131
carbon dioxide	0.36896	271.03	1.9637	1.97741
nitrous oxide	0.36763	272.01	1.96549	1.9721
sulfur dioxide	0.36696	272.51	2.927	2.9057
year	1842	1842	1809	1847/53

gas	$c_{p}$	$c_{p}$	${\bar{c}}_{p}$	${\bar{c}}_{p}$	${\bar{c}}_{p}$
air	0.2669	0.2375	1	1	1
oxygen	0.2361	0.2175	0.9765	1	1.0126
nitrogen	0.2754	0.2438	1	1	0.9971
hydrogen	3.2936	3.4090	0.9033	1	0.9933
carbon monoxide	0.2884	0.2450	1.0340		0.9979
nitric oxide		0.2317			1.0131
hydrogen chloride		0.1852			0.9823
carbon dioxide	0.2210	0.2163	1.2583	1.222	1.3924
nitrous oxide	0.2369	0.2262	1.3503		1.4514
sulfur dioxide		0.1544			1.4375
ethylene	0.4207	0.4040	1.5530	1.531	1.7288
year	1813	1853	1813	1835	1853

author	year	γ	$v_{s}$	$θ$
CD ^[45][45] C.B. Desormes and N. Clément, Journal de Physique 89, 321 (1819)., Laplace ^[24][24] P.S. Laplace, Connaissance des Tems ou des Mouvements Célestes à l'Usage des Astronomes et des Navigateurs pour l'an 1825, p. 219 (1822), available in https://gallica.bnf.fr/ark:/12148/bpt6k65057931. https://gallica.bnf.fr/ark:/12148/bpt6k6...	1822	1.354	332.9	12.5
GLW, Laplace ^[24][24] P.S. Laplace, Connaissance des Tems ou des Mouvements Célestes à l'Usage des Astronomes et des Navigateurs pour l'an 1825, p. 219 (1822), available in https://gallica.bnf.fr/ark:/12148/bpt6k65057931. https://gallica.bnf.fr/ark:/12148/bpt6k6...	1822	1.37244	335.2	13
Masson [⁴⁶[46] E. Péclet, Traité Élémentaire de Physique (Hachette, Paris, 1847), v.1, 4ème éd.,⁴⁷[47] A. Masson, Annales de Chimie et de Physique 53, 257 (1858).]	1847	1.4196
Weisbach ^[48][48] J. Weisbach, Civilingenieur 5, 46 (1859).	1859	1.4025
Cazin ^[49][49] A. Cazin, Annales de Chimie et de Physique 66, 206 (1862).	1862	1.408
Dulong ^[36][36] P.L. Dulong, Annales de Chimie et de Physique 41, 113 (1829).	1829	1.421	333	0
MVB ^[51][51] G. Moll and A. Van Beek, Philosophical Transactions 114, 424 (1824)., Simons ^[50][50] G. Simons, Philosophical Transactions 120, 209 (1830).	1830	1.4152	332.244	0
MVB ^[51][51] G. Moll and A. Van Beek, Philosophical Transactions 114, 424 (1824)., Rankine ^[29][29] W.J.M. Rankine, Transactions of the Royal Society of Edinburgh 20, 147 (1853).	1850	1.40853	332.25	0
BM ^[52][52] A. Bravais and C.F. Martins, Comptes Rendus des Séances de l'Académie des Sciences 20, 1164 (1844)., Rankine ^[29][29] W.J.M. Rankine, Transactions of the Royal Society of Edinburgh 20, 147 (1853).	1850	1.40955	332.37	0
D & B ^[32][32] F. Delaroche and J.E. Bérard, Annales de Chimie 85, 72 (1813)., Joule ^[58][58] J.P. Joule, Philosophical Magazine 31, 114 (1847).	1847	1.347	333.8	0
Regnault ^[28][28] V. Regnault, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 36, 676 (1853)., Rankine ^[59][59] W.J.M. Rankine, Philosophical Magazine 5, 483 (1853).	1853	1.4094	332,35	0
Regnault ^[28][28] V. Regnault, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 36, 676 (1853)., Le Conte ^[60][60] J. Le Conte, Philosophical Magazine 27, 1 (1864).	1864	1.41047	332.490	0

author	Dulong	Rankine	Masson	Cazin
year	1829	1853	1853	1862
air	1.421	1.4094	1.4148	1.408
oxygen	1.415	1.4014	1.4148	1.408
nitrogen			1.4148	1.408
hydrogen	1.407	1.4150	1.4148	1.408
carbon monoxide	1.428	1.4047	1.4248	1.408
nitric oxide			1.4049
hydrogen chloride			1.4084
carbon dioxide	1.348	1.2714	1.2886	1.291
nitrous oxide	1.343		1.2813	1.285
sulfur dioxide			1.2619	1.262
ammonia			1.3146	1.328
ethylene	1.240		1.2708	1.257

gas	$ρ$	$μ^{*}$	$μ$	cf
oxygen	1.10359	15.074	32	$O^{2}$
nitrogen	0.96913	13.238	28	$N^{2}$
hydrogen	0.07321	1	2	$H^{2}$
carbon monoxide	0.96782	13.22	28	CO
nitric oxide	1.03636	14.156	30	NO
chlorine	2.468	33.74	71	${Cl}^{2}$
hydrogen chloride	1.278	17.45	36.5	HCl
carbon dioxide	1.5196	20.75	44	${CO}^{2}$
nitrous oxide	1.52092	20.775	44	$N^{2} O$
sulfur dioxide	2.265	30.94	64	${SO}^{2}$
ammonia	0.59438	8.119	17	${NH}^{3}$
year	1809	1811	1858	1858

[1] ^*Correspondence email address: oliveira@if.usp.br.

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Equipartition of energy, Avogadro law and ratio of specific heats

Abstract