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FROM THE HISTORY OF PHYSICS
D. I. MENDELEEV’S IDEAL GAS EQUATION OF STATE
V. N. Goloushkin
“1874. I consider this formula
(given by me) essential and important
in the physicochemical sense.”D. I. Mendeleev, 1899.[^1]
Clapeyron (1799–1864), a French engineer and scholar who in 1820–1830 was a professor at the St. Petersburg Institute of Transportation Engineers, published in 1834 an article “On the Motive Power of Heat”[^3].
In this article, in particular, he gave the relation between the pressure, volume, and temperature of a gas, obtained from a combination of the Mariotte–Gay-Lussac law:
\[ pv = R(267 + t), \quad \text{where } R = \frac{p_0 v_0}{267 + t_0}. \]
(At that time the coefficient of volumetric expansion of a gas was taken to be equal to \(\frac{1}{267}\); \((267 + t)\) corresponds to the absolute temperature, the concept of which had not yet been introduced at that time.)
This equation later came to be called Clapeyron’s equation.
Mendeleev drew attention to the fact that the basic information about the properties of ideal gases is expressed by three laws: Boyle–Mariotte’s, Gay-Lussac’s, and Avogadro–Ampère–Gerhardt’s, which are not entirely exact. In his opinion, explanations of deviations from these laws were doubtful because they had not been verified experimentally, while the questions raised thereby were of primary importance for all natural science. Therefore he often discussed projects for obtaining complete and accurate experimental data on changes in gas pressure with variation of their volumes, temperatures, and nature.
For the implementation of the projected plans, long years of work were required, and, most importantly, large monetary funds, which Mendeleev did not have.
Finally, in 1872, with the help of the Russian Technical Society, he succeeded in obtaining the necessary funds and in beginning the investigations of the elasticity of gases that had earlier been conceived. The result of this great work was the capital work On the Elasticity of Gases, published in 1875.
Mendeleev also regarded his work as a possible reason to induce young scientists to study the nature of gases and to disseminate exact methods for their investigation. Valuing the direct practical applications of the exact sciences, Mendeleev, as it were apologizing, says that little of what he had done could have applied significance, although his work had another aim.
Mentioning that technology is drawing ever closer to the practice of the experimental sciences and that laboratory methods very often pass as a whole into factory and, in general, technical methods, Mendeleev expresses the hope that his mastic (§§ 29, 30), mercury pump without stopcocks (§ 33), new method of making barometers (§ 42), differential barometer (Chapter V), especially its application to leveling, the method of calibrating tubes (§§ 64—71), experiments on the resistance of tubes to rupture (§§ 73—74), the new construction of cathetometers (§ 80), and methods of observation with their aid (§ 79) will be useful in a technical respect. Practice confirmed Mendeleev’s confidence.
In this work, in addition to enormous experimental material, Mendeleev set forth a new equation of state for ideal gases, which arose in the course of his investigations. Thus, the needs of practice gave rise to the appearance of a new gas equation, more perfect and more general than Clapeyron’s equation.
The equation of state derived by Mendeleev in 1872–1874 made it possible, by elementary methods, to analyze the possible errors in experiments on the compressibility of gases, and this was necessary for evaluating the method of observation. Moreover, assuming the nature of the gas and its mass to be unchanged, Mendeleev obtained Clapeyron’s formula from his equation.
With this equation Mendeleev first acquainted Russian scientists, reporting it on September 12, 1874[^3] at a meeting of the Chemical Society, and on September 17[^4]—at a meeting of the Physical Society at St. Petersburg University.
In the minutes of the society’s meetings it is recorded that Mendeleev reported a general formula for gases, based on the totality of the laws of Mariotte, Gay-Lussac, and Avogadro (Ampère–Gerard). His formula is fuller and more general than the well-known formula of Clapeyron, since
if in the latter the constant changes with the nature and mass of the gas, then in Mendeleev’s formula it is constant for all gases. The formula simplifies all approximate calculations relating to gases and vapors, when the exact applicability of the three named laws may be assumed. It is justified by existing data and may be valid even when there is a deviation from one of the three laws.
After this Mendeleev sent a note about this to the Comptes rendus of the Paris Academy of Sciences ^5, and in January 1877, at the request of the editors of the journal Nature ^6, communicating to the scientific world of England the results of his investigations of the Boyle–Mariotte law, he presented his equation.
He also introduces this same equation in his Principles of Chemistry, beginning with the 3rd edition ^7 in Chapter VII, “Particles and Atoms,” and in the supplements to it ^8. Mendeleev himself first publishes the equation and gives its derivation in his work “On the Elasticity of Gases” (§ 20 of this work ^10 is entitled “General Equations for Gases...”):
\[ KAP=\frac{M}{V}(C+T). \]
“In order to express, in absolute quantities, the dependence between volume and elasticity, it is necessary to take into account the mass and temperature of the gas; however, in separate observations, where the temperature and mass remain constant, the task of determining the desired dependence can be greatly simplified, as indeed was done in the majority of investigations carried out up to now. If ... one denotes by \(P\) the pressure or elasticity, by \(T\) the temperature, by \(V\) the volume, and by \(M\) the mass or, under constant acceleration of gravity, the weight of the gas, then for a given gas the dependence of these quantities, as is known, is expressed by the equality
\[ K_iPV=M(C_0+T), \]
where the coefficient \(K_i\) is constant for a given gas and depends on its nature, while the constant \(C_0\) (the so-called temperature of absolute zero) remains the same for all permanent gases. Taking into account Avogadro’s law (Ampère and Gerhardt), according to which the densities of gases are proportional to the weight of their particles, and denoting by \(A_i\) the particle (molecular.—V. G.) weight of a gas, exactly determined by chemical investigation, we have, according to this law, that
\[ K_i : A_i = K_0 : 1, \]
where \(K_0\) for all gases (regarding them as perfect*) is a quantity
\[ \text{________________} \]
*) In the Russian physico-chemical literature of the third quarter of the nineteenth century, instead of the modern name “ideal gas,” the term “perfect gas” was used.
constant, and therefore
\[ K_0 A_i P = M(C_0 + T). \tag{I} \]
Such is the most general formula expressing the properties of a perfect gas; in it, with a variable nature of the gas, only \(A\) changes,”[^9] (the whole discharge, except for the words “of a perfect gas,” is mine.—V. G.). “In an experiment one may vary: \(A_i\), i.e., the nature of the gas, for then its particle weight changes; \(P\), \(V\), \(M\), and \(T\); the coefficients \(K_0\) and \(C_0\) will remain constant, if the above-enumerated laws are exactly valid. The numerical value of \(K_0\) and \(C_0\) will in this case change only with the change of the units in which \(A_i\), \(P\), \(V\), \(M\), and \(T\) are expressed.”[^9]
Further, giving for different gases values of \(K_0\) that differ only slightly from one another, Mendeleev explains this by errors in the determination of \(A\), \(P\), \(V\), \(M\), and \(T\), and by inaccuracies in the laws used for deriving formula (I).
Mendeleev gives the derivation of the formula in two ways, one of which, the simpler, is presented below:
“If at volume \(=1\) the mass of the gas \(=K\), then at volume \(=V\) it will be \(M=KV\). If at pressure \(=1\) the mass of the gas \(=K_1\), then, according to Mariotte’s law, at pressure \(P\) it will be \(=K_1P\). If at particle weight \(=1\) the mass of the gas \(=K_2\), then, according to Avogadro–Gerhardt’s law, at particle weight \(A\) the mass of the gas will be \(=K_2A\). If at absolute temperature \(=1\) the mass of the gas \(=K_3\), then at absolute temperature \(C+T\) it will, according to Gay-Lussac’s law, be equal to
\[ \frac{K_3}{C+T}. \]
Therefore, if \(V\), \(A\), \(P\), and \(C+T\) are varied simultaneously, then the mass of the gas will be
\[ M=\frac{K \cdot K_1 \cdot K_2 \cdot K_3}{C+T} VAP, \]
and therefore, taking \(K \cdot K_1 \cdot K_2 \cdot K_3 = K\), we obtain
\[ \frac{M}{V}(C+T)=KAP, \]
where \(K\) is a quantity constant for all gases, if the above-mentioned laws are true. The quantity \(K\) is the mass or weight of a unit volume of gas having particle weight \(=1\), at pressure \(=1\) and at absolute temperature \(=1\).”[^10]
In the 5th edition of Principles of Chemistry he gives formula (I) a somewhat different form, since he uses different designations and expresses the quantities in another system of units.
Giving it the following form: \(pv = 6255 \dfrac{m}{M}(273 + t)\), Mendeleev says that “instead of the formula \(pv = R(273 + t)\), where \(R\) varies with the mass and nature of the gas, one may use the above formula, and, taking the weight of the gas \(m\), equal to the weight of its particles (gram-molecule.—V. G.), we obtain \(PV = 6255(273 + t)\) for all gases”[^11].
In the 8th edition of Principles of Chemistry (1906), Mendeleev, repeating the exposition of the derivation of the equation given by him in the 5th edition, gives for the constant quantity the corrected value 6200 instead of the former 6255, and then, expressing the molecular weight in grams and taking into account that the volume of a gram-molecule for ideal gases is equal to 22.412 liters (D. Berthelot, 1904) and that the temperature of absolute zero is equal to \(273.09^\circ\)C, obtains the equation of state for a given weight of an ideal permanent gas in the form
\[ pv = \frac{22.412}{273.09}(273.09 + t), \tag{A} \]
where \(p\) is expressed in atmospheres and \(v\)—in liters.
Passing from this to the form of the formula given by him much earlier, expressing the pressure in atmospheres and the volumes in liters, and denoting the molecular weight by the letter \(M\), Mendeleev obtains the equation of state for \(m\) grams of an ideal gas
\[ pv = 0.08207\,\frac{m}{M}(273.09 + t). \tag{B} \]
The latter formula is “more general, broader, and permits more applications”[^12].
If we pass to modern notation, we obtain the widely known formula \(pv = \dfrac{m}{\mu}RT\), whose creator’s name has proved to be forgotten.
What has been set forth above makes it possible to correct the error that was previously made and henceforth to call the equation of state of ideal gases (derived by Mendeleev as early as 1874)
\[ pV = \frac{m}{\mu}RT \]
the Mendeleev equation, and \(R\)—the gas constant for a gram-molecule, whose magnitude is the same for all gases—the Mendeleev gas constant.
The formula combining Boyle–Mariotte’s and Gay-Lussac’s laws and having lost its significance with the discovery of Mendeleev’s equation may be assigned no name at all.
Only the inattentive attitude of the official science of tsarist Russia toward the work of Russian scholars can explain the historical injustice that occurred in the naming of one of the fundamental equations of the modern doctrine of the gaseous state of matter*).
Cited Literature
- D. I. Mendeleev, Literary Heritage, vol. I, 1939, p. 68.
- Clapeyron, Ann. d. Physique 59, 451, 464, 569 (1843). (From Jour. de l’école polyt., 1834, 14.)
- ZhRFKhO 6 (chemical section), issue 7, sect. I, p. 208.
- ZhRFKhO 6 (physical section), issue 7, sect. I, p. 121.
- Comptes Rendus Paris 82, 412 (1876).
- Nature 15, 500 (1877).
- D. I. Mendeleev, Principles of Chemistry, 3rd ed., 1877, pp. 440, 442.
- D. I. Mendeleev, Principles of Chemistry, 8th ed., 1905 (according to the 13th ed., 1947), vol. I, 235, 535.
- D. I. Mendeleev, On the Elasticity of Gases, part I, § 20, 1875.
- Ibid., note to § 20.
- D. I. Mendeleev, Principles of Chemistry, 5th ed., 1889, p. 240.
- Ibid., 13th ed. (reprint from the 8th ed.), 535, 536.
- Advances in Chemistry 20, issue 1, 132, 133 (1951).
*) While collecting material for the history of the physical section of the RFKhO in December 1949 and drawing attention to Mendeleev’s report on the equation of state, I began a more detailed investigation of this question.