ALEKSEI IOSIFOVICH BACHINSKY

(1877–1944)

M. P. Volarovich

On July 31, 1944, in Moscow, the outstanding Russian physicist Aleksei Iosifovich Bachinsky, professor of Moscow State University, died; he was widely known both in the USSR and abroad for his works in the field of molecular physics and thermodynamics.

A. I. Bachinsky was born on March 21 (old style), 1877, in the town of Kholm, former Lublin Province, into the family of a mathematics teacher, a Ukrainian who had moved to Russia from Austro-Hungary, where Slavs at that time were subjected to persecution by the Austrian government. From 1886 he studied at the Kholm gymnasium, from which he graduated in 1895. In the same year he came to Moscow and entered the mathematical division of the Faculty of Physics and Mathematics of Moscow University. At that time, courses in physics there were given by outstanding Russian scientists: Aleksandr Grigoryevich Stoletov in experimental physics and Nikolai Aleksandrovich Umov in theoretical physics. Talented young people gathered around them. A. G. Stoletov organized the first physical laboratory of Moscow University, in which P. N. Lebedev worked from 1891.

The lectures of A. G. Stoletov and N. A. Umov made a great impression on A. I., and contact with the leading Moscow physicists determined his subsequent activity. He chose physics as his specialty and decided to devote himself to scientific and pedagogical work, to which he dedicated his entire life.

After graduating from the university in 1899, A. I. was retained at the Department of Physics to prepare for a professorship, and already in 1900 his first two scientific papers were printed in the Proceedings of the Society of Lovers of Natural Science. One of them, entitled “Toward a Dynamic Theory of Electricity,” connected with the interpretation of Maxwell’s experiments, was devoted to finding the most favorable conditions for setting up further experiments on the detection of the inertia of electric charges (a phenomenon later established experimentally by Tolman). In the other article, the question of the dependence of the viscosity of mercury on temperature was considered.

In 1907 A. I. was appointed privatdozent of Moscow University. In 1918 he became professor of physics at Moscow University and for a number of years gave special courses in thermodynamics, statistical physics, and certain other branches of physics for senior students. In 1930, as a result of a serious illness, A. I. lost his sight and was forced to cease teaching. However, with the help of his daughter O. A. Bachinskaya, he continued his scientific work to the end of his life.

In the years immediately following A. I. Bachinskii’s graduation from the university, the two areas of physics on which his subsequent scientific activity was concentrated became defined: molecular physics and the study of the thermodynamic properties of matter.

Questions of molecular physics at the end of the nineteenth century attracted very great attention from physicists. At that time the kinetic theory of gases had received brilliant development in the well-known works of Maxwell, Boltzmann, Clausius, and other scientists. The molecular theory of liquids, however, was at that time very little developed, and the empirical regularities of the liquid state of matter had as yet scarcely been studied. A. I. Bachinskii became interested in this little-investigated area of physics. He devoted many works to the study of the nature of the liquid state of matter. The very simple regularities established by him, confirmed with great accuracy by experimental data, still have fundamental importance in the corresponding branches of molecular physics. The area of physics chosen by A. I. Bachinskii for his investigations is very difficult to develop. This follows, for example, from the fact that the molecular-kinetic theory of the liquid state of matter has advanced comparatively little up to the present time, whereas the atomic theory of the crystalline solid, as is known, had already achieved very great successes at the beginning of the present century.

It is impossible in a journal article to set forth in detail the results of all A. I.’s scientific works, the number of which exceeds 60. However, before turning to a consideration of his most important works on questions of the viscosity of liquids, it is necessary to dwell briefly on other investigations in the molecular physics of liquids, which are closely intertwined with his work in thermodynamics. Their content and significance are characterized in detail in the reports of P. A. Rebinder¹ and K. A. Putilov² at the meeting devoted to the twenty-fifth anniversary of Bachinskii’s law of viscosity. On the basis of a large body of experimental material, A. I. established that the surface tension of liquids \(\sigma\) is, to a high degree of accuracy, proportional to the fourth power of the difference between the densities of the liquid \((\Delta)\) and its saturated vapor \((\delta)\), namely:

\[ \sigma = C(\Delta - \delta)^4, \tag{1} \]

where \(C\) is the coefficient of proportionality. It should be recalled that

Laplace proposed an analogous equation with exponent 2, and van der Waals—with exponent 3. For temperatures far from the critical point, $\delta$ is very small in comparison with $\Delta$, and instead of formula (1) one may, according to Bachinskii, write:

\[ \delta=\frac{C}{v^{4}}, \tag{2} \]

where $v$ is the specific volume of the liquid.

From this formula there is obtained directly the equation for the parachor $P$,

\[ P=s^{\frac{1}{4}}v, \tag{3} \]

which Sugden and other authors applied to the study of the character of bonding in chemical compounds and to the solution of a number of structural problems.

At the end of the nineteenth century and at the beginning of the present century, much attention in physics was devoted to the formulation of an equation of state of matter that would cover, with sufficient accuracy, the region of gases and liquids. In this connection A. I. set himself the goal of refining the van der Waals equation, which, as is known, while correctly expressing the qualitative aspect of the phenomenon, at the same time proves quantitatively to be little satisfactory for real substances. Having analyzed a large number of works in which several dozen different equations of state of matter had been proposed, and having processed a series of experimental data, mainly those of Young, A. I. proposed the following equation of state:

\[ pvV=aT-\beta+\gamma\sqrt{T_{\mathrm{кр}}-T}, \tag{4} \]

where $p$ is the pressure, $v$ is the volume of the liquid, $V$ is the volume of the saturated vapor, $T$ is the absolute temperature, $T_{\mathrm{кр}}$ is the critical temperature, and $\alpha$, $\beta$, and $\gamma$ are constants. Associated with this equation is Bachinskii’s “law of parabolas” (in a definite system of coordinates equation (4) gives parabolas on graphs), which is confirmed by experiment over a wide interval of temperatures with an accuracy up to 0.5%.

In connection with investigations of the equation of state, A. I. introduced a new concept: the orthometric density of a substance, corresponding to that point on the state diagram where the van der Waals isotherm intersects the isotherm of an ideal gas. He showed that the orthometric density decreases with temperature according to a linear law. The introduction of the concept of the orthometric state of a substance enabled A. I., in a number of cases, to obtain good agreement of experimental data with the van der Waals equation.

A. I. returned repeatedly to the question of the equation of state throughout the course of his scientific activity. Thus, after the appearance of Bridgman’s outstanding works^3 on the compressibility of matter at high-

pressures up to 12,000 kg/cm², A. I., already ill after the loss of his sight, undertook the solution of the very difficult problem of establishing an equation of state for the region of high pressures. In doing so he succeeded in obtaining a very simple equation for the isotherm, namely:

\[ p+K=\frac{L}{v^{7}}, \tag{5} \]

which satisfies Bridgman’s experiments with a high degree of accuracy \((K\) and \(L\) are constants for a given temperature).

Of great interest are the works of A. I. Bachinskii devoted to the question of the additivity of molar volumes. He was the first to use, in calculating the quantity \(b\) in the van der Waals equation, the molar volumes at the boiling point and analogous constants of a substance instead of atomic increments—bond increments. For fatty, as well as for aromatic hydrocarbons, A. I. showed that the volume corresponding to one bond \((\mathrm{C—H}\) or \(\mathrm{C—C})\) is, with considerable accuracy, the same, and that the corresponding increment is equal to 7.37. The same number was obtained for analogous bonds in haloid derivatives of hydrocarbons, whereas for the \(\mathrm{C—Cl}\) bond the increment proved to be 25.2, for the \(\mathrm{C—Br}\) bond 31.1, etc. On the basis of the bond increments established by him, A. I. calculated the molar volumes at the boiling point, the critical volumes, etc., for a large number of compounds, and very good agreement with the values observed experimentally was obtained.

A. I. devoted considerable attention to the study of the relation between the pressure of saturated vapors of a substance and temperature and, over a number of years, published several works on this question. In one of them he showed that the well-known Nernst formula, obtained by a peculiar integration of the Clapeyron equation, does not satisfy the experimental data over a wide interval of temperatures and pressures. Thus, on the curve \(p=f(T)\), according to Nernst, there is a maximum, whereas the results of experiments give a steeply rising curve which breaks off at the critical point. In 1927, in a report at the 3rd All-Union Thermotechnical Congress, A. I. proposed the following equation:

\[ \sqrt[4]{p}=a+bt e^{-\frac{\beta}{\ln \frac{p_{\mathrm{cr}}}{p}}}, \tag{6} \]

which represents the Jarolimek formula supplemented, on the basis of a careful study of experimental data, by an exponential term. Equation (6), in which \(a\), \(b\), and \(\beta\) are constants, has a rather complicated form, but it expresses very accurately the dependence of the saturated vapor pressure \(p\) on the temperature \(t\).

The best known among A. I. Bachinsky’s works are his investigations in the field of the viscosity of liquids. It is no accident that he chose as the epigraph to his principal work (Investigation of the Internal Friction of Liquids, 1913) the following words of D. I. Mendeleev[^4]: “The connection that exists between viscosity and other physical and chemical properties leads one to think that the magnitude of internal friction will play an important role in molecular mechanics.” The words of D. I. Mendeleev have now been fully justified. Every investigator concerned with questions of the molecular-kinetic theory of liquids first of all tests theoretical conclusions on the regularities expressing the dependence of the viscosity of liquids on various physical and chemical parameters (temperature, pressure, chemical structure, etc.). Enormous credit for the correct formulation of work in this direction and for its broad development belongs to A. I., who was the first to approach the consideration of this problem of microrheology in a fully scientific manner.

At the meeting of the University of Physicochemistry named after Academician N. D. Zelinsky in 1938, devoted to the twenty-fifth anniversary of Bachinsky’s law of viscosity, I. A. Kablukov, N. D. Zelinsky, and other speakers gave a very high assessment of these works of A. I. Bachinsky[^5]. Being interested in the theory of the liquid state of matter, A. I. noted that viscosity is the most characteristic property of liquids. Its changes with temperature and pressure are many times greater than for other properties of matter. Over a sufficiently broad interval of temperatures and pressures the viscosity of liquids \(\eta\) changes by hundreds, thousands, tens of thousands of times and more. Before the appearance of A. I.’s works there existed interpolation formulas \(\eta=f(t)\), having in the denominator one or another power series; however, they did not reproduce sufficiently well the most accurate experimental data then known, those of Thorpe and Rodger[^6].

A. I. proposed expressing the viscosity of liquids as a function of the specific volume \(v\), which in turn depends on temperature and pressure. The basis for this was his correct, from the point of view of rheology, conception that the mechanism of the internal friction of liquids differs essentially from that for gases. Bearing in mind that the viscosity of liquids is determined by the character of molecular interaction, and that the latter depends on the distances between molecules, A. I. made the entirely correct conclusion that viscosity is a function of the density of the liquid or of its specific volume. Thus, in 1912 there was established the formula which is now called Bachinsky’s law of viscosity, namely:

\[ \eta=\frac{c}{v-\omega}. \tag{7} \]

In this equation \(c\) and \(\omega\) are constants, and it is noteworthy that \(\omega\) in a number of cases is close to the constant \(b\) in the van der Waals equation.

A. I. called the quantity \(\omega\) the “limiting volume,” and the difference \(v-\omega\) the “free volume” of the liquid. In particular, A. I. established that the value of \(\omega\) for many substances amounts to \(0.3\) of the critical volume.

In his very thorough classical work, published in 1913 in the Vremennik of the Scientific Society named after Ledentsov, A. I. carried out a most detailed verification of equation (7) on the extensive experimental material of Thorpe and Rodger and other authors. Later repeated verification on new experimental data confirmed his conclusion that formula (7) agrees very well with experiments for all non-associated liquids. If equation (7) is valid, then on graphs \(v=f\left(\dfrac{1}{\eta}\right)\), where \(\dfrac{1}{\eta}\) is the fluidity of the liquid, straight lines should be obtained. Considering the numerous graphs of this type presented in A. I.’s work, one cannot but be surprised at the high degree of accuracy with which this regularity is justified. From several dozen tables calculated by A. I., it is evident that the discrepancies between the results of measurements and formula (7) usually amount to several hundredths or tenths of a percent, and only in very rare cases exceed \(1\%\).

Further, A. I. showed that the molecular limiting volume \(M\omega\), established on the basis of formula (7), where \(M\) is the molecular weight, possesses an additive property. At the same time, in accordance with what had been set forth somewhat earlier, he also took into account bond increments. In his detailed work (1913) he also considered the change of viscosity as a function of volume connected with an increase in pressure, the relation of the quantity \(c\) to the parameters of the van der Waals equation of state, etc.

On questions connected with Bachinsky’s law of viscosity, very many works have been published over the past three decades, both in our country, in the USSR, and abroad. Bingham devoted great attention to Bachinsky’s law in his interesting book devoted to the problems of rheology, i.e. the fluidity and plasticity of substances\(^7\). He quite correctly assessed the role which equation (7) was to play in the theory of the viscosity of liquids. Bingham repeatedly returned to the consideration of Bachinsky’s law, applying it in studies of the structure of liquids and their association\(^8\). He introduced a correction term into equation (7), which made it possible to calculate the limiting volume more accurately.

On the other hand, Hatschek, in the book The Viscosity of Liquids\(^{10}\), incorrectly presented the state of the question. Having devoted a little attention to Bachinsky’s law, he dwelt in detail on a consideration of MacLeod’s formula\(^ {11}\). The latter, in 1923, i.e. 10 years later than A. I., proposed a formula analogous to Bachinsky’s law. MacLeod’s formula is obtained from equation (7) if, instead of the specific

of the volume \(v\) to substitute into it a polynomial expressing the expansion of the liquid as a function of temperature. A. I. wrote precisely about this in his work of 1913. He, however, did not use such an expression, rightly considering that, being considerably more complicated than formula (7), it would not introduce anything new. Indeed, Mak-Leod’s further derivations are essentially a repetition of A. I.’s work. It is curious that equation (1), established by A. I. for the relation between the surface tension of a liquid and its density, was likewise again “discovered” by Mak-Leod and published in the journal of the Faraday Society two years after A. I.’s work had appeared in print.

In fact, Mak-Leod wrote his formula for viscosity in the form \(^{11}\):

\[ \eta = \frac{c}{(v-b)^n}, \tag{8} \]

where \(n\) is a quantity characterizing the degree of association of liquids. However, for nonassociated liquids he took, in accordance with formula (7), \(n = 1\).

In view of the fact that Batschinsky’s law of viscosity agrees excellently with experimental data, a large number of works have been published devoted to its theoretical justification \(^{10,12}\). A. S. Predvoditelev \(^{13}\) gave a derivation of formula (7), proceeding from the conception of fluctuations and taking as his basis the usual equations of hydrodynamics, modified in connection with Maxwell’s relaxation theory. M. F. Shirokov \(^{14}\) took another path and derived Batschinsky’s law from molecular-kinetic considerations, introducing into the equation for the viscosity of gases corrections analogous to van der Waals’ corrections to the equation of state of ideal gases. A theoretical derivation of formula (7) is also found in the works of Herzog and Kudár \(^{15}\), Goldhammer \(^{16}\), and others.

Ya. I. Frenkel \(^{17}\) compared equation (7) with the exponential formula derived by him, expressing the dependence of the viscosity of a liquid on temperature. In doing so, he obtained a number of interesting conclusions; in particular, he showed (earlier than Powell and Eyring \(^{9}\)) that the conceptions developed by A. I. in 1913 correspond to modern views on the nature of the liquid state of matter, namely to the so-called “hole” theory of liquids. P. P. Lazarev \(^{18}\) compared Batschinsky’s law with the equation proposed by Le Chatelier for expressing the dependence of the viscosity of molten glasses on temperature. It was thereby found that agreement is obtained only within a certain temperature interval, and that thereafter a divergence is observed.

A further application of Batschinsky’s law in the region of high temperatures is found in the works of M. P. Volarovich. It was established \(^{19}\) that the experimental values of the viscosity obtained by Dantuma for potassium nitrate in the temperature range \(350\text{--}540^\circ\mathrm{C}\) and for sodium chloride at temperatures \(820\text{--}1000^\circ\mathrm{C}\) are well satisfied

equation (7). Later, Ipatov \(^{20}\), repeating this work, came to the same conclusion. M. P. Volarovich \(^{21}\) undertook special investigations of the density, at high temperatures up to \(1300^\circ\) C, of a number of molten salts and silicates solidifying in the form of glasses, and showed that in a certain temperature interval, where the viscosity is comparatively small, Batschinski’s law is justified. With an increase in viscosity as the temperature is lowered, these glassy substances exhibit deviations from equation (7). Batschinski’s law revealed the same regularity when applied to molten fats \(^{22}\).

Later A. I. developed his formula for the case of the viscosity of binary liquid mixtures (1920). In doing so he showed that in the absence of compression upon mixing, i.e., for “ideal” mixtures, the free volume \(v - \omega\) is an additive quantity. If, however, compression of volume is observed upon mixing, then, according to A. I., a correction term accounting for the decrease in volume should be introduced into the formula for the viscosity of mixtures. Independently of A. I. Batschinski, a similar equation for the viscosity of binary liquid mixtures was proposed somewhat later by Meyer and Milius \(^{23}\), who also carried out an experimental verification of this equation. A. I.’s formula for the viscosity of mixtures was widely used by G. P. Luchinskii \(^{24}\), Kotler \(^{25}\), and others.

In the present article we have been able to touch only very briefly upon some of the principal works of A. I. Batschinski. However, from the foregoing it is clear what great interest A. I.’s investigations represent, and what broad application they have found in the works of numerous authors. It is clear that not only have they not lost their significance at the present time, but, on the contrary, in connection with the development of the theory of the liquid state of matter they are attracting ever more and more attention from researchers and arousing ever greater interest.

Special mention should be made of the excellent textbooks written by A. I. Batschinski for secondary school, from which our schoolchildren studied for many years. A. I. was an outstanding teacher. All who heard his lectures remember well the impeccable form of exposition, as well as the clarity and precision with which he conveyed the content of lectures and reports. A. I. always showed great interest in the work of beginners, young scientists. Many scientific workers often turned to him for advice and consultations, and always met with the warmest reception from A. I. and received answers to the questions that interested them. All who met A. I. Batschinski will preserve the brightest memory of him.

In conclusion I wish to express my deep gratitude to A. I. Batschinski’s daughter, Olga Alekseevna Batschinskaia, as well as to K. A. Putilov, for giving me the opportunity to use A. I.’s archive in preparing this article.

Cited Literature

1. P. A. Rebinder, Modern Problems of Physical Chemistry and Chemical Technology, Coll. 2, 133 (1938).
2. K. A. Putylov, Idem, p. 141.
3. P. W. Bridgman, The Physics of High Pressure, trans. from English, ONTI, Moscow—Leningrad, 1935.
4. D. I. Mendeleev, Principles of Chemistry, 7th ed., p. 209, 1903.
5. Modern Problems of Physical Chemistry and Chemical Technology, Coll. 2, 109 (1938).
6. T. E. Thorpe and J. W. Rodger, Trans. Roy. Soc., A 135, 397 (1894); A 189, 71 (1897).
7. E. C. Bingham, Fluidity and Plasticity, New York, 1922.
8. E. C. Bingham and H. J. Farnwalt, Journ. of Rheol. 1, No. 4, 372 (1930); E. C. Bingham and P. W. Kinney, Journ. Appl. Phys., 2, No. 3, 192 (1940).
9. E. R. Pouehl and G. Eyring, J. Phys. Chem. 27, 278 (1945).
10. E. Hatschek, Viscosity of Liquids, trans. from English with suppl. by M. P. Volarovich, 2nd ed., ONTI, Moscow—Leningrad (1935).
11. D. B. Mac Leod, Trans. Farad. Soc., 19, 6 (1923).
12. B. V. Bak, J. Phys. Chem. 15, 1902 (1935).
13. A. Predwoditelew, Zschr. f. Phys. 49, 279 (1928).
14. M. F. Shirokov, Journ. Phys. Chem. 3, 173 (1932); Journ. Exper. and Theor. Phys. 3, 237 (1933); Proceedings of the Conf. on the Viscosity of Liquids and Colloids, 1, 25 (1941).
15. R. O. Herzog and H. C. Kudar, Zschr. f. Phys., 80, 217 (1933); 83, 28 (1933); Physik. Zschr. 35, 437 (1934).
16. A. D. Goldgammer, DAN 3, No. 7, 484 (1934).
17. Ya. I. Frenkel, Proceedings of the Conf. on the Viscosity of Liquids and Colloids, 1, 11 (1941); Kinetic Theory of Liquids, Publ. House Acad. Sci. USSR, 1945, p. 192.
18. P. P. Lazarev, DAN, A, No. 3, 27 (1927).
19. M. P. Volarovich, Izv. Acad. Sci. USSR; Dept. of Mathematical and Natural Sciences, p. 1431 (1933).
20. Ipat’ev, Modern Problems of Physical Chemistry and Chemical Technology, Coll. 2, 118 (1938).
21. M. P. Volarovich, Izv. Acad. Sci. USSR, Dept. of Mathematical and Natural Sciences, p. 663 (1933); M. P. Volarovich and A. A. Leont’eva, Dokl. Acad. Sci. USSR 2, No. 8—9, 535 (1935).
22. M. R. Wolagowitsch and G. B. Rawitsch, Koll. Zschr. 73, 339 (1935); G. B. Ravnich, Colloid Journal 5, 13 (1939).
23. J. Meyer and Milian, Zschr. phys. Chem., 95, 349 (1920).
24. G. P. Luchinskii, Journ. Phys. Chem. 6, 700 (1935); 8, No. 6 (1936); G. P. Luchinskii and A. I. Likhacheva, Idem 7, 546 (1936).
25. G. Kottler, Journ. phys. Chem. 47, 277 (1943); 48, 77 (1944); Rheology Bull. 16, No. 1, 20 (1945).

List of Scientific Works of A. I. Bachinskii

1. On the dynamical theory of electricity, Proceedings of the Department of Physical Sciences of the Moscow Society of Lovers of Natural Science, Anthropology, and Ethnography (MOLEAE), 10 (1900).
2. On the law of variation of the viscosity of mercury with temperature, Proceedings of the Department of Physical Sciences (MOLEAE), 10 (1900).
3. On the dependence of the internal friction of liquids on their chemical nature. Supplement to the Protocols of the Moscow Society of Naturalists (MOIP), No. 7 (1900).
4. Studien zur Kenntnis der Abhängigkeit der Viskosität der flüssigen Körper von der Temperatur und ihrer chemischen Konstitution, Abh. 1, Bull. de la Soc. des Nat. de Moscou, No. 1 (1901).

M. P. VOLAROVICH

1. On the relation between the viscosity parameter and certain other physical constants, Zschr. phys. Chemie 37 (1901).

2. On Maxwell’s law \(K = n^2\), in relation to the theory of the molecular structure of bodies, Zschr. phys. Chemie 38, 119 (1901).

3. Studies on the dependence of liquid bodies on temperature and on their chemical constitution, Part II, Bull. de la Soc. des Nat. de Moscou, No. 3 (1902).

4. On an extension of the concept of critical quantities, Zschr. phys. Chemie 40, No. 5, 629 (1902).

5. An attempt at a physical interpretation of the periodic law of the chemical elements, Proceedings of the Moscow Society of Naturalists, No. 2 (1902).

6. Note on the law of the straight middle line, Zschr. phys. Chemie 41, 741 (1902).

7. On the relation between the internal-friction parameter and certain other physical constants. Journal of the Physico-Chemical Society at St. Petersburg University 33 (1901).

8. On the relation between the heat of vaporization and the critical quantities, Zschr. phys. Chemie 43, No. 3, 369 (1903).

9. On the polyresistance of orthomeric liquids, in particular of acetic acid, Bull. de la Soc. des Nat. de Moscou, No. 2–3, 188 (1903).

10. The speaking flame (with Gabritschewski), Ann. der Phys. 11, 223 (1903).

11. On the speaking flame. Reply to Mr. Ruhmer, Ann. der Phys. 12, 1169 (1903); Zschr. phys. Chemie 47, 743 (1904).

12. Relations for the chemical properties of substances, Ann. der Phys. 14, 288 (1904).

13. Treatise on the equation of state, Parts I–II, Ann. d. Phys. 19, 307 (1906).

14. Treatise on the equation of state, Part III, Ann. d. Phys. 21, 1001 (1906).

15. On the conditions of sensitivity of balances, Fiz. obozr. 11, 1910; Zschr. phys. Chem., Unterricht. 24, issue 1, 24 (1911).

16. On the determination of the degree of molecular association of liquids, Zschr. phys. Chemie 75 (1911).

17. The old and the new in physics (speech), Russkaya mysl’ (December 1911).

18. In memory of M. V. Lomonosov, Russkaya mysl’, 1911.

19. a) A precursor of modern science; b) Lomonosov, Russkoe slovo, No. 257 (1911).

20. Lomonosov’s activity and the significance of his works, Vremennik of the Society for Promoting the Advancement of Experimental Sciences named after Ledentsov (1912).

21. The law of the viscosity of liquids, Phys. Zschr. 13, 1157 (1912).

22. The molecular association of liquids I–II, Zschr. phys. Chemie 82 (1913).

23. Investigations on the internal friction of liquids II, Zschr. phys. Chemie 84, 643 (1913).

24. Investigations on the internal friction of liquids, Vremennik of the Society named after Ledentsov, Supplement No. 3 (1913).

25. Work on load lifting, journal Fizika (1913).

26. The internal friction of liquids, systematically presented. Supplement to the minutes of the meetings of the Moscow Society of Lovers of Natural Science, Anthropology and Ethnography, No. 3 (1913).

27. Viscosity and the molecular structure of liquids, Contemporary Problems of Electromagnetism, p. 27 (1913).

28. The needs of the physics teacher in the field of mathematics, journal Mathematics Education, No. 2 (1914).

29. On one geometric theorem, journal Mathematics Education (1914).

30. Biographical sketch of N. A. Umov, Vremennik of the Society named after Ledentsov (1915).

31. N. A. Umov, Priroda, 1915.

32. A remarkable Russian scientist, Russkaya mysl’ (1915).

33. Sketch of the life and works of N. A. Umov, Report of Moscow University, Part 1 (1915).

1. Umov’s characteristics as a scientist, thinker, and person, Proceedings of the Moscow Society of Naturalists, 1915; Bull. de la Soc. des Nat. de Moscou (1916).
2. Do saturated vapors obey Mariotte’s law, journal Fizika, Nos. 3–4, 18 (1916).
3. Wilhelm Weber—the creator of electronic theory, Priroda (1916).
4. Preface and notes to the Collected Works of N. A. Umov, Moscow (1916).
5. Scientific and literary activity of P. I. Vyazemsky, Proceedings of the Karadag Scientific Station (1917).
6. The fideism of Prof. Khvolson, Philosophical Yearbook (1917).
7. Molecular fields and their volumes, Izv. Akad. Nauk (1918).
8. Note on the occasion of the 300th anniversary of Kepler’s discovery of the third law of planetary motion, Uspekhi fiz. nauk (1918).
9. On the viscosity of binary liquid mixtures, Izv. Fiz. instit. Mosk. nauchn. instit. (1921).
10. On formulas of surface tension, Izv. Fiz. instit. Mosk. nauchn. instit., p. 60 (1922).
11. From the history of Russian science, Uspekhi fiz. nauk (1923).
12. On the orthometric curve (jointly with Zharkov), Zhurn. fiz.-khim. obshch., p. 175 (1924).
13. On the principles of mechanics, Collection of articles on questions of the physico-mathematical sciences (1924).
14. From the history of theories of light, Collection of articles on questions of the physico-mathematical sciences (1924).
15. On the approximate replacement of a sinusoid by a hyperbola, Collection of articles on questions of the physico-mathematical sciences (1924).
16. Formula for the elasticity of vapors (jointly with A. Ya. Modestov), Zhurn. prikl. fiz., 5, issue 2, 177 (1926).
17. Bemerkung über die Differenz \(C_p - C_v\), Zschr. f. Phys. 45, H. 11/12, 892 (1927).
18. Isaac Newton, Narodnyi uchitel’ (1927).
19. A new formula for the elasticity of vapors, Proceedings of the Third All-Union Heat-Engineering Congress (1927); Nature (1927).
20. Physico-technical equipment of the school and methods of using it (jointly with Pavshei), Pedagogical Encyclopedia (1927).
21. On the dependence of the pressure of saturated vapors on temperature (jointly with A. Ya. Modestov), Zhurn. prikl. fiz. 5, issue, no. 81 (1928).
22. How great is the air pressure on the human body, Iskra (1928).
23. Preface and notes to the translation of Andrews’s work “On the Continuity of the Gaseous and Liquid States of Matter,” M.—L. (1933).
24. Preliminary communications and discussions concerning Luchinsky’s article, Zhurn. fiz. khim. 11, issue 4, 557 (1938).
25. Twenty-five years of the law of viscosity. Sovrem. probl. fiz.-khim. i khim. tekhnologii, Univ. fiz.-khim. i khim.-tekhn. im. akad. N. D. Zelinskogo, collection 2, 113 (1938).
26. Articles on physics in the Granat Encyclopedic Dictionary (atom, weights, galvanism, and others).
27. Some prospects of the theory of the viscosity of liquids, Report at the conference on viscous liquids and colloids at the Institute of Mechanical Engineering, Academy of Sciences of the USSR, 15/V 1941, Proceedings of the Conference, vol. 2, 104 (1944).

BOOKS BY A. I. BACHINSKY

1. Introduction to the kinetic theory of gases. A course of lectures delivered at Moscow State University during the autumn semester of 1907, Moscow, 1908.
2. The doctrine of forces and motion (with a preface by N. E. Zhukovsky), Moscow, published by the Sytin partnership, 1914; 2nd ed. 1918; 3rd ed. under the title Introduction to Theoretical Mechanics. The Doctrine of Forces and Motion, M.—L., Gosizdat, 1924.
3. Physics for secondary educational institutions, issues 1, 2, 3, published by the Sytin partnership, Moscow, 1915–1917; 4th ed., issue 1, 1919.

1. Abridged Physics for Use in the Second-Stage School, Moscow, Gosizdat, 1922; 5th ed., 1924.

2. Electricity and Magnetism, Moscow, Gosizdat, 1922; 2nd ed., 1923.

3. Collection of Questions and Problems in Elementary Physics, Moscow, Gosizdat, 1923; 2nd ed., 1923; 3rd–6th eds., 1925–1928.

4. Textbook of Physics on a Production Basis; pt. 1—mechanics, heat; pt. 2—oscillations, optics, electricity; Moscow, Gosizdat, 1925; 5th ed., 1927.

5. Physics in Three Books (a textbook for schools), books 1–3, Moscow–Leningrad, Gosizdat, 1925–1926; 10th ed., 1931.

6. Physico-Technical Reference Tables, Moscow–Leningrad, Gosizdat, 1926; 2nd ed., under the title Reference Tables in Physics (jointly with K. A. Putilov), Moscow, State Educational-Pedagogical Publishing House, 1934.

7. Dictionary-Handbook of Physics, Moscow, “Rab. prosveshcheniya” Publishing House, 1928.

8. Workbook in Physics, Compiled According to the Program of the Scientific-Pedagogical Section of GUS, issues 1–3, Moscow–Leningrad, Gosizdat, 1927; 4th ed., 1929.

9. Editing, translated from German: E. Grimsehl, Physics, pts. 1–4; eds. 1–3, 1927–1938.

10. New Workbook in Physics, for the Labor School, 5th ed., issues 1–3, Moscow, Gosizdat, 1930.

11. Physics (jointly with K. A. Putilov), Moscow, 1930.

12. Physics for Engineering-Economic Higher Educational Institutions, pt. 1 (jointly with A. Ya. Modestov), Moscow–Leningrad, State Publishing House of Technical-Theoretical Literature, 1932.

13. Physics for Engineering-Economic Higher Educational Institutions; Elements of Thermodynamics (based on material from the physics course by A. I. Bachinsky and A. Ya. Modestov for engineering-economic higher educational institutions), Moscow, 1938.

14. Handbook of Physics (jointly with K. A. Putilov and N. P. Suvorov), Moscow, Uchpedgiz, 1941.

15. Physics, a Textbook for Vocational and Railway Schools (jointly with S. M. Ilyashenko), Moscow–Leningrad, State Publishing House of Technical-Theoretical Literature, 1941.

16. Physics, a Textbook for Self-Education (jointly with S. M. Ilyashenko), 2nd stereotype ed., Moscow–Leningrad, 1945.