home Physical laws and equations Boyle's equation. Boyle-Mariotte's law. Gas laws. Isotherm

Boyle's equation. Boyle-Mariotte's law. Gas laws. Isotherm

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Boyle-Mariotte lawlinking changes in the volume of gas at constant temperature with changes in its elasticity. This law, discovered in 1660 English. physicist Boyle and later, but, independently of him, Mariotte in France, in its simplicity and certainty occupies a very important place in science, although later studies showed the existence of deviations from it and that the law actually refers to the so-called ideal gas. The history of its discovery is very instructive. Franciscus Linus, professor of mathematics at Luttich (1595-1675), did not admit that air, such a mobile and light substance, could support a column of mercury in a barometric tube, although Galileo's student Evangelist Torricelli (1608-1647) undoubtedly proved that that it is the pressure of the atmosphere that is the cause of this phenomenon. Until that time, everyone assumed that nature does not tolerate emptiness (horror vacui) and that therefore mercury, water and all other liquids rush into empty tubes. When it turned out that the water in the tube follows the pump piston only up to a height of slightly more than 30 feet, Galileo assumed that the fear of emptiness had a limit. Lin explained that mercury is held in the tube by invisible threads (funiculus) and that he himself felt these threads when he closed the upper opening of the tube with his finger, which was then filled with mercury and turned over with its lower end into a cup with mercury; at the same time, the mercury in a sufficiently long tube descended, but stopped at known height... This interpretation of the experiment by Torricelli by Lin prompted Boyle to make several new experiments, which he described in his "A defense of the doctrine touching spring and weight of the air" (London, 1662). To prove that air has the ability to resist, Boyle took a siphon-shaped tube sealed at the short end (Fig. 1). When mercury was poured into the long knee, it compressed the air contained in the short knee, the more the more mercury was poured in the other. When the mercury in the short knee reached the AB level, in the long one it was at the CD level, which means that the elasticity of the compressed air was such that it could maintain the pressure of the mercury column with a height from AB to CD. And since this height in the first experiments of B. was equal to the height of mercury in the barometer, this proved that the mercury column in the barometer was supported by atmospheric air. Pouring various, more and more large quantities of mercury into the long elbow of the tube, B. recorded the heights of the mercury column and the corresponding volumes of compressed air, but at first did not pay attention to their numerical ratios. His student Richard Townley, looking at the numbers on the table, noticed that the volumes of trapped air are inversely proportional to the pressures produced on it. If the air first occupied 12 inches in length in the tube, and the mercury in both knees was at the same height, then when so much mercury was poured into the long knee that the air occupied only 6 inches in length, it turned out that the height of the supported column of mercury was 29 Eng. inches. At first, the air trapped in the short knee had the same elasticity as the atmosphere, which could support the mercury in the barometer at 29 inches in height, and in the second case, the trapped air was subjected to atmospheric pressure and the pressure of the mercury column at 29 inches, i.e., in total - pressure 29 x 2 inches: which means that when the volume of air was half, its elasticity became twice as much. After that, Boyle repeated and diversified the experiments many times and proved that the same law applies to cases of increasing air volume.

For this he used a cylindrical vessel (Fig. 2), which was filled with mercury; immersing tube A with open ends there until a part of AB, equal to 1 inch in length, remained above the mercury, B. closed and sealed hole A and then lifted the tube. At the same time, the volume of AB increased and finally turned into the volume of AD - twice as large; the mercury rose to a height of B "D, which was almost half, 29 р", the then height of mercury in the barometer. Obviously, the air contained in DA did not have sufficient elasticity to press on the surface of D with such force as it did before pressed on B; the difference in elasticity in both positions of the tube is measured by the column DB ", which is 15⅜" long. Therefore, the elasticity of air in the doubled volume AD is 29¾ without 15⅜, that is, 14⅜ or almost exactly half of the former. When the tube was lifted so much that the volume AD took up a length of 10 inches, the height of the mercury DB "was 26¾, therefore the air elasticity was measured by a difference of 29¾-26¾, that is, 3 inches, which is almost exactly 1/10 of the original elasticity A description of these experiments is found in New Experiments touching the spring of the air (Oxf. 1660); Continuation of Experiments (Oxf. 1669), On the rarefaction of air (London, 1671); Second continuation "(London, 1681)," General history of the air "(London, 1692). The French scientist Mariotte (Edme Mariotte, 1620-1684) made a series of experiments in exactly the same way and found the same law, which is usually called his name; only the English call it Boyle's law See the works of Mariotte: "Essay sur la nature de l'Air" (Paris, 1676), "Du mouvement des eaux et des autres fluides" (part. II, disc. 2). Did Mariotte know about Boyle's experiments - it is impossible to give a positive answer to this, although it is known that Marriott was in relations with England Liysk scientists already in 1668. Be that as it may, Marriott made the same experiments and measurements as Boyle, only with greater precision, and his experiments became better known. The tube (Fig. 1) received, as a device, the name Mariotte, and the law is named after him, although for some time it has been justly called the Boyle-Mariotte law; perhaps it would be even fairer to add Townley's name. In any case, Marriott is so famous for his other works that, despite the evidence of numbers, it is difficult to suspect him of the lack of independence of the works that led to reopening important physical law. The history of physics shows that very important laws discovered in one country may have been unknown for a long time in another; thus, an important law concerning the strength of a galvanic current, discovered by Ohm in Germany, was rediscovered a few years later in France by the physicist Poulier.

With a decrease in the volume of air by two or three times, its density must also increase in the same ratio; the temperature of the gas when measuring its volume must be constant, otherwise its cooling or heating itself can change the volume and elasticity; in addition, the air must not contain water or other liquids. Subject to all these conditions, the Boyle-Mariotte law should be expressed as follows: the volumes of a certain amount of dry air at a constant temperature are inversely proportional to the pressures produced on it, and, consequently, to its elasticities, the density of the air is directly proportional to this pressure; or, in short, the volume of air is inversely proportional to the pressure produced on it. If we denote the initial volume of the gas by the letter v, and the pressure under which it is located, by the letter p, if the compressed volume of the gas is v ", and the pressure, always measured by the height of the mercury column, will be p"; then the law of B.-M. expressed by the proportion: v: v "\u003d p": p; whence pv \u003d p "v", ie the product of the gas volume and the corresponding pressure is a constant value at a constant temperature. Other gases, as will be explained later, follow the same law. No matter how simple the experiments of Boyle and Mariotte seem, however, even with the small degree of accuracy of the device design that was available at that time, they required many experimental precautions. Failure to follow the proper rules was probably the cause of the various conflicting statements by later observers. For example, Bez observed under the equator in his experiments a decrease in the volume of air in a smaller ratio than an increase in its elasticity. Bugar's numerous experiments in the same latitudes, on the contrary, confirmed the law of B.-M .; in addition, the experiments of Amonton, Sgravesand, Fontana, Shukburg led to the same conclusion.

But all the experiments of that time did not reach high pressures and were not so accurate that there was no doubt about the correctness of the law. Sulzer ("Mém. De Berlin", v. IX, 1753), and then Robison, concluded from their experiments that at pressures 7 or 8 times higher than atmospheric, the elasticity increases in a much smaller ratio than the volume decreases; but the experiments of Winkler (1765) again prove the applicability of the B.-M. up to 8 atmosphere pressure. In the present century (1826) Danish scientists Oerstedt and Svensen once again confirmed the correctness of the law up to 8 atm. pressure; their other experiments, extending up to 70 atmospheres, were done in a less credible method. But even within these close limits (up to 8 atm), some gases do not follow the B.-M. In the second half of the XVIII table. Van Marum made sure that ammonia gas decreases in volume much faster than air; similar to that, Oerstedt and Svendsen found sulfurous acid for the gas much later. In addition, it was discovered that both gases pass into a liquid state at a slightly higher pressure; this property was later proved for other gases as well. Despretz, with even more precise experiments ("Ann. De Chim. Et de phys.", 2, XXXIV, 1827), made sure that many gases do not follow the B.-M. even at pressures that are far from those at which gases liquefy. Despres made experiments in a manner similar to that first used by Van Marum. Two glass tubes sealed at one end, of which one was filled with air and the other with another gas, were immersed with their open ends in a bath filled with mercury, placed at the bottom of a glass cylinder filled with water. Pressure was applied to the water by means of a piston placed in the upper bottom of the cylinder, the water pressed on the mercury, which, entering the tubes, compressed the gases. The experiments made with such a device led Despres to the conclusion that ammonia, sulfurous, hydrogen sulphide and synergistic gases at the same pressure value occupy a smaller volume than air. The measurement accuracy was so great that the difference between the compression of these gases and air was noticeable even when the volume of the latter was reduced by only half; the volumes of the named gases were less than half of the initial volume. According to the experiments of Despres, hydrogen gas is compressed in the same way as air to 1/15 of the original volume, but at twenty atmospheres of pressure the volume of hydrogen was more than the corresponding volume of air. Dulong and Arago (Mémoires de l'Académie des Sciences, vol. X, Annales de Chim. et de Phys. ", vol. XLIII, 1830) measured the compression of air to 27 atmospheres of pressure; their device consisted of a tube 1.7 m long, in which the air was compressed, and connected to it another, composed of 13 parts, each 2 meters long. This long, composite tube was attached to a wooden mast mounted inside a tall tower. Dulong and Arago found that B.-M. true for air even when compressed to 1/24 of its original volume. Later, the French physicist Poulier did experiments in a manner similar to that used by Oerstedt and Despres, but at high pressures, and concluded that oxygen, nitrogen, hydrogen, carbon monoxide and nitric oxide follow the same compression law up to 100 atmospheres as air, but that the six gases named below are compressed more than air and that the difference between their volumes and the air volume increases with increasing pressure. These gases are: sulfurous acid, ammonia, carbon dioxide, nitrous oxide, oil and bog gases.

In 1847, Regnault's extensive and precise research on this subject was published (Mémoires de l'Académie des sciences de Paris, XXI, 1847), which, together with other physical works performed on behalf of the French government, are described in these memoirs under the title "Relation des expériences entreprises par ordre de M. le ministre des travaux publics etc". Taking advantage of the improvements in instruments and methods of observation introduced by his predecessors, Regnault added significant new improvements, eliminating the main difficulty in the accuracy of measuring gradually decreasing volumes of gas. No matter how significant the length of the tube in which the gas was compressed in the experiments of Arago and Dulong (1.7 meters), nevertheless, at high pressures, the volume of the gas became very small, and then every small inaccuracy in measuring the position of the mercury blocking the gas becomes more and more and more perceptible relative to the constantly decreasing volume being measured. Regnault used in his experiments a tube of 3 meters in length for compressing gases, and after measuring the total volume of the gas and then compressed to half the volume at a certain corresponding pressure, he again pumped gas into this tube until it was completely filled. The thus obtained again a large volume of gas under pressure, b aboutlighter than the original one, it was brought back to half the volume by increasing the height of the mercury column in the long tube. Using this method, Regnault at very high pressures (for 25 atmospheres for air) always measured large volumes; in addition, he took into account many other experimental precautions that ensured the accuracy of his conclusions. Regnault's experiments proved that the important law of nature, indicated by Boyle and Mariotte, is not formulated mathematically exactly by those simple relationshipwhich they gave him that the compression or decrease in the volume of air and nitrogen occurs in a somewhat greater ratio than the increase in pressure on the gas or than the elasticity of the latter, and that for hydrogen the compression, on the contrary, is somewhat weaker than would be expected in the case of exact applicability to to him the law B.-M. Several numbers taken from Regnault's memoirs, placed in the next tablet, show that the observed deviations are generally small, but clearly increase with increasing pressure. The first two columns of the table show the heights of the mercury column pressing on the gas, expressed in atmospheres (for Regnault, in millimeters), and the height of 760 million mercury is taken as a measure of the normal atmospheric pressure. The numbers in the third column show quotients obtained from dividing the ratio of the initial gas volume to the volume reduced by compression by the ratio last pressure to the original. If we call the letters v, v 1 the volumes of the initial and reduced gas, and the letters. p and p 1 are the corresponding pressures on the gas, then according to the B.-M. should be: v: v 1 \u003d p 1: p, hence (v: v 1): (p 1: p) \u003d 1, that is, if both written relations are really equal, then the quotient from dividing one relation into another should be equal to 1. But the numbers in the third column are more and more 1 and slowly but constantly increase:

Any number in the third column shows the quotient relating to a halving of the air volume when the pressure changes from p (number of the first column) to p 1 (second column). From these numbers it can be seen that the decrease in the volume of air occurs in a greater ratio than the increase in the corresponding pressure or elasticity of the gas. At first, both relationships differ little from each other, but when moving from 12 atm. by 24, the decrease in volume is 1.006366 times greater than the increase in pressure. A little calculation allows us to conclude that 10,000 cubic meters. sant. air at a pressure of 0.972 atm, being subjected to a pressure of 24.9 times greater, will occupy a volume of 396 cubic meters. sant. instead of 401 k. s., as it would follow, if the law of B.-M. accurately expressed the law of nature.

Compression of nitrogen represents the same, but slightly smaller deviations from the B.-M. law, and since atmospheric air consists of oxygen and nitrogen, Regnault concluded that oxygen is compressed more than nitrogen and air. The next plate contains the numbers obtained in experiments I with hydrogen; the column numbers have the same meaning as in Table A.

Since all the numbers in the third column are less than one and are constantly decreasing, the volume of compressed hydrogen is constantly more than it would follow according to the B.-M. law, and this deviation increases with increasing pressure. Like Regnault, hydrogen compresses like a spring, less and less as the pressure increases. As for carbon dioxide , which is comparatively easily compressing, which, like air, represents a faster decrease in volume than an increase in elasticity, then it deviates from the law even at relatively low pressures at ordinary temperature, but, being heated to the boiling point of water (100 ° C.), shows much smaller deviations. If it should be concluded from the extremely accurate experiments of Regnault that the law of B.-M. with very imperceptible deviations, it is applied only to some gases at pressures far from the liquefaction point, and at a significantly high temperature, the study of the issue is not limited to these results. Boyle's and Regnault's experiments are separated by a time interval of nearly 200 years. The properties of gases were studied in many respects during this period of time, the list of liquefying gases was constantly increasing, and a few years ago, the works of Pictet and Cailletet made the final generalization that with a decrease in the volume of gases through pressure and with a decrease in their temperature, they all turn into liquid ... At the same time, research on the compression of gases was supplemented by other scientists who compressed gas at pressures far exceeding 25 and 30 atmospheres, at which Regnault and his closest predecessors stopped. It was mentioned above that Poulier had already brought the pressure up to 100 atm., But his experiments were not so arranged so that in them it was possible to find the answer to the meaning of the law of B.-M. at high pressures. Such an answer is given by the experiments of Natterer, Calhete and Amag for strong pressures and the experiments of DI Mendeleev - for weak ones. Amaga installed his device at the bottom of a mine, which was about 400 meters (about 190 fathoms) deep. Measurements of the volume of gas at such a depth and the enormous height of the pressing mercury column were accompanied by such great technical difficulties that only the compressibility of nitrogen was directly studied. The law of compression of other gases in comparison with nitrogen was found by Amag by the method of Despres and Poulier. In Amag's experiments, the pressure reached 430¾ atmosphere, and the volume of nitrogen decreased only 335 только times. Calhete lowered his instrument into an artesian well 500 meters deep (about 230 fathoms); the height of the pressing mercury column was gradually increased as the device was lowered. The tube in which the gas was compressed was gilded inside; mercury, entering it, amalgamated gold, so that a trace remained on the gilding, a limit between gas and mercury, by which it was possible to measure the volume occupied by the compressed gas. In addition, Calhete carried out experiments on the compression of air and hydrogen in a special device in which pressures were brought to 605 atmospheres. These experiments were preceded by research by Natterer (1851-1854), who, with the help of a special injection pump device, brought the pressure on the gas to 2790 atmospheres. The gas was concentrated in a thick-walled steel vessel, which was equipped with a well-made valve, which was gradually loaded as the elasticity of the gas increased, which was measured by the weight of the weight on the valve. At the end of the compression of the gas, it was passed in parts into another vessel of a certain volume, where it assumed an elasticity equal to one atmosphere, and a successive decrease in the elasticity of the compressed gas was determined, at first a rapid, then more and more slowing down. The numbers obtained from these measurements provided a means to determine the elasticities of gases corresponding to its compression. The combination of all these experiments, in comparison with the experiments of Regnault, led to the conclusion that all gases, with the exception of hydrogen, undergo such changes in volume v and elasticity p, starting from one atmosphere, that the product vp decreases until the pressure or elasticity reaches a certain limit, and that with a further increase in pressure this product vp increases. In the first period, the gases are compressed more than they should according to the law of B.-M., in the second period - less. The limits, that is, the number of atmospheres of pressure at which the amount of compression should be obtained according to the B.-M. law, are not shown by different researchers in the same way, but there is no doubt that for each gas there is a special such limit; only hydrogen at all tested pressures is compressed less than it follows according to the law of B.-M. It remained to supplement these studies by studying the relationship between the elasticity and the volume of gases at pressures lower than atmospheric, that is, in rarefied air; according to the low-precision experiments of Boyle and Mariotte, and their law is true for rarefied air. An exact study of the law of compression of rarefied gases was made by DI Mendeleev with the cooperation of ML Kirpichev (experiments of the Imperial Russian Technical Society, "On the elasticity of gases" by D. Mendeleev, part 1, St. Petersburg, 1875, in 4 °). This work and others related to it were carried out at the expense of the Technical Society; with the same funds, the named essay was published, which describes the author's methods and devices for measuring the elasticity and volumes of gases. The experiments were carried out on air, hydrogen and carbon dioxide. Below is one series of experiments, from which the relationship between the volumes of very rarefied air and its elasticity is visible.

Hence, it can be seen that with a decrease in pressure on the gas, its volume increases in a smaller ratio than the elasticity decreases, consequently, and vice versa: with an increase in pressure, the volume decreases in a smaller ratio. Indeed: the second pressure is 7.71 times less than the first, and the second volume is only 7.38 times more than the first; the third pressure is 2.35 times less than the second, and the third volume is 1.92 times more than the second. This means that the compression and expansion of air at very low pressures deviates from the law of B.-M. in the same direction as at very strong pressures; a similar thing happened for carbon dioxide. Amaga and Zilleshtrom worked on the same issue, Regno also made several measurements with air at an elasticity of 300 millimeters. Regno and Ziljestrom came to the conclusion that thin air deviates from the law of B.-M. in the same direction as at pressures slightly higher than atmospheric; Amag's experiments did not lead him to reliable results (see the critical assessment of the experiments of R. and Z., made by D. I. Mendeleev in the essay "On the elasticity of gases", §§ 82, 92, 94.)

Summarizing everything that has been said about air, it can be seen that in a rarefied state it contracts less than it follows according to the B.-M. law, that at a density near atmospheric and greater than its air it compresses more than according to the B.-M. law, and, finally, with very high density he again retreats in the same direction as with a very small one. In the transition from retreats in one direction to retreats in the other, the air must be compressed according to the law of B.-M., and this happens only twice in the range from the lowest studied elasticity (about ⅓ mil.) To the highest (2700 atmospheres). Other gases probably follow the same variable compression law, except hydrogen, which is constantly compressed less than the B.-M. law.

Doubts have long been raised as to whether gases could follow the law of B.-M. at very high pressures. Since during compression, the density of a gas constantly increases to the same extent, it would be possible to reach the point that the compressed gas would be denser than the densest metal, i.e., that the gas brought by compression to a certain volume would be heavier, for example, platinum taken in the same amount. The unlimited compaction of the gas cannot be allowed for the reason that the substance of the gas, which itself occupies a certain part of the space, thereby supplies the limit of compression. The newest chemistry (see Mendeleev, "On the elasticity of gases", pp. 8-12) leads to considerations that do not allow a gas to be reduced by compression to a very high density. But in reality, the observed fact that all the gases tested at high pressures occupy a volume not so small as would follow according to the B.-M. law, and that the deviations from this law are the more significant, the greater the pressure; this fact shows that the decrease in volume approaches a certain limit. For some gases at ordinary temperatures, such a limit has been found, since these gases turn into liquid, and liquids at the strongest pressures only very slightly decrease in volume. Other gases, which do not turn into liquid from one compression without a more or less significant decrease in temperature, deviate more and more from the law of B.-M. Hydrogen at 3000 atm. pressure occupies a volume only 1000 times smaller than the initial one, that is, at this pressure its volume is three times more than one would expect if the B.-M. law is accurate. Several of Regnault's experiments on the compression of gases at the boiling point of water show that as the temperature rises, deviations from the B.-M. become less; this circumstance led him to the conclusion that an increase in temperature brings the gas closer to ideal state, in which he follows the B.-M. law, but this concept of an ideal gas is not yet sufficiently substantiated. In conclusion, it must be said that the B.-M. law, actually expressing the compression of gases only in certain limiting cases, nevertheless served as a starting point for studying their properties. Together with Gay-Lussac's law relating to the expansion of gases from heat, he presents a mathematical formula that needs to be modified in order to represent in its entirety the phenomenon of change in the volume of gases. Van der Waltz's formula (see this word) already penetrates deeper into the nature of gases.

Despite a lot of experimental work on the compression of gases, science can expect more new, even more extensive research. It would be desirable to see the exact and difficult studies of highly expanded gases made by D.I.Mendeleev, leading to important conclusions, repeated and widespread. Regnault's experiments will remain guiding for a long time, but the accuracy of our time may seem insufficient in the near future.

Describes the behavior of a gas in an isothermal process. The law is a consequence of the Clapeyron equation.

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Physics for Dummies. Lecture 26. Boyle-Mariotte Law

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The wording

The Boyle-Mariotte law is as follows:

In mathematical form, this statement is written in the form of the formula

p V \u003d C, (\\ displaystyle pV \u003d C,)

where p (\\ displaystyle p) - gas pressure; V (\\ displaystyle V) - gas volume, and C (\\ displaystyle C) - constant value under the agreed conditions. In general, the value C (\\ displaystyle C) is determined by the chemical nature, mass and temperature of the gas.

Obviously, if the index 1 denote the quantities related to the initial state of the gas, and the index 2 - to the final, then the given formula can be written in the form

p 1 V 1 \u003d p 2 V 2 (\\ displaystyle p_ (1) V_ (1) \u003d p_ (2) V_ (2)).

From the above and the above formulas, the form of the dependence of the gas pressure on its volume in the isothermal process follows:

p \u003d C V. (\\ displaystyle p \u003d (\\ frac (C) (V)).)

This dependence is another, equivalent to the first, expression of the content of the Boyle - Mariotte law. It means that

The pressure of a certain mass of gas at a constant temperature is inversely proportional to its volume.

Then the connection between the initial and final states of the gas participating in the isothermal process can be expressed as:

p 1 p 2 \u003d V 2 V 1. (\\ displaystyle (\\ frac (p_ (1)) (p_ (2))) \u003d (\\ frac (V_ (2)) (V_ (1))).)

It should be noted that the applicability of this and the above formula, which connects the initial and final pressures and gas volumes with each other, is not limited to the case of isothermal processes. The formulas remain valid in those cases when the temperature changes during the process, but as a result of the process, the final temperature turns out to be equal to the initial one.

It is important to clarify that this law is valid only in cases where the gas in question can be considered ideal. In particular, the Boyle - Mariotte law is fulfilled with high accuracy in relation to rarefied gases. If the gas is strongly compressed, then significant deviations from this law are observed.

Consequences

Boyle's Law - Mariotte states that the pressure of a gas in an isothermal process is inversely proportional to the volume occupied by the gas. If we take into account that the density of a gas is also inversely proportional to the volume it occupies, then we come to the conclusion:

In an isothermal process, the gas pressure changes in direct proportion to its density.

β T \u003d 1 p. (\\ displaystyle \\ beta _ (T) \u003d (\\ frac (1) (p)).)

Thus, we come to the conclusion:

The isothermal coefficient of compressibility of an ideal gas is equal to the reciprocal of its pressure.

The study of the relationship between the parameters characterizing the state of a given mass of gas, we begin with the study of gas processes occurring when one of the parameters remains unchanged. English scientist Boyle (in 1669) and French scientist Marriott (in 1676) discovered a law that expresses the dependence of pressure change on changes in gas volume at constant temperature. Let's carry out the following experiment.

By rotating the handle, we will change the volume of gas (air) in cylinder A (Fig. 11, a). According to the reading of the manometer, we note that the gas pressure also changes. We will change the volume of gas in the vessel (the volume is determined on the B scale) and, noticing the pressure, we will write them down in table. 1. It can be seen from it that the product of the gas volume by its pressure was almost constant: how many times the gas volume decreased, its pressure increased by the same amount.

As a result of similar, more accurate, experiments, it was discovered: for a given mass of gas at a constant temperature, the gas pressure changes inversely with the change in the volume of the gas. This is the formulation of the Boyle-Mariotte law. Mathematically, it will be written for two states as follows:

The process of changing the state of a gas at a constant temperature is called isothermal. The Boyle-Mariotte law formula is the equation for the isothermal state of a gas. At a constant temperature, the average speed of molecular movement does not change. A change in the volume of gas causes a change in the number of impacts of molecules on the walls of the vessel. This is the reason for the change in gas pressure.

Let us depict this process graphically, for example, for the case V \u003d 12 liters, p \u003d 1 at.... We will plot the gas volume on the abscissa axis, and its pressure on the ordinate axis (Fig. 11, b). Let's find the points corresponding to each pair of values \u200b\u200bof V and p, and, connecting them together, we get a graph of the isothermal process. The line depicting the relationship between the volume and pressure of a gas at constant temperature is called an isotherm. Pure isothermal processes do not occur. But there are often cases when the gas temperature changes little, for example, when the compressor is pumping air into the cylinders, when the combustible mixture is injected into the cylinder of an internal combustion engine. In such cases, the calculations of the volume and pressure of the gas are made according to the Boyle-Mariotte law *.

At constant temperature, the volume occupied by a gas is inversely proportional to its pressure.

Robert Boyle is a vivid example of a scientist-gentleman, the son of a long-gone era, when science was the lot of exceptionally wealthy people who devoted their leisure time to it. Most of Boyle's studies are, according to modern classification, to the category of chemical experiments, although he himself, for sure, considered natural philosopher (theoretical physicist) and natural scientist (experimental physicist). Apparently, he became interested in the behavior of gases after seeing the project of one of the world's first air pumps. Having designed and built another, improved version of his two-way air-vacuum pump, he decided to investigate how the increased and decreased gas pressure in a sealed vessel to which he was connected new apparatus, affects the properties of gases. Being a gifted experimenter, Boyle simultaneously adhered to very new and unusual views for that era, believing that science should proceed from empirical observations, and not be based solely on speculative and philosophical constructions.

In Boyle's formulation, the law literally sounded like this: "Under the influence of an external force, the gas is elastically compressed, and in its absence it expands, while the linear compression or expansion is proportional to the elastic force of the gas." Imagine that you are squeezing an inflated balloon. Insofar as free space there is enough air between the air molecules, you without much difficulty, applying some force and doing some work, will squeeze the ball, reducing the volume of gas inside it. This is one of the main differences between gas and liquid. In a ball of liquid water, for example, the molecules are packed tightly, as if the ball were filled with microscopic pellets. Therefore, water does not lend itself, unlike air, to elastic compression. (If you don’t believe, try pushing a tight-fitting cork inside the neck of a bottle filled with water up to the cork.) Boyle's law, along with Charles's law, formed the basis for the ideal gas equation of state.

J. Trefil calls it "Boyle's law", but we preferred the name of the law adopted in the Russian tradition. - Approx. translator.

Analysis of data on the pressure and volume of air during its compression

Table 3-1, which contains Boyle's experimental data on the relationship between pressure and volume for atmospheric airlocated under the spoiler.

After the researcher receives data similar to those shown in table. 3-1 he tries to find mathematical equationlinking together the two dependent quantities that he measured. One way to obtain such an equation is to graphically plot the dependence of various degrees of one quantity on another in the hope of obtaining a straight line graph. The general equation of a straight line is:

y \u003d ax + b (3-2)

where x and y are related variables, and a and b are constant numbers. If b is zero, a straight line goes through the origin.

In fig. 3-3 show different ways of graphing data for pressure P and volume V, shown in table. 3-1. Plots of P versus 1 / K and V versus 1 / P are straight lines passing through the origin. The plot of the dependence of the logarithm of P on the logarithm of V is also a straight line with a negative slope, the tangent of the angle is -1. All these three plots lead to equivalent equations:

P \u003d a / V (3-3a)
V \u003d a / P (3-3b)

lg V \u003d lg a - lg P (3-3v)

Each of these equations represents one of the options boyle-Mariotte lawwhich is usually phrased like this: for a given number of moles of gas, its pressure is proportional to the volume, provided that the gas temperature remains constant.

By the way, you are probably wondering why the Boyle-Mariotte law is called by a double name. This happened because this law, independently of Robert Boyle, who discovered it in 1662, was rediscovered by Edm Mariotte in 1676. So that's it.

When the relationship between two measured quantities is as simple as in this case, it can also be established numerically. If each pressure value P is multiplied by the corresponding value of the volume V, it is easy to make sure that all products for a given gas sample at constant temperature are approximately the same (see Table 3-1). Thus, we can write that

P V \u003d a ≈ 1410 (3-3g)

Equation (З-Зг) describes the hyperbolic relationship between the values \u200b\u200bof P and V (see Fig. 3-3, a). To check that the plot of the dependence of P on V constructed from the experimental data really corresponds to the hyperbola, we will construct an additional plot of the dependence of the product P V on P and make sure that it is a horizontal straight line (see Fig. 3-3, e) ...

Boyle found that for a given amount of any gas at a constant temperature, the relationship between pressure P and volume V is quite
is satisfactorily described by the relation

P V \u003d const (at constant T and n) (3-4)

Boyle-Mariotte formula

To compare volumes and pressures of one and the same gas sample under different conditions (but constant temperature), it is convenient to represent boyle-Mariotte law in the following formula:

P 1 V 1 \u003d P 2 V 2 (3-5)

where indices 1 and 2 correspond to two different conditions.

Example 4. Plastic bags with foodstuffs delivered to the Colorado plateau (see example 3) often burst, because the air in them, when rising from sea level to an altitude of 2500 m, under conditions of reduced atmospheric pressure, expands. If we assume that there is 100 cm 3 of air inside the bag at atmospheric pressure corresponding to sea level, what volume should this air occupy at the same temperature on the Colorado Plateau? (Assume that shrunken pouches are used to deliver products that do not restrict air expansion; missing data should be taken from Example 3.)

Decision
We will use Boyle's law in the form of equation (3-5), where index 1 will refer to conditions at sea level, and index 2 - to conditions at an altitude of 2500 m above sea level. Then P 1 \u003d 1,000 atm, V 1 \u003d 100 cm 3, P 2 \u003d 0.750 atm, and V 2 should be calculated. So,

P 1 V 1 \u003d P 2 V 2
1,000 atm 100 cm 3 \u003d 0.750 atm V 2

V 2 \u003d 133 cm 3

I hope that after studying lesson 25 "" you will remember the dependence of gas volume and pressure on each other .. If you have any questions, write them in the comments. If there are no questions, then proceed to the next lesson.