We begin the study of the relationship between the parameters characterizing the state of a given mass of gas by studying gas processes that occur while 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 changes 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 pressure gauge reading, we note that the gas pressure also changes. We will change the volume of gas in the vessel (the volume is determined by scale B) and, noticing the pressure, we will write them down in the table. 1. It can be seen from it that the product of the volume of a gas and its pressure was almost constant: no matter how many times the volume of the gas decreased, the same number of times its pressure increased.
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 in inverse proportion to the change in gas volume. This is the formulation of the Boyle-Mariotte law. Mathematically, for two states it will be written as follows:
The process of changing the state of a gas at a constant temperature is called isothermal. The formula of the Boyle-Mariotte law is the equation of the isothermal state of a gas. At constant temperature, the average speed of molecules does not change. A change in the volume of a gas causes a change in the number of impacts of molecules on the walls of the container. This is the reason for the change in gas pressure.
Let us depict this process graphically, for example for the case V = 12 l, p = 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 of V and p, and by connecting them together, we will obtain 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. Isothermal processes do not occur in their pure form. But there are often cases when the gas temperature changes little, for example, when a compressor pumps air into cylinders, or when a combustible mixture is injected into the cylinder of an internal combustion engine. In such cases, calculations of gas volume and pressure are made according to the Boyle-Mariotte law *.
Change in one of the macroscopic parameters of a substance of a certain mass - pressure R, volume V or temperature t - causes changes to other parameters.
If all the quantities characterizing the state of the gas change simultaneously, then it is difficult to establish any definite patterns experimentally. It’s easier to first study processes in which mass and one of three parameters - R,V or t - remain unchanged. Quantitative relationships between two parameters of a gas of the same mass with a constant value of the third parameter are called gas laws.
Boyle-Mariotte law
The first gas law was discovered by the English scientist R. Boyle (1627-1691) in 1660. Boyle’s work was called “New Experiments Concerning an Air Spring.” Indeed, gas behaves like a compressed spring; this can be verified by compressing air in a regular bicycle pump.
Boyle studied the change in gas pressure as a function of volume at constant temperature. The process of changing the state of a thermodynamic system at a constant temperature is called isothermal (from the Greek words isos - equal, therme - heat). To maintain a constant temperature of a gas, it is necessary that it can exchange heat with a large system in which a constant temperature is maintained - a thermostat. Atmospheric air can serve as a thermostat if its temperature does not change noticeably during the experiment.
Boyle observed the change in the volume of air trapped in a long curved tube by a column of mercury (Fig. 3.6, a). Initially, the mercury levels in both legs of the tube were the same and the air pressure was equal to atmospheric pressure (760 mm Hg). While adding mercury to the long elbow of the tube, Boyle noticed that the volume of air was halved when the difference in levels in both elbows turned out to be equal h = 760 mm, and, consequently, the air pressure doubled (Fig. 3.6, b). This led Boyle to the idea that the volume of a given mass of gas and its pressure are inversely proportional.
A) b)
Further observations of the change in volume when adding different portions of mercury confirmed this conclusion.
Independently of Boyle, somewhat later, the French scientist E. Marriott (1620-1684) came to the same conclusions. Therefore, the found law was called the Boyle-Mariotte law. According to this law, the pressure of a given mass (or amount) of gas at a constant temperature is inversely proportional to the volume of the gas: 
.
If p 1 - gas pressure at volume V 1 , And p 2 - its pressure at volume V 2 , That
(3.5.1)
It follows that p 1 V l = p 2 V 2 , or
(3.5.2)
at t = const.
The product of the pressure of a gas of a given mass and its volume is constant if the temperature does not change.
This law is valid for any gases, as well as for mixtures of gases (for example, air).
You can verify the validity of the Boyle-Mariotte law using the device shown in Figure 3.7. The sealed corrugated vessel is connected to a pressure gauge that records the pressure inside the vessel. By rotating the screw you can change the volume of the vessel. The volume can be judged using a ruler. By changing the volume and measuring the pressure, you can see that equation (3.5.2) is satisfied.
Like other physical laws, the Boyle-Mariotte law is approximate. At pressures several hundred times greater than atmospheric pressure, deviations from this law become significant.
On a graph of pressure versus volume, each state of a gas corresponds to one point.
Isotherms
The process of changing gas pressure depending on volume is depicted graphically using a curve called an isotherm (Fig. 3.8). The gas isotherm expresses the inverse relationship between pressure and volume. A curve of this kind is called a hyperbola. Different isotherms correspond to different constant temperatures, since a higher temperature at the same volume corresponds to a higher pressure*. Therefore, the isotherm corresponding to a higher temperature t2, lies above the isotherm corresponding to the lower temperature t 1.
* This will be discussed in more detail later.
The law is formulated as follows: the product of the volume of a given mass of gas and its pressure at a constant temperature is a constant value. Mathematically, this law can be written like this:
P 1 V 1 = P 2 V 2 or PV = const (1)
The following consequences follow from the Boyle-Marriott law: the density and concentration of a gas at a constant temperature is directly proportional to the pressure under which the gas is located:
(2);
(3) ,
Where d 1 – density, C 1 – gas concentration under pressure P 1; d 2 and C 2 are the corresponding values under pressure P 2 .
Example 1. A gas cylinder with a capacity of 0.02 m 3 contains gas under a pressure of 20 atm. What volume will the gas occupy if the cylinder valve is opened without changing its temperature? Final pressure 1 atm.
Example 2. Compressed air is supplied to a gas holder (gas collection tank) with a volume of 10 m3. How long will it take to pump it up to a pressure of 15 atm if the compressor sucks in 5.5 m 3 of atmospheric air per minute at a pressure of 1 atm? The temperature is assumed to be constant.
Example 3. 112 g of nitrogen under a pressure of 4 atm occupy a volume of 20 liters. What pressure must be applied so that the nitrogen concentration becomes 0.5 mol/l, provided that the temperature remains unchanged?
1.1.2 Gay-Lussac's and Charles' laws
Gay-Lussac found that at constant pressure, with an increase in temperature of 1°C, the volume of a given mass of gas increases by 1/273 of its volume at 0°C.
Mathematically, this law is written:
(4)
,
Where V- volume of gas at temperature t°С, a V 0 – volume of gas at 0°C.
Charles showed that the pressure of a given mass of gas, when heated by 1°C at a constant volume, increases by 1/273 of the pressure that the gas has at 0°C. Mathematically, this law is written as follows:
(5)
,
where P 0 and P are gas pressures, respectively, at temperatures 0С and tС.
When replacing the Celsius scale with the Kelvin scale, the connection between them is established by the relation T = 273 + t, the formulas of Gay-Lussac's and Charles's laws are significantly simplified.
Gay-Lussac's Law: at constant pressure, the volume of a given mass of gas is directly proportional to its absolute temperature:
(6)
.
Charles's Law: at constant volume, the pressure of a given mass of gas is directly proportional to its absolute temperature:
(7)
.
From the laws of Gay-Lussac and Charles it follows that at constant pressure the density and concentration of a gas are inversely proportional to its absolute temperature:
(8)
,
(9) .
Where d 1 and C 1 - density and concentration of gas at absolute temperature T 1, d 2 and C 2 are the corresponding values at absolute temperature T 2 .
Example 4. At 20ºC the volume of gas is 20.4 ml. What volume will the gas occupy when it is cooled to 0°C if the pressure remains constant?
Primep 5. At 9°C, the pressure inside the oxygen cylinder was 94 atm. Calculate how much the pressure in the cylinder increased if the temperature rose to 27ºC?
Example 6. Density of chlorine gas at 0ºС and pressure 760 mm Hg. Art. equal to 3.220 g/l. Find the density of chlorine, taking it as an ideal gas, at 27ºС at the same pressure.
Example 7. Under normal conditions, the concentration of carbon monoxide is 0.03 kmol/m3. Calculate at what temperature the mass of 10 m 3 of carbon monoxide will be equal to 7 kg?
Combined Boyle-Mariotte-Charles-Gay-Lussac law.
The formulation of this law: for a given mass of gas, the product of pressure and volume divided by absolute temperature is constant for all changes occurring in the gas. Mathematical notation:
(10)
where V 1 is the volume and P 1 is the pressure of a given mass of gas at absolute temperature T 1 , V 2 - volume and P 2 - pressure of the same mass of gas at absolute temperature T 2.
One of the most important applications of the unified gas law is “bringing the volume of gas to normal conditions.”
Example 8. Gas at 15°C and pressure 760 mmHg. Art. occupies a volume of 2 liters. Bring the volume of gas to normal conditions.
To facilitate such calculations, you can use the conversion factors given in the tables.
Example 9. In the gasometer above the water there is 7.4 liters of oxygen at a temperature of 23°C and a pressure of 781 mm Hg. Art. The water vapor pressure at this temperature is 21 mmHg. Art. What volume will the oxygen in the gasometer occupy under normal conditions?
Scientists studying thermodynamic systems have found that a change in one macroparameter of the system leads to a change in the rest. For example, an increase in pressure inside a rubber ball when it is heated causes an increase in its volume; An increase in the temperature of a solid leads to an increase in its size, etc.
These dependencies can be quite complex. Therefore, first we will consider the existing connections between macroparameters using the example of the simplest thermodynamic systems, for example, for rarefied gases. The experimentally established functional relationships between physical quantities for them are called gas laws.
Robert Boyle (1627-1691). A famous English physicist and chemist who studied the properties of air (mass and elasticity of air, the degree of its rarefaction). Experience has shown that the boiling point of water depends on the pressure of the environment. He also studied the elasticity of solids, hydrostatics, light and electrical phenomena, and for the first time expressed an opinion about the complex spectrum of white light. Introduced the concept of “chemical element”.
The first gas law was discovered by the English scientist R. Boylem in 1662 while studying the elasticity of air. He took a long bent glass tube, sealed at one end, and began to pour mercury into it until a small closed volume of air formed in the short elbow (Fig. 1.5). Then he added mercury to the long elbow, studying the relationship between the volume of air in the sealed end of the tube and the pressure created by the mercury in the left elbow. The scientist’s assumption that there is a certain relationship between them was confirmed. Comparing the results obtained, Boyle formulated the following position:
There is an inverse relationship between the pressure and volume of a given mass of gas at a constant temperature:p ~ 1/V.
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| Edm Marriott |
Edm Marriott(1620—1684) . French physicist who studied the properties of liquids and gases, collisions of elastic bodies, pendulum oscillations, and natural optical phenomena. He established the relationship between the pressure and volume of gases at a constant temperature and explained on its basis various applications, in particular, how to find the altitude of an area using barometer readings. It has been proven that the volume of water increases when it freezes.
A little later, in 1676, the French scientist E. Marriott independently of R. Boyle, he generally formulated the gas law, which is now called Boyle-Mariotte law. According to him, if at a certain temperature a given mass of gas occupies a volume V 1 at pressure p1, and in another state at the same temperature its pressure and volume are equal p2 And V 2, then the following relationship is true:
p 1 /p 2 =V 2 /V 1 or p 1V 1 = p2V 2.
Boyle-Mariotte law : if at a constant temperature a thermodynamic process occurs, as a result of which the gas changes from one state (p 1 andV 1)to another (p2iV 2),then the product of pressure and the volume of a given mass of gas at a constant temperature is constant:
pV = const.Material from the site
A thermodynamic process that occurs at a constant temperature is called isothermal(from the gr. isos - equal, therme - warmth). Graphically on the coordinate plane pV it is represented by a hyperbole called isotherm(Fig. 1.6). Different isotherms correspond to different temperatures - the higher the temperature, the higher on the coordinate plane pV there is a hyperbola (T 2 >T 1). It is obvious that on the coordinate plane pT And VT isotherms are depicted as straight lines, perpendicular to the temperature axis.
Boyle-Mariotte law installs relationship between pressure and volume of gas for isothermal processes: at constant temperature, the volume V of a given mass of gas is inversely proportional to its pressure p.
Let us now move on to a more detailed study of the question of how the pressure of a certain mass of gas changes if its temperature remains unchanged and only the volume of the gas changes. We have already found out that this isothermal the process is carried out under the condition that the temperature of the bodies surrounding the gas is constant and the volume of the gas changes so slowly that the temperature of the gas at any moment of the process does not differ from the temperature of the surrounding bodies. We thus pose the question: how are volume and pressure related to each other during an isothermal change in the state of a gas? Daily experience teaches us that when the volume of a certain mass of gas decreases, its pressure increases. An example is the increase in elasticity when inflating a soccer ball, bicycle or car tire. The question arises: how exactly does the pressure of a gas increase with a decrease in volume if the temperature of the gas remains unchanged?
The answer to this question was given by research carried out in the 17th century by the English physicist and chemist Robert Boyle (1627-1691) and the French physicist Eden Marriott (1620-1684).
Experiments establishing the relationship between gas volume and pressure can be reproduced: on a vertical stand , equipped with divisions, there are glass tubes A And IN, connected by a rubber tube C. Mercury is poured into the tubes. Tube B is open at the top, and tube A has a tap. Let's close this tap, thus locking a certain mass of air in the tube A. As long as we do not move the tubes, the mercury level in both tubes is the same. This means that the pressure of the air trapped in the tube A, the same as the ambient air pressure.
Let's now slowly pick up the phone IN. We will see that the mercury in both tubes will rise, but not equally: in the tube IN the mercury level will always be higher than in A. If you lower tube B, then the mercury level in both elbows decreases, but in tube IN the decrease is greater than in A. Volume of air trapped in the tube A, can be counted by tube divisions A. The pressure of this air will differ from atmospheric pressure by the pressure of a column of mercury, the height of which is equal to the difference in the levels of mercury in tubes A and B. At. picking up the phone IN The pressure of the mercury column is added to atmospheric pressure. The volume of air in A decreases. When the handset goes down IN the level of mercury in it turns out to be lower than in A, and the pressure of the mercury column is subtracted from the atmospheric pressure; air volume in A
increases accordingly. Comparing the values obtained in this way for the pressure and volume of air locked in tube A, we will be convinced that when the volume of a certain mass of air increases by a certain number of times, its pressure decreases by the same amount, and vice versa. The air temperature in the tube can be considered constant in our experiments. Similar experiments can be carried out with other gases. The results are the same. So,
the pressure of a certain mass of gas at a constant temperature is inversely proportional to the volume of the gas (Boyle-Mariotte law). For rarefied gases, the Boyle-Mariotte law is satisfied to a high degree
accuracy. For highly compressed or cooled gases, noticeable deviations from this law are found. Formula expressing the Boyle-Mariotte law.
