Let's make sure that the gas molecules are really located far enough from each other, and therefore the gases are well compressible. Let's take a syringe and place its piston approximately in the middle of the cylinder. Connect the hole of the syringe to a tube, the second end of which is tightly closed. Thus, some air will be enclosed in the syringe barrel under the piston and in the tube.In the syringe barrel under the piston some air will be enclosed. Now let’s place a weight on the movable piston of the syringe. It is easy to notice that the piston will drop slightly. This means that the volume of air has decreased. In other words, gases are easily compressed. Thus, there are quite large gaps between gas molecules. Placing a weight on the piston causes the volume of gas to decrease. On the other hand, after installing the load, the piston, having dropped slightly, stops in a new equilibrium position. This means that air pressure force on the piston increases and again balances the increased weight of the piston with the load. And since the area of the piston remains unchanged, we come to an important conclusion.
As the volume of a gas decreases, its pressure increases.
Let us remember at the same time that the mass of the gas and its temperature remained unchanged during the experiment. The dependence of pressure on volume can be explained as follows. As the volume of a gas increases, the distance between its molecules increases. Each molecule now needs to travel a greater distance from one impact with the wall of the vessel to the next. The average speed of movement of molecules remains unchanged. Consequently, gas molecules hit the walls of the container less often, and this leads to a decrease in gas pressure. And, conversely, when the volume of a gas decreases, its molecules hit the walls of the container more often, and the gas pressure increases. As the volume of a gas decreases, the distance between its molecules decreases
Dependence of gas pressure on temperature
In previous experiments, the temperature of the gas remained constant, and we studied the change in pressure due to a change in the volume of the gas. Now consider the case when the volume of gas remains constant, but the temperature of the gas changes. The mass also remains unchanged. Such conditions can be created by placing a certain amount of gas in a cylinder with a piston and securing the piston
Change in temperature of a given mass of gas at a constant volume
The higher the temperature, the faster gas molecules move.
Therefore,
Firstly, molecules hit the walls of the vessel more often;
Secondly, the average force of impact of each molecule on the wall becomes greater. This brings us to another important conclusion. As the temperature of a gas increases, its pressure increases. Let us remember that this statement is true if the mass and volume of the gas remain unchanged as its temperature changes.
Storage and transportation of gases.
The dependence of gas pressure on volume and temperature is often used in technology and in everyday life. If it is necessary to transport a significant amount of gas from one place to another, or when gases need to be stored for a long time, they are placed in special durable metal vessels. These vessels can withstand high pressures, so with the help of special pumps, significant masses of gas can be pumped into them, which under normal conditions would occupy hundreds of times more volume. Since the gas pressure in the cylinders is very high even at room temperature, they should never be heated or in any way attempted to make a hole in them, even after use.
Gas laws physics.
Real world physics in calculations is often reduced to somewhat simplified models. This approach is most applicable to describing the behavior of gases. The rules established experimentally were compiled by various researchers into the gas laws of physics and gave rise to the concept of “isoprocess.” This is a passage of an experiment in which one parameter remains constant. The gas laws of physics operate on the basic parameters of gas, or more precisely, its physical state. Temperature, occupied volume and pressure. All processes that relate to changes in one or more parameters are called thermodynamic. The concept of an isostatic process comes down to the statement that during any change in state, one of the parameters remains unchanged. This is the behavior of the so-called “ideal gas”, which, with some reservations, can be applied to real matter. As noted above, the reality is somewhat more complicated. However, with high reliability the behavior of a gas at a constant temperature is characterized using the Boyle-Mariotte law, which states:
The product of volume and gas pressure is a constant value. This statement is considered true in the case when the temperature does not change.
This process is called “isothermal”. In this case, two of the three parameters under study change. Physically everything looks simple. Squeeze inflated balloon. The temperature can be considered constant. As a result, the pressure inside the ball will increase as the volume decreases. The value of the product of two parameters will remain unchanged. Knowing the initial value of at least one of them, you can easily find out the indicators of the second. Another rule in the list of “gas laws of physics” is the change in the volume of a gas and its temperature at the same pressure. This is called an "isobaric process" and is described using Gay-Lusac's law. The ratio of gas volume and temperature is unchanged. This is true provided that the pressure in a given mass of substance is constant. Physically, everything is simple too. If you have ever charged a gas lighter or used carbon dioxide fire extinguisher, saw the effect of this law “live”. Gas coming out of a can or fire extinguisher expands rapidly. His temperature drops sharply. You can freeze the skin of your hands. In the case of a fire extinguisher, whole flakes of carbon dioxide snow are formed when the gas, under the influence of low temperature, quickly turns into a solid state from a gaseous state. Thanks to Gay-Lusac's law, you can easily find out the temperature of a gas by knowing its volume at any given time. The gas laws of physics also describe behavior under the condition of a constant occupied volume. Such a process is called isochoric and is described by Charles’s law, which states: With a constant occupied volume, the ratio of pressure to temperature of the gas remains unchanged at any time. In reality, everyone knows the rule: you cannot heat air freshener cans and other vessels containing gas under pressure. It ends with an explosion. What happens is exactly what Charles's law describes. The temperature is rising. At the same time, the pressure increases, since the volume does not change. The cylinder is destroyed at the moment when the indicators exceed the permissible values. So, knowing the occupied volume and one of the parameters, you can easily set the value of the second. Although the gas laws of physics describe the behavior of an ideal model, they can be easily applied to predict the behavior of gases in real systems. Especially in everyday life, isoprocesses can easily explain how a refrigerator works, why a cold stream of air flies out of a air freshener can, why a chamber or ball bursts, how a sprinkler works, and so on.
Fundamentals of MCT.
Molecular kinetic theory of matter- way of explanation thermal phenomena, which connects the occurrence of thermal phenomena and processes with the characteristics of the internal structure of matter and studies the reasons that determine thermal movement. This theory gained recognition only in the 20th century, although it comes from the ancient Greek atomic theory of the structure of matter.
explains thermal phenomena by the peculiarities of the movement and interaction of microparticles of matter
It is based on the laws of classical mechanics of I. Newton, which allow us to derive the equation of motion of microparticles. However, due to their huge number (there are about 10 23 molecules in 1 cm 3 of substance), it is impossible every second to unambiguously describe the movement of each molecule or atom using the laws of classical mechanics. Therefore, to construct modern theory heat methods use methods of mathematical statistics that explain the course of thermal phenomena based on the patterns of behavior of a significant number of microparticles.
Molecular kinetic theory built on the basis of generalized equations of motion for a huge number of molecules.
Molecular kinetic theory explains thermal phenomena from the standpoint of ideas about internal structure substances, that is, finds out their nature. This is a deeper, albeit more complex theory that explains the essence of thermal phenomena and determines the laws of thermodynamics.
Both existing approaches - thermodynamic approach And molecular kinetic theory- scientifically proven and mutually complement each other, and do not contradict each other. In this regard, the study of thermal phenomena and processes is usually considered from the standpoint of either molecular physics or thermodynamics, depending on how it is easier to present the material.
Thermodynamic and molecular-kinetic approaches complement each other in explaining thermal phenomena and processes.
The amount of air in cylinders depends on the volume of the cylinder, air pressure and its temperature. The relationship between air pressure and its volume at a constant temperature is determined by the relationship
where р1 and р2 are the initial and final absolute pressure, kgf/cm²;
V1 and V2 - initial and final volume of air, l. The relationship between air pressure and its temperature at a constant volume is determined by the relationship
where t1 and t2 are the initial and final air temperatures.
Using these dependencies, you can solve various problems that you encounter in the process of charging and operating air-breathing apparatus.
Example 4.1. The total capacity of the apparatus cylinders is 14 liters, the excess air pressure in them (according to the pressure gauge) is 200 kgf/cm². Determine the volume of free air, i.e. the volume reduced to normal (atmospheric) conditions.
Solution. Initial absolute atmospheric air pressure p1 = 1 kgf/cm². Final absolute pressure of compressed air p2 = 200 + 1 = 201 kgf/cm². The final volume of compressed air V 2 = 14 l. Volume of free air in cylinders according to (4.1)
Example 4.2. From a transport cylinder with a capacity of 40 liters with a pressure of 200 kgf/cm² (absolute pressure 201 kgf/cm²), air was transferred into the apparatus cylinders with a total capacity of 14 liters and a residual pressure of 30 kgf/cm² (absolute pressure 31 kgf/cm²). Determine the air pressure in the cylinders after air bypass.
Solution. Total volume of free air in the system of transport and equipment cylinders according to (4.1)
Total volume of compressed air in the cylinder system
Absolute pressure in the cylinder system after air bypass
excess pressure = 156 kgf/cm².
This example can be solved in one step by calculating the absolute pressure using the formula
Example 4.3. When measuring the air pressure in the apparatus cylinders in a room with a temperature of +17° C, the pressure gauge showed 200 kgf/cm². The device was taken outside, where a few hours later, during a working check, a pressure drop on the pressure gauge was discovered to 179 kgf/cm². The outside air temperature is -13° C. There is a suspicion of air leakage from the cylinders. Check the validity of this suspicion using calculations.
Solution. The initial absolute air pressure in the cylinders is p1 = 200 + 1 = 201 kgf/cm², the final absolute pressure p2 = 179 + 1 = 180 kgf/cm². Initial air temperature in cylinders t1 = + 17° C, final temperature t2 = - 13° C. Calculated final absolute air pressure in cylinders according to (4.2)
Suspicions are unfounded, since the actual and calculated pressures are equal.
Example 4.4. A submarine swimmer underwater consumes 30 l/min of air compressed to a pressure of a diving depth of 40 m. Determine the free air consumption, i.e., convert to atmospheric pressure.
Solution. Initial (atmospheric) absolute air pressure p1 = l kgf/cm². The final absolute pressure of compressed air according to (1.2) р2 =1 + 0.1*40 = 5 kgf/cm². Final compressed air flow V2 = 30 l/min. Free air flow according to (4.1)
Topics of the Unified State Examination codifier: isoprocesses - isothermal, isochoric, isobaric processes.
Throughout this paper we will adhere to the following assumption: mass and chemical composition gas remain unchanged. In other words, we believe that:
That is, there is no gas leakage from the vessel or, conversely, gas inflow into the vessel;
That is, the gas particles do not experience any changes (say, there is no dissociation - the breakdown of molecules into atoms).
These two conditions are satisfied in very many physically interesting situations (for example, in simple models heat engines) and therefore deserve separate consideration.
If the mass of a gas and its molar mass are fixed, then the state of the gas is determined three macroscopic parameters: pressure, volume And temperature. These parameters are related to each other by the equation of state (Mendeleev-Clapeyron equation).
Thermodynamic process(or just process) is a change in the state of a gas over time. During the thermodynamic process, the values of macroscopic parameters - pressure, volume and temperature - change.
Of particular interest are isoprocesses- thermodynamic processes in which the value of one of the macroscopic parameters remains unchanged. By fixing each of the three parameters in turn, we obtain three types of isoprocesses.
1. Isothermal process runs at a constant gas temperature: .
2. Isobaric process runs at constant gas pressure: .
3. Isochoric process occurs at a constant volume of gas: .
Isoprocesses are described by very simple laws of Boyle - Mariotte, Gay-Lussac and Charles. Let's move on to studying them.
Isothermal process
Let an ideal gas undergo an isothermal process at temperature . During the process, only the gas pressure and its volume change.
Let us consider two arbitrary states of the gas: in one of them the values of macroscopic parameters are equal, and in the second - . These values are related by the Mendeleev-Clapeyron equation:
As we said from the beginning, mass and molar mass are assumed to be constant.
Therefore, the right sides of the written equations are equal. Therefore, the left sides are also equal:
(1)
Since the two states of the gas were chosen arbitrarily, we can conclude that During an isothermal process, the product of gas pressure and its volume remains constant:
(2)
This statement is called Boyle-Mariotte law.
Having written the Boyle-Mariotte law in the form
(3)
You can also give this formulation: in an isothermal process, the gas pressure is inversely proportional to its volume. If, for example, during isothermal expansion of a gas its volume increases three times, then the gas pressure decreases three times.
How to explain the inverse relationship between pressure and volume from a physical point of view? At a constant temperature, the average kinetic energy of gas molecules remains unchanged, that is, simply put, the force of impacts of molecules on the walls of the vessel does not change. As the volume increases, the concentration of molecules decreases, and accordingly the number of impacts of molecules per unit time per unit wall area decreases - the gas pressure drops. On the contrary, as the volume decreases, the concentration of molecules increases, their impacts occur more frequently and the gas pressure increases.
Isothermal process graphs
In general, graphics thermodynamic processes It is customary to depict in the following coordinate systems:
-diagram: abscissa axis, ordinate axis;
-diagram: abscissa axis, ordinate axis.
The graph of an isothermal process is called isotherm.
An isotherm on an -diagram is a graph of an inversely proportional relationship.
Such a graph is a hyperbola (remember algebra - the graph of a function). The hyperbola isotherm is shown in Fig. 1.
Rice. 1. Isotherm on -diagram
Each isotherm corresponds to a certain fixed temperature value. It turns out that the higher the temperature, the higher the corresponding isotherm lies on -diagram.
In fact, let us consider two isothermal processes performed by the same gas (Fig. 2). The first process occurs at temperature, the second - at temperature.
Rice. 2. The higher the temperature, the higher the isotherm
We fix a certain volume value. On the first isotherm it corresponds to pressure, on the second - class="tex" alt="p_2 > p_1"> . Но при фиксированном объёме давление тем больше, чем выше температура (молекулы начинают сильнее бить по стенкам). Значит, class="tex" alt="T_2 > T_1"> .!}
In the remaining two coordinate systems, the isotherm looks very simple: it is a straight line perpendicular to the axis (Fig. 3):
Rice. 3. Isotherms on and -diagrams
Isobaric process
Let us recall once again that an isobaric process is a process taking place at constant pressure. During the isobaric process, only the volume of the gas and its temperature change.
A typical example of an isobaric process: gas is located under a massive piston that can move freely. If the mass of the piston and the cross section of the piston are , then the gas pressure is constant all the time and equal to
where is atmospheric pressure.
Let an ideal gas undergo an isobaric process at pressure . Consider again two arbitrary states of the gas; this time the values of the macroscopic parameters will be equal to and .
Let's write down the equations of state:
Dividing them by each other, we get:
In principle, this could already be enough, but we will go a little further. Let us rewrite the resulting relationship so that in one part only the parameters of the first state appear, and in the other part - only the parameters of the second state (in other words, we “spread the indices” across different parts):
(4)
And from here now - due to the arbitrariness of the choice of states! - we get Gay-Lussac's law:
(5)
In other words, at constant gas pressure, its volume is directly proportional to temperature:
(6)
Why does volume increase with increasing temperature? As the temperature rises, the molecules begin to beat harder and lift the piston. At the same time, the concentration of molecules drops, the impacts become less frequent, so that in the end the pressure remains the same.
Isobaric process graphs
The graph of an isobaric process is called isobar. On the -diagram, the isobar is a straight line (Fig. 4):
Rice. 4. Isobar on the -diagram
The dotted section of the graph means that in the case of real gas at sufficiently low temperatures the ideal gas model (and with it the Gay-Lussac law) stops working. In fact, as the temperature decreases, gas particles move more and more slowly, and the forces of intermolecular interaction have an increasingly significant influence on their movement (analogy: a slow ball is easier to catch than a fast one). Well, at very low temperatures, gases completely turn into liquids.
Let us now understand how the position of the isobar changes with pressure changes. It turns out that the higher the pressure, the lower the isobar goes on -diagram.
To verify this, consider two isobars with pressures and (Fig. 5):
Rice. 5. The lower the isobar, the greater the pressure
Let us fix a certain temperature value. We see that . But at a fixed temperature, the greater the pressure, the smaller the volume (Boyle-Mariotte law!).
Therefore, class="tex" alt="p_2 > p_1"> .!}
In the remaining two coordinate systems, the isobar is a straight line perpendicular to the axis (Fig. 6):
Rice. 6. Isobars on and -diagrams
Isochoric process
An isochoric process, recall, is a process that takes place at a constant volume. In an isochoric process, only the gas pressure and its temperature change.
It is very simple to imagine an isochoric process: it is a process taking place in a rigid vessel of a fixed volume (or in a cylinder under a piston when the piston is fixed).
Let an ideal gas undergo an isochoric process in a vessel with a volume of . Again, consider two arbitrary gas states with parameters and . We have:
Divide these equations by each other:
As in the derivation of Gay-Lussac’s law, we “split” the indices into different parts:
(7)
Due to the arbitrariness of the choice of states, we arrive at Charles' law:
(8)
In other words, at a constant volume of gas, its pressure is directly proportional to temperature:
(9)
An increase in the pressure of a gas of a fixed volume when it is heated is a completely obvious thing from a physical point of view. You can easily explain this yourself.
Graphs of an isochoric process
The graph of an isochoric process is called isochore. On the -diagram, the isochore is a straight line (Fig. 7):
Rice. 7. Isochore on the -diagram
The meaning of the dotted section is the same: the inadequacy of the ideal gas model at low temperatures.
Rice. 8. The lower the isochore, the greater the volume
The proof is similar to the previous one. We fix the temperature and see that . But at a fixed temperature, the lower the pressure, the larger the volume (again, the Boyle-Mariotte law). Therefore, class="tex" alt="V_2 > V_1"> .!}
In the remaining two coordinate systems, an isochore is a straight line perpendicular to the axis (Fig. 9):
Rice. 9. Isochores on and -diagrams
Boyle's laws - Mariotte, Gay-Lussac and Charles's laws are also called gas laws.
We derived gas laws from the Mendeleev-Clapeyron equation. But historically, everything was the other way around: gas laws were established experimentally, and much earlier. The equation of state subsequently appeared as their generalization.
Studies of the dependence of gas pressure on temperature under the condition of a constant volume of a certain mass of gas were first carried out in 1787 by Jacques Alexandre Cesar Charles (1746 - 1823). These experiments can be reproduced in a simplified form by heating the gas in a large flask connected to a mercury manometer M in the form of a narrow curved tube (Fig. 6).
Let us neglect the insignificant increase in the volume of the flask when heated and the insignificant change in volume when the mercury is displaced in a narrow manometric tube. Thus, the volume of gas can be considered constant. By heating the water in the vessel surrounding the flask, we will note the temperature of the gas using a thermometer T, and the corresponding pressure is indicated by the pressure gauge M. Fill the vessel with melting ice and measure the pressure p 0, corresponding to a temperature of 0 °C.
Experiments of this kind showed the following.
1. The pressure increment of a certain mass is certain part α the pressure that a given mass of gas had at a temperature of 0 °C. If the pressure at 0 °C is denoted by p 0, then the increase in gas pressure when heated by 1 °C is p 0 +αp 0 .
When heated by τ, the pressure increase will be τ times greater, i.e. pressure increase is proportional to temperature increase.
2. Magnitude α, showing by what part of the pressure at 0 °C the gas pressure increases when heated by 1 °C, has the same value (more precisely, almost the same) for all gases, namely 1/273 °C -1. Size α called temperature coefficient of pressure. Thus, the temperature coefficient of pressure for all gases has the same value, equal to 1/273 °C -1.
The pressure of a certain mass of gas when heated to 1 °C with a constant volume increases by 1/273 part of the pressure that this mass of gas had at 0°C ( Charles's law).
It should, however, be borne in mind that the temperature coefficient of gas pressure obtained by measuring temperature with a mercury manometer is not exactly the same for different temperatures: Charles’s law is satisfied only approximately, although with a very high degree of accuracy.
Formula expressing Charles's law. Charles's law allows you to calculate the pressure of a gas at any temperature if its pressure at temperature is known
0°C. Let the pressure of a given mass of gas at 0 °C in a given volume be p 0, and the pressure of the same gas at temperature t There is p. There is a temperature increase t, therefore, the pressure increment is equal to αp 0 t and the desired pressure
This formula can also be used if the gas is cooled below 0 °C; at the same time t will have negative values. At very low temperatures, when the gas approaches the state of liquefaction, as well as in the case of highly compressed gases, Charles’ law is not applicable and formula (2) ceases to be valid.
Charles's law from the point of view of molecular theory. What happens in the microcosm of molecules when the temperature of a gas changes, for example, when the temperature of the gas rises and its pressure increases? From the point of view of molecular theory, there are two possible reasons for the increase in the pressure of a given gas: firstly, the number of impacts of molecules per unit time per unit area could increase, and secondly, the impulse transmitted when one molecule hits the wall could increase. Both reasons require an increase in the speed of molecules (we remind you that the volume of a given mass of gas remains unchanged). From here it becomes clear that an increase in gas temperature (in the macrocosm) is an increase in the average speed of the random movement of molecules (in the microcosm).
Some types of incandescent electric lamps are filled with a mixture of nitrogen and argon. When the lamp operates, the gas in it heats up to approximately 100 °C. What should be the pressure of the gas mixture at 20 °C if it is desirable that the gas pressure in it does not exceed atmospheric pressure when the lamp is operating? (answer: 0.78 kgf/cm2)
A red line is placed on the pressure gauges, indicating the limit above which an increase in gas is dangerous. At a temperature of 0 °C, the pressure gauge shows that the excess gas pressure over the outside air pressure is 120 kgf/cm2. Will the redline be reached when the temperature rises to 50 °C if the redline is at 135 kgf/cm2? Take the outside air pressure equal to 1 kgf/cm2 (answer: the pressure gauge needle goes beyond the red line)
Annotation: traditional presentation of the topic, supplemented by a demonstration on a computer model.
Of the three aggregate states of matter, the simplest is the gaseous state. In gases, the forces acting between molecules are small and, under certain conditions, can be neglected.
Gas is called perfect , If:
The sizes of the molecules can be neglected, i.e. molecules can be considered material points;
The forces of interaction between molecules can be neglected (the potential energy of interaction of molecules is much less than their kinetic energy);
The collisions of molecules with each other and with the walls of the vessel can be considered absolutely elastic.
Real gases are close in properties to ideal gases when:
Conditions close to normal conditions (t = 0 0 C, p = 1.03·10 5 Pa);
At high temperatures.
The laws governing the behavior of ideal gases were discovered empirically long enough. Thus, the Boyle-Mariotte law was established back in the 17th century. Let us give the formulations of these laws.
Boyle's Law - Mariotte. Let the gas be in conditions where its temperature is maintained constant (such conditions are called isothermal ).Then for a given mass of gas, the product of pressure and volume is a constant:
This formula is called isotherm equation. Graphically the dependence of p on V for different temperatures shown in the figure.
The property of a body to change pressure when volume changes is called compressibility. If the volume change occurs at T=const, then the compressibility is characterized isothermal compressibility coefficient which is defined as the relative change in volume causing a unit change in pressure.
For an ideal gas it is easy to calculate its value. From the isotherm equation we obtain:
The minus sign indicates that as volume increases, pressure decreases. Thus, the isothermal compressibility coefficient of an ideal gas is equal to the reciprocal of its pressure. As pressure increases, it decreases, because The higher the pressure, the less opportunity the gas has for further compression.
Gay-Lussac's law. Let the gas be in conditions where its pressure is maintained constant (such conditions are called isobaric ). They can be achieved by placing gas in a cylinder closed by a movable piston. Then a change in gas temperature will lead to movement of the piston and a change in volume. The gas pressure will remain constant. In this case, for a given mass of gas, its volume will be proportional to the temperature:
where V 0 is the volume at temperature t = 0 0 C, - volumetric expansion coefficient gases It can be represented in a form similar to the compressibility coefficient:
Graphically, the dependence of V on T for various pressures is shown in the figure.
Moving from temperature in Celsius to absolute temperature, Gay-Lussac's law can be written as:
Charles's law. If a gas is in conditions where its volume remains constant ( isochoric conditions), then for a given mass of gas the pressure will be proportional to the temperature:
where p 0 - pressure at temperature t = 0 0 C, - pressure coefficient. It shows the relative increase in gas pressure when it is heated by 1 0:
Charles's law can also be written as:
Avogadro's Law: One mole of any ideal gas at the same temperature and pressure occupies the same volume. At normal conditions(t = 0 0 C, p = 1.03·10 5 Pa) this volume is equal to m -3 /mol.
The number of particles contained in 1 mole of various substances is called. Avogadro's constant :
It is easy to calculate the number n0 of particles per 1 m3 under normal conditions:
This number is called Loschmidt number.
Dalton's Law: the pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the gases entering it, i.e.
Where - partial pressures- the pressure that the components of the mixture would exert if each of them occupied a volume equal to the volume of the mixture at the same temperature.
Clapeyron - Mendeleev equation. From the ideal gas laws we can obtain equation of state , connecting T, p and V of an ideal gas in a state of equilibrium. This equation was first obtained by the French physicist and engineer B. Clapeyron and Russian scientists D.I. Mendeleev, therefore bears their name.
Let a certain mass of gas occupy a volume V 1, have a pressure p 1 and be at a temperature T 1. The same mass of gas in a different state is characterized by the parameters V 2, p 2, T 2 (see figure). The transition from state 1 to state 2 occurs in the form of two processes: isothermal (1 - 1") and isochoric (1" - 2).
For these processes, we can write the laws of Boyle - Mariotte and Gay - Lussac:
Eliminating p 1 " from the equations, we obtain
Since states 1 and 2 were chosen arbitrarily, the last equation can be written as:
This equation is called Clapeyron equation , in which B is a constant, different for different masses of gases.
Mendeleev combined Clapeyron's equation with Avogadro's law. According to Avogadro's law, 1 mole of any ideal gas with the same p and T occupies the same volume V m, therefore the constant B will be the same for all gases. This constant common to all gases is denoted by R and is called universal gas constant. Then
This equation is ideal gas equation of state , which is also called Clapeyron-Mendeleev equation .
The numerical value of the universal gas constant can be determined by substituting the values of p, T and V m into the Clapeyron-Mendeleev equation under normal conditions:
The Clapeyron-Mendeleev equation can be written for any mass of gas. To do this, remember that the volume of a gas of mass m is related to the volume of one mole by the formula V = (m/M)V m, where M is molar mass of gas. Then the Clapeyron-Mendeleev equation for a gas of mass m will have the form:
where is the number of moles.
Often the equation of state of an ideal gas is written in terms of Boltzmann constant :
Based on this, the equation of state can be represented as
where is the concentration of molecules. From the last equation it is clear that the pressure of an ideal gas is directly proportional to its temperature and concentration of molecules.
Small demonstration ideal gas laws. After pressing the button "Let's get started" You will see the presenter's comments on what is happening on the screen (black color) and a description of the computer's actions after you press the button "Next" (brown). When the computer is “busy” (that is, testing is in progress), this button is inactive. Move on to the next frame only after comprehending the result obtained in the current experiment. (If your perception does not coincide with the presenter’s comments, write!)
You can verify the validity of the ideal gas laws on the existing
