The clapeyron-mendeleev equation is as follows. Ideal gases

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 they can be neglected.

The gas is called perfect , if:

The size of molecules can be neglected, i.e. molecules can be considered material points;

We can neglect the forces of interaction between molecules (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 the ideal at:

Conditions close to normal conditions (t = 0 0 C, p = 1.03 10 5 Pa);

At high temperatures.

The laws that govern the behavior of ideal gases were discovered experimentally quite a long time ago. So, Boyle's law - Mariotte was established in the 17th century. We give the formulations of these laws.

Boyle's Law - Mariotte. Let the gas be under conditions where its temperature is kept constant (such conditions are called isothermal ). Then for a given mass of gas, the product of pressure and volume is a constant value:

This formula is called isotherm equation. Graphically, the dependence of p on V for various temperatures is shown in the figure.

The property of a body to change pressure with a change in volume is called compressibility. If the change in volume occurs at T=const, then the compressibility is characterized by isothermal compressibility factor which is defined as the relative change in volume that causes a change in pressure per unit.

For an ideal gas, it is easy to calculate its value. From the isotherm equation we get:

The minus sign indicates that as the volume increases, the pressure decreases. Thus, the isothermal compressibility of an ideal gas is equal to the reciprocal of its pressure. With increasing pressure, it decreases, because. the greater the pressure, the less the gas has the ability to further compress.

Gay-Lussac law. Let the gas be under conditions where its pressure is maintained constant (such conditions are called isobaric ). They can be carried out by placing gas in a cylinder closed by a movable piston. Then a change in the temperature of the gas will move the piston and change the volume. The pressure of the gas will remain constant. In this case, for a given mass of gas, its volume will be proportional to the temperature:

where V 0 - volume at temperature t = 0 0 C, - volume expansion coefficient gases. It can be represented in a form similar to the compressibility factor:

Graphically, the dependence of V on T for various pressures is shown in the figure.

Moving from temperature in the Celsius scale to absolute temperature, Gay-Lussac's law can be written as:

Charles' Law. If the gas is under 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 \u003d 0 0 C, - pressure coefficient. It shows the relative increase in gas pressure when it is heated by 10:

Charles' 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. Under 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, called. Avogadro's constant :

It is easy to calculate the number n 0 particles in 1 m 3 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 included in 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.

Equation of Clapeyron - Mendeleev. From the laws of an ideal gas, one can obtain equation of state , linking 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 some mass of gas occupies a volume V 1 , has a pressure p 1 and is 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 is carried out in the form of two processes: isothermal (1 - 1") and isochoric (1" - 2).

For these processes, one can write down the laws of Boyle - Mariotte and Gay - Lussac:

Eliminating p 1 " from the equations, we get

Since states 1 and 2 were chosen arbitrarily, the last equation can be written as:

This equation is called Clapeyron's 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 at the same p and T occupies the same volume V m, so the constant B will be the same for all gases. This common constant for all gases is denoted 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, recall that the volume of a gas of mass m is related to the volume of one mole by the formula V \u003d (m / M) V m, where M is molar mass of gas. Then the Clapeyron - Mendeleev equation for a gas of mass m will look like:

where is the number of moles.

The equation of state for an ideal gas is often written in terms of Boltzmann's constant :

Based on this, the equation of state can be represented as

where is the concentration of molecules. From the last equation it can be seen that the pressure of an ideal gas is directly proportional to its temperature and concentration of molecules.

Small demo ideal gas laws. After pressing the button "Let's start" You will see the host's comments on what is happening on the screen (black color) and a description of the computer's actions after you press the button "Further"(Brown color). When the computer is "busy" (i.e., experience is in progress), this button is not active. Move on to the next frame only after understanding the result obtained in the current experiment. (If your perception does not match the host's comments, write!)

You can verify the validity of the ideal gas laws on the existing

It is known that rarefied gases obey the laws of Boyle and Ge-Lussac. Boyle's law states that when a gas is compressed isothermally, pressure changes inversely with volume. Therefore, when

According to Gae-Lussac's law, heating a gas at constant pressure entails its expansion by the volume that it occupies at and at the same constant pressure.

Therefore, if there is a volume occupied by a gas at 0 ° C and at pressure, there is a volume occupied by this gas at

and at the same pressure

We will depict the state of the gas as a point on the diagram (the coordinates of any point in this diagram indicate the numerical values ​​​​of pressure and volume or 1 mole of gas; lines are plotted in Fig. 184, for each of which these are gas isotherms).

Let us imagine that the gas was taken in some arbitrarily chosen state C, at which its temperature is the pressure p and the volume occupied by it

Rice. 184 Gas isotherms according to Boyle's law.

Rice. 185 Diagram explaining the derivation of the Clapeyron equation from the laws of Boyle and Ge-Lussac.

Cool it down to without changing the pressure (Fig. 185). Based on Gay-Lussac's law, we can write that

Now, while maintaining the temperature, we will compress the gas or, if necessary, let it expand until its pressure becomes equal to one physical atmosphere. This pressure will be denoted by and the volume, which as a result will be occupied by the gas (at through (point in Fig. 185). Based on Boyle's law

Multiplying term by term the first equality by the second and reducing by we get:

This equation was first derived by B.P. Clapeyron, an outstanding French engineer who worked in Russia as a professor at the Institute of Communications from 1820 to 1830. The constant value 27516 is known to be the gas constant.

According to the law discovered in 1811 by the Italian scientist Avogadro, all gases, regardless of their chemical nature, occupy the same volume at the same pressure if they are taken in quantities proportional to their molecular weight. Using the mole as a unit of mass (or, which is the same, a gram-molecule, gram-mole), Avogadro's law can be formulated as follows: at a certain temperature and a certain pressure, a mole of any gas will occupy the same volume. So, for example, at and at pressure, a mole of any gas occupies

The laws of Boyle, Ge-Lussac and Avogadro, found experimentally, were later theoretically derived from molecular kinetic concepts (Kroenig in 1856, Clausius in 1857 and Maxwell in 1860). From a molecular kinetic point of view, Avogadro's law (which, like other gas laws, is exact for ideal gases and approximate for real ones) means that equal volumes of two gases contain the same number of molecules if these gases are at the same temperature and the same pressure.

Let there be the mass (in grams) of an oxygen atom, the mass of a molecule of any substance, the molecular weight of this substance: Obviously, the number of molecules contained in a mole of any substance is equal to:

i.e., a mole of any substance contains the same number of molecules. This number is equal to it is called Avogadro's number.

D. I. Mendeleev in 1874 pointed out that, thanks to Avogadro’s law, Clapeyron’s equation, which synthesizes the laws of Boyle and Ge-Lussac, acquires the greatest generality when it is related not to an ordinary weight unit (gram or kilogram), but to a mole of gases. Indeed, since a mole of any gas at occupies a volume equal to the numerical value of the gas constant for all gases taken in the amount of 1 gram-molecule, it should be the same regardless of their chemical nature.

The gas constant for 1 mole of gas is usually denoted by a letter and is called the universal gas constant:

If the volume y (and hence contains not 1 mole of gas, but moles), then, obviously,

The numerical value of the universal gas constant depends on the units in which the values ​​​​on the left side of the Clapeyron equation are measured. For example, if pressure is measured in and volume in then from here

In table. 3 (p. 316) gives the values ​​of the gas constant, expressed in various commonly used units.

When the gas constant is included in a formula, all terms of which are expressed in caloric units of energy, then the gas constant must also be expressed in calories; approximately, exactly

The calculation of the universal gas constant is based, as we have seen, on Avogadro's law, according to which all gases, regardless of their chemical nature, occupy a volume

In fact, the volume occupied by 1 mole of gas under normal conditions is not exactly equal for most gases (for example, for oxygen and nitrogen it is slightly less, for hydrogen it is slightly more). If this is taken into account in the calculation, then there will be some discrepancy in the numerical value for gases of different chemical nature. So, for oxygen instead it turns out for nitrogen. This discrepancy is due to the fact that all gases in general at ordinary density do not follow Boyle's and Gay-Lussac's laws quite exactly.

In technical calculations, instead of measuring the mass of a gas in moles, the mass of a gas is usually measured in kilograms. Let the volume contain gas. The coefficient in the Clapeyron equation means the number of moles contained in the volume, i.e. in this case

As already mentioned, the state of a certain mass of gas is determined by three thermodynamic parameters: pressure R, volume V and temperature T. There is a certain relationship between these parameters, called the equation of state, which is generally given by the expression

where each of the variables is a function of the other two.

The French physicist and engineer B. Clapeyron (1799-1864) derived the equation of state for an ideal gas by combining the laws of Boyle - Mariotte and Gay-Lussac. Let some mass of gas occupy a volume V 1 , has pressure p 1 and is at temperature T 1 . The same mass of gas in another arbitrary state is characterized by the parameters p 2 , V 2 , T 2 (Fig. 63). The transition from state 1 to state 2 is carried out in the form of two processes: 1) isothermal (isotherm 1 - 1¢, 2) isochoric (isochore 1¢ - 2).

In accordance with the laws of Boyle - Mariotte (41.1) and Gay-Lussac (41.5), we write:

(42.1) (42.2)

Eliminating from equations (42.1) and (42.2) p¢ 1 , we get

Since states 1 and 2 were chosen arbitrarily, for a given mass of gas, the value pV/T remains constant, i.e.

Expression (42.3) is the Clapeyron equation, in which IN is the gas constant, different for different gases.

The Russian scientist D. I. Mendeleev (1834-1907) combined Clapeyron's equation with Avogadro's law, referring equation (42.3) to one mole, using the molar volume V m . According to Avogadro's law, for the same R And T moles of all gases occupy the same molar volume V m , so constant B will the same for all gases. This common constant for all gases is denoted R and is called the molar gas constant. Equation

(42.4)

satisfies only an ideal gas, and it is the equation of state of an ideal gas, also called the Clapeyron-Mendeleev equation.

The numerical value of the molar gas constant is determined from formula (42.4), assuming that a mole of gas is under normal conditions (p 0 = 1.013×10 5 Pa, T 0 = 273.15 K, V m = 22.41×10 -3 me/mol): R = 8.31 J/(mol×K).

From equation (42.4) for a mole of gas, one can pass to the Clapeyron-Mendeleev equation for an arbitrary mass of gas. If, at some given pressure and temperature, one mole of a gas occupies a molar volume V m , then under the same conditions the mass m of the gas will occupy the volume V \u003d (t / M) × V m, where M- molar mass (mass of one mole of a substance). The unit of molar mass is the kilogram per mole (kg/mol). Clapeyron - Mendeleev equation for mass T gas

(42.5)

where v=m/M- amount of substance.

Often they use a slightly different form of the ideal gas equation of state, introducing the Boltzmann constant:

Proceeding from this, we write the equation of state (42.4) in the form

where N A /V m \u003d n is the concentration of molecules (the number of molecules per unit volume). Thus, from the equation

it follows that the pressure of an ideal gas at a given temperature is directly proportional to the concentration of its molecules (or the density of the gas). At the same temperature and pressure, all gases contain the same number of molecules per unit volume. The number of molecules contained in 1 m 3 of gas at normal conditions is called the Loschmant number*:

Basic Equation

Molecular Kinetic Theory

Ideal gases

To derive the basic equation of the molecular kinetic theory, we consider a one-atomic ideal gas. Let us assume that gas molecules move randomly, the number of mutual collisions between gas molecules is negligibly small compared to the number of impacts on the walls of the vessel, and the collisions of molecules with the walls of the vessel are absolutely elastic. On the wall of the vessel, we single out some elementary area D S(Fig. 64) and calculate the pressure exerted on this area. With each collision, a molecule moving perpendicular to the site transfers momentum to it m 0 v -(- t 0) = 2t 0 v, where m 0 is the mass of the molecule, v is its speed. For time D t sites D S only those molecules are reached that are enclosed in the volume of the cylinder with the base D S and height vDt (Fig. 64). The number of these molecules is equal to nDSvDt (n is the concentration of molecules).

However, it must be taken into account that the molecules actually move towards the DS area at different angles and have different velocities, and the molecular velocity changes with each collision. To simplify the calculations, the chaotic movement of molecules is replaced by movement along three mutually perpendicular directions, so that at any time 1/3 of the molecules move along each of them, and half of the molecules - 1/6 - move along this direction in one direction, half - in the opposite direction. . Then the number of impacts of molecules moving in a given direction on the site D S will

l/6nDSvDt . When colliding with the platform, these molecules will transfer momentum to it.

Then the pressure of the gas exerted by it on the wall of the vessel,

If the gas is in volume V contains N molecules moving with velocities v 1 ,v 2 , ..., v n , then it is advisable to consider the root mean square velocity

(43.2)

characterizing the entire set of pelvic molecules. Equation (43.1), taking into account (43.2), takes the form

(43.3)

Expression (43.3) is called the basic equation of the molecular-kinetic theory of ideal gases. Exact calculation, taking into account the movement of molecules in all possible directions, gives the same formula.

Given that n=N/V, we get

where E is the total kinetic energy of the translational motion of all gas molecules.

Since the mass of the gas m=Nm 0 , then equation (43.4) can be rewritten as

For one mole of gas t = M(M- molar mass), so

where F m is the molar volume. On the other hand, according to the Clapeyron-Mendeleev equation, pV m = RT. In this way,

(43.6)

Since M \u003d m 0 N A is the mass of one molecule, and N A is Avogadro's constant, it follows from equation (43.6) that

(43.7)

where k=R/N A is Boltzmann's constant. From here we find that at room temperature the oxygen molecules have a root-mean-square velocity of 480 m/s, hydrogen - 1900 m/s. At the temperature of liquid helium, the same velocities will be 40 and 160 m/s, respectively.

Average kinetic energy of the translational motion of one molecule of an ideal gas

(we used formulas (43.5) and (43.7)) is proportional to the thermodynamic temperature and depends only on it. From this equation it follows that at T=0 = 0, i.e., at 0 K, the translational motion of gas molecules stops, and, consequently, its pressure is zero. Thus, the thermodynamic temperature is a measure of the average kinetic energy of the translational motion of ideal gas molecules, and formula (43.8) reveals the molecular-kinetic interpretation of temperature.

Gas is one of the four states of aggregation in which matter can be.

The particles that make up a gas are very mobile. They move almost freely and randomly, periodically colliding with each other like billiard balls. Such a collision is called elastic collision . During a collision, they dramatically change the nature of their movement.

Since in gaseous substances the distance between molecules, atoms and ions is much greater than their size, these particles interact very weakly with each other, and their potential energy of interaction is very small compared to the kinetic one.

The bonds between molecules in a real gas are complex. Therefore, it is also quite difficult to describe the dependence of its temperature, pressure, volume on the properties of the molecules themselves, their quantity, and the speed of their movement. But the task is greatly simplified if, instead of a real gas, we consider its mathematical model - ideal gas .

It is assumed that in the ideal gas model there are no forces of attraction and repulsion between molecules. They all move independently of each other. And the laws of classical Newtonian mechanics can be applied to each of them. And they interact with each other only during elastic collisions. The time of the collision itself is very short compared to the time between collisions.

Classical ideal gas

Let's try to imagine the molecules of an ideal gas as small balls located in a huge cube at a great distance from each other. Because of this distance, they cannot interact with each other. Therefore, their potential energy is zero. But these balls move with great speed. This means they have kinetic energy. When they collide with each other and with the walls of the cube, they behave like balls, that is, they rebound elastically. At the same time, they change the direction of their movement, but do not change their speed. This is what the movement of molecules in an ideal gas looks like.

1. The potential energy of interaction between molecules of an ideal gas is so small that it is neglected in comparison with the kinetic energy.
2. Molecules in an ideal gas are also so small that they can be considered material points. And this means that they total volume is also negligible compared to the volume of the container containing the gas. And this volume is also neglected.
3. The average time between collisions of molecules is much longer than the time of their interaction during a collision. Therefore, the interaction time is also neglected.

A gas always takes the shape of the container it is in. The moving particles collide with each other and with the walls of the vessel. During the impact, each molecule acts on the wall with some force for a very short period of time. This is how pressure . The total gas pressure is the sum of the pressures of all molecules.

Ideal gas equation of state

The state of an ideal gas is characterized by three parameters: pressure, volume And temperature. The relationship between them is described by the equation:

where R - pressure,

V M - molar volume,

R is the universal gas constant,

T - absolute temperature (degrees Kelvin).

Because V M = V / n , where V - volume, n is the amount of substance, and n= m/M , then

where m - mass of gas, M - molar mass. This equation is called the Mendeleev-Claiperon equation .

At constant mass, the equation takes the form:

This equation is called unified gas law .

Using the Mendeleev-Klaiperon law, one of the gas parameters can be determined if the other two are known.

isoprocesses

With the help of the unified gas law equation, it is possible to study processes in which the mass of the gas and one of the most important parameters - pressure, temperature or volume - remain constant. In physics, such processes are called isoprocesses .

From From the unified gas law, other important gas laws follow: boyle-mariotte law, Gay-Lussac's law, Charles' law, or Gay-Lussac's second law.

Isothermal process

A process in which pressure or volume changes but the temperature remains constant is called isothermal process .

In an isothermal process T = const, m = const .

The behavior of a gas in an isothermal process describes boyle-mariotte law . This law was discovered experimentally English physicist Robert Boyle in 1662 and French physicist Edme Mariotte in 1679. And they did it independently of each other. Boyle-Mariotte's law is formulated as follows: In an ideal gas at constant temperature, the product of the pressure of the gas and its volume is also constant.

The Boyle-Mariotte equation can be derived from the unified gas law. Substituting into the formula T = const , we get

p · V = const

That's what it is boyle-mariotte law . It can be seen from the formula that The pressure of a gas at constant temperature is inversely proportional to its volume.. The higher the pressure, the lower the volume, and vice versa.

How to explain this phenomenon? Why does the pressure decrease as the volume of a gas increases?

Since the temperature of the gas does not change, the frequency of impacts of molecules on the walls of the vessel does not change either. If the volume increases, then the concentration of molecules becomes smaller. Consequently, per unit area there will be a smaller number of molecules that collide with the walls per unit time. The pressure drops. As the volume decreases, the number of collisions, on the contrary, increases. Accordingly, the pressure also increases.

Graphically, the isothermal process is displayed on the plane of the curve, which is called isotherm . She has the shape hyperbole.

Each temperature value has its own isotherm. The higher the temperature, the higher is the corresponding isotherm.

isobaric process

The processes of changing the temperature and volume of a gas at constant pressure are called isobaric . For this process m = const, P = const.

The dependence of the volume of gas on its temperature at a constant pressure was also established experimentally French chemist and physicist Joseph Louis Gay-Lussac who published it in 1802. Therefore, it is called Gay-Lussac's law : " Etc and constant pressure, the ratio of the volume of a constant mass of a gas to its absolute temperature is a constant value.

At P = const the unified gas law equation becomes Gay-Lussac equation .

An example of an isobaric process is a gas inside a cylinder in which a piston moves. As the temperature rises, the frequency of molecular collisions with the walls increases. The pressure increases and the piston rises. As a result, the volume occupied by the gas in the cylinder increases.

Graphically, the isobaric process is represented by a straight line called isobar .

The higher the pressure in the gas, the lower the corresponding isobar is located on the graph.

Isochoric process

isochoric, or isochoric, called the process of changing the pressure and temperature of an ideal gas at a constant volume.

For isochoric process m = const, V = const.

It is very easy to imagine such a process. It takes place in a vessel of a fixed volume. For example, in a cylinder, the piston in which does not move, but is rigidly fixed.

The isochoric process is described Charles law : « For a given mass of gas at constant volume, its pressure is proportional to temperature". The French inventor and scientist Jacques Alexander Cesar Charles established this relationship with the help of experiments in 1787. In 1802 Gay-Lussac specified it. Therefore, this law is sometimes called Gay-Lussac's second law.

At V = const from the unified gas law equation we get the equation charles law, or Gay-Lussac's second law .

At constant volume, the pressure of a gas increases when its temperature increases. .

On the graphs, the isochoric process is displayed by a line called isochore .

The larger the volume occupied by the gas, the lower is the isochore corresponding to this volume.

In reality, no gas parameter can be kept constant. This can only be done in laboratory conditions.

Of course, an ideal gas does not exist in nature. But in real rarefied gases at very low temperatures and pressures not exceeding 200 atmospheres, the distance between molecules is much greater than their size. Therefore, their properties approach those of an ideal gas.

State equationideal gas(sometimes the equationClapeyron or the equationMendeleev - Clapeyron) is a formula that establishes the relationship between pressure, molar volume and absolute temperature of an ideal gas. The equation looks like:

Since , where is the amount of substance, and , where is the mass, is the molar mass, the equation of state can be written:

This form of writing is named after the equation (law) of Mendeleev - Clapeyron.

In the case of a constant gas mass, the equation can be written as:

The last equation is called unified gas law. From it are obtained the laws of Boyle - Mariotte, Charles and Gay-Lussac:

- Boyle's Law - Mariotte.

- Gay-Lussac's law.

- lawCharles(second law of Gay-Lussac, 1808). And in the form of a proportion this law is convenient for calculating the transition of a gas from one state to another. From the point of view of a chemist, this law may sound somewhat different: The volumes of reacting gases under the same conditions (temperature, pressure) are related to each other and to the volumes of gaseous compounds formed as simple integers. For example, 1 volume of hydrogen combines with 1 volume of chlorine to form 2 volumes of hydrogen chloride:

1 volume of nitrogen combines with 3 volumes of hydrogen to form 2 volumes of ammonia:

- Boyle's Law - Mariotte. Boyle's law - Mariotte is named after the Irish physicist, chemist and philosopher Robert Boyle (1627-1691), who discovered it in 1662, and also after the French physicist Edme Mariotte (1620-1684), who discovered this law independently of Boyle in 1677. In some cases (in gas dynamics), it is convenient to write the equation of state for an ideal gas in the form

where is the adiabatic exponent, is the internal energy of a unit mass of a substance. Emil Amaga discovered that at high pressures, the behavior of gases deviates from the Boyle-Mariotte law. And this circumstance can be clarified on the basis of molecular concepts.

On the one hand, in highly compressed gases, the sizes of the molecules themselves are comparable with the distances between the molecules. Thus, the free space in which the molecules move is less than the total volume of the gas. This circumstance increases the number of molecular impacts on the wall, since it reduces the distance that a molecule must travel to reach the wall. On the other hand, in a highly compressed and therefore denser gas, molecules are noticeably attracted to other molecules much more of the time than molecules in a rarefied gas. This, on the contrary, reduces the number of molecular impacts on the wall, since in the presence of attraction to other molecules, the gas molecules move towards the wall at a lower speed than in the absence of attraction. At not too high pressures, the second circumstance is more significant and the product decreases slightly. At very high pressures, the first circumstance plays an important role and the product increases.

5. Basic equation of the molecular-kinetic theory of ideal gases

To derive the basic equation of the molecular kinetic theory, we consider a monatomic ideal gas. Let us assume that gas molecules move randomly, the number of mutual collisions between gas molecules is negligibly small compared to the number of impacts on the walls of the vessel, and the collisions of molecules with the walls of the vessel are absolutely elastic. Let us single out some elementary area DS on the vessel wall and calculate the pressure exerted on this area. With each collision, a molecule moving perpendicular to the site transfers momentum to it m 0 v-(-m 0 v)=2m 0 v, where T 0 is the mass of the molecule, v - her speed.

During the time Dt of the platform DS, only those molecules are reached that are enclosed in the volume of a cylinder with a base DS and a height v D t .The number of these molecules is n D Sv D t (n- concentration of molecules).

However, it is necessary to take into account that the molecules actually move towards the area

DS at different angles and have different speeds, and the speed of the molecules changes with each collision. To simplify the calculations, the chaotic motion of molecules is replaced by motion along three mutually perpendicular directions, so that at any time 1/3 of the molecules move along each of them, and half of the molecules (1/6) move along this direction in one direction, half in the opposite direction. . Then the number of impacts of molecules moving in a given direction on the platform DS will be 1/6 nDSvDt. When colliding with the platform, these molecules will transfer momentum to it.

D R = 2m 0 v 1 / 6 n D Sv D t= 1 / 3n m 0 v 2D S D t.

Then the pressure of the gas exerted by it on the wall of the vessel,

p=DP/(DtDS)= 1 / 3 nm 0 v 2 . (3.1)

If the gas is in volume V contains N molecules,

moving at speeds v 1 , v 2 , ..., v N, then

appropriate to consider root mean square speed

characterizes the totality of gas molecules.

Equation (3.1), taking into account (3.2), takes the form

p = 1 / 3 Fri 0 2 . (3.3)

Expression (3.3) is called the basic equation of the molecular-kinetic theory of ideal gases. Accurate calculation, taking into account the movement of molecules throughout

possible directions gives the same formula.

Given that n = N/V, we get

where E is the total kinetic energy of the translational motion of all gas molecules.

Since the mass of the gas m =Nm 0 , then equation (3.4) can be rewritten as

pV= 1 / 3 m 2 .

For one mole of gas t = M (M - molar mass), so

pV m = 1 / 3 M 2 ,

where V m - molar volume. On the other hand, according to the Clapeyron-Mendeleev equation, pV m =RT. In this way,

RT= 1 / 3 M 2 , whence

Since M \u003d m 0 N A, where m 0 is the mass of one molecule, and N A is the Avogadro constant, it follows from equation (3.6) that

where k = R/N A is the Boltzmann constant. From here we find that at room temperature the oxygen molecules have a root-mean-square velocity of 480 m/s, hydrogen - 1900 m/s. At the temperature of liquid helium, the same velocities will be 40 and 160 m/s, respectively.

Average kinetic energy of the translational motion of one molecule of an ideal gas

) 2 /2 = 3 / 2 kT(43.8)

(we used formulas (3.5) and (3.7)) is proportional to the thermodynamic temperature and depends only on it. It follows from this equation that at T=0 =0,t. That is, at 0 K, the translational motion of gas molecules stops, and, consequently, its pressure is zero. Thus, the thermodynamic temperature is a measure of the average kinetic energy of the translational motion of ideal gas molecules, and formula (3.8) reveals the molecular-kinetic interpretation of temperature.