Use of the periodic clapeyron formula in car engines. Mendeleev-Clapeyron equation

Gas is one of four states of aggregation in which a substance can exist.

The particles that make up the gas are very mobile. They move almost freely and chaotically, 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 sizes, these particles interact very weakly with each other, and their potential interaction energy is very small compared to the kinetic energy.

The connections 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 real gas, we consider it mathematical model - ideal gas .

It is assumed that in the model ideal gas There are no forces of attraction or 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 at 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 bounce elastically. At the same time, they change the direction of their movement, but do not change their speed. This is roughly 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 compared to 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 vessel in which the gas is located. And this volume is also neglected.
3. The average time between collisions of molecules is much greater than the time of their interaction during a collision. Therefore, the interaction time is also neglected.

Gas always takes the shape of the container in which it is located. Moving particles collide with each other and with the walls of the container. During an impact, each molecule exerts some force on the wall for a very short period of time. This is how it arises 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 - universal gas constant,

T - absolute temperature (degrees Kelvin).

Because V M = V / n , Where V - volume, n - the amount of substance, and n= m/M , That

Where m - gas mass, M - molar mass. This equation is called Mendeleev-Clayperon equation .

At constant mass the equation becomes:

This equation is called united gas law .

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

Isoprocesses

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

From The unified gas law leads to other important gas laws: Boyle-Mariotte law, Gay-Lussac's law, Charles's law, or Gay-Lussac's second law.

Isothermal process

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

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

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

The Boyle-Marriott 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 . From the formula it is clear 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 of a gas decrease as the volume of a gas increases?

Since the temperature of the gas does not change, the frequency of collisions of molecules with the walls of the vessel does not change. If the volume increases, the concentration of molecules becomes less. Consequently, per unit area there will be fewer molecules that collide with the walls per unit time. The pressure drops. As the volume decreases, the number of collisions, on the contrary, increases. The pressure increases accordingly.

Graphically, an isothermal process is displayed on a curve plane, which is called isotherm . She has a shape hyperboles.

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

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

At P = const the equation of the unified gas law turns into Gay-Lussac equation .

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

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

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

Isochoric process

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

For an isochoric process m = const, V = const.

It is very simple to imagine such a process. It occurs 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's law : « For a given mass of gas at constant volume, its pressure is proportional to temperature" The French inventor and scientist Jacques Alexandre César Charles established this relationship through experiments in 1787. In 1802, it was clarified by Gay-Lussac. Therefore this law is sometimes called Gay-Lussac's second law.

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

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

On graphs, an isochoric process is represented by a line called isochore .

How more volume occupied by the gas, the lower is the isochore corresponding to this volume.

In reality, no gas parameter can be maintained unchanged. 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 the molecules is much greater than their sizes. Therefore, their properties approach those of an ideal gas.

1. An ideal gas is a gas in which there are no intermolecular interaction forces. With a sufficient degree of accuracy, gases can be considered ideal in cases where their states are considered that are far from the regions of phase transformations.
2. For ideal gases the following laws are valid:

a) Boyle’s Law - Mapuomma: at constant temperature and mass, the product of the numerical values ​​of pressure and volume of a gas is constant:
pV = const

Graphically, this law in PV coordinates is depicted by a line called an isotherm (Fig. 1).

b) Gay-Lussac's law: at constant pressure, the volume of a given mass of gas is directly proportional to its absolute temperature:
V = V0(1 + at)

where V is the volume of gas at temperature t, °C; V0 is its volume at 0°C. The quantity a is called the temperature coefficient of volumetric expansion. For all gases a = (1/273°С-1). Hence,
V = V0(1 +(1/273)t)

Graphically, the dependence of volume on temperature is depicted by a straight line - an isobar (Fig. 2). At very low temperatures (close to -273°C), Gay-Lussac's law is not satisfied, therefore solid line replaced by a dotted line on the graph.

c) Charles’s law: at constant volume, the pressure of a given mass of gas is directly proportional to its absolute temperature:
p = p0(1+gt)

where p0 is the gas pressure at temperature t = 273.15 K.
The quantity g is called the temperature coefficient of pressure. Its value does not depend on the nature of the gas; for all gases = 1/273 °C-1. Thus,
p = p0(1 +(1/273)t)

The graphical dependence of pressure on temperature is depicted by a straight line - an isochore (Fig. 3).

d) Avogadro's law: at the same pressures and the same temperatures and equal volumes various ideal gases contained same number molecules; or, what is the same: at the same pressures and the same temperatures, the gram molecules of different ideal gases occupy the same volumes.
So, for example, when normal conditions(t = 0°C and p = 1 atm = 760 mm Hg) gram molecules of all ideal gases occupy a volume Vm = 22.414 l. · The number of molecules contained in 1 cm3 of an ideal gas under normal conditions is called the Loschmidt number; it is equal to 2.687*1019> 1/cm3
3. The equation of state of an ideal gas has the form:
pVm = RT

where p, Vm and T are the pressure, molar volume and absolute temperature of the gas, and R is the universal gas constant, numerically equal to the work done by 1 mole of an ideal gas when heated isobarically by one degree:
R = 8.31*103 J/(kmol*deg)

For an arbitrary mass M of gas, the volume will be V = (M/m)*Vm and the equation of state has the form:
pV = (M/m) RT

This equation is called the Mendeleev-Clapeyron equation.
4. From the Mendeleev-Clapeyron equation it follows that the number n0 of molecules contained in a unit volume of an ideal gas is equal to
n0 = NA/Vm = p*NA /(R*T) = p/(kT)

where k = R/NA = 1/38*1023 J/deg - Boltzmann's constant, NA - Avogadro's number.

We take the formula and substitute it into it. We get:

p= nkT.

Recall now that A, where ν - number of moles of gas:

,

pV= νRT.(3)

Relationship (3) is called Mendeleev - Clapeyron equation. It gives the relationship between the three most important macroscopic parameters that describe the state of an ideal gas - pressure, volume and temperature. Therefore, the Mendeleev-Clapeyron equation is also called ideal gas equation of state.

Considering that , Where m- gas mass, we obtain another form of the Mendeleev-Clapeyron equation:

(4)

There is another useful variation of this equation. Let's divide both parts by V:

But - gas density. From here

(5)

In physics problems, all three forms of notation (3)-(5) are actively used.

Isoprocesses

Throughout this section we will make the following assumption: mass and chemical composition gas remain unchanged. In other words, we believe that:

m= const, that is, there is no gas leakage from the vessel or, conversely, gas inflow into the vessel;

µ = const, 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

Thermodynamic process(or simply 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 occurs at a constant gas temperature: T= const.

2. Isobaric process runs at constant gas pressure: p= const.

3. Isochoric process occurs at a constant volume of gas: V= const.

Isoprocesses are described very simple laws Boyle - Mariotte, Gay-Lussac and Charles. Let's move on to studying them.

Isothermal process

In an isothermal process, the gas temperature is constant. During the process, only the gas pressure and its volume change.

Let's establish a connection between pressure p and volume V gas in an isothermal process. Let the gas temperature be T. Let us consider two arbitrary states of the gas: in one of them the values ​​of macroscopic parameters are equal p 1 ,V 1 ,T, and in the second - p 2 ,V 2 ,T. These values ​​are related by the Mendeleev-Clapeyron equation:

As we said from the very beginning, the mass of gas m and its molar mass µ are assumed unchanged. Therefore, the right sides of the written equations are equal. Therefore, the left sides are also equal: p 1V 1 = p 2V 2.

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:

pV= const .

This statement is called Boyle-Mariotte law. Having written the Boyle-Mariotte law in the form

p= ,

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.

It is known that rarefied gases are subject to the laws of Boyle and Guey-Lussac. Boyle's law states that during isothermal compression of a gas, the pressure changes in inverse proportion to the volume. Therefore, when

According to the Gay-Lussac law, heating a gas at constant pressure entails its expansion by the volume that it occupies 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 then

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; Fig. 184 shows lines, for each of which these are gas isotherms).

Let us imagine that the gas was taken in some arbitrarily chosen state C, in 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 Guey-Lussac.

Let's cool it to without changing the pressure (Fig. 185). Based on Guey-Lussac's law, we can write that

Now, maintaining the temperature, we will compress the gas or, if necessary, provide it with the opportunity to expand until its pressure becomes equal to one physical atmosphere. We denote this pressure by a the volume that will eventually be occupied by gas (at through (point in Fig. 185). Based on Boyle’s law

Multiplying the first equality term by term 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 Railways from 1820 to 1830. The constant value 27516 is called 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 (or, what is the same, the gram-molecule, gram-mole) as the unit of mass, Avogadro’s law can be formulated as follows: when certain temperature and at 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, Guey-Lussac and Avogadro, found experimentally, were later derived theoretically from molecular kinetic concepts (by 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 a substance, the molecular weight of this substance: Obviously, the number of molecules contained in a mole of a substance is equal to:

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

D.I. Mendeleev pointed out in 1874 that thanks to Avogadro’s law, the Clapeyron equation, synthesizing the laws of Boyle and Guey-Lussac, acquires the greatest generality when it is related not to the usual weight unit (gram or kilogram), but to the 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 must 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 volume y (and therefore 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 quantities 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 energy units, then the gas constant must also be expressed in calories; approximately, more precisely

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 of

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 a little less, for hydrogen it is a little more). If we take this into account when calculating, we will find some discrepancy in the numerical value for different chemical nature gases So, for oxygen instead it turns out for nitrogen. This discrepancy is due to the fact that all gases at ordinary densities do not quite accurately follow the laws of Boyle and Guey-Lussac.

In technical calculations, instead of measuring the mass of a gas in moles, the mass of the 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

Every student in the tenth grade, in one of the physics lessons, studies the Clapeyron-Mendeleev law, its formula, formulation, and learns how to apply it in solving problems. IN technical universities this topic is also included in the course of lectures and practical work, and in several disciplines, not just physics. The Clapeyron-Mendeleev law is actively used in thermodynamics when drawing up equations of state for an ideal gas.

Thermodynamics, thermodynamic states and processes

Thermodynamics is a branch of physics that is devoted to the study general properties bodies and thermal phenomena in these bodies without taking into account their molecular structure. Pressure, volume and temperature are the main quantities taken into account when describing thermal processes in bodies. A thermodynamic process is a change in the state of a system, i.e. a change in its basic quantities (pressure, volume, temperature). Depending on whether changes in basic quantities occur, systems can be equilibrium or nonequilibrium. Thermal (thermodynamic) processes can be classified as follows. That is, if a system passes from one equilibrium state to another, then such processes are called, accordingly, equilibrium. Nonequilibrium processes, in turn, are characterized by transitions of nonequilibrium states, that is, the main quantities undergo changes. However, they (the processes) can be divided into reversible (a reverse transition through the same states is possible) and irreversible. All states of the system can be described by certain equations. To simplify calculations in thermodynamics, the concept of an ideal gas is introduced - a certain abstraction that is characterized by the absence of interaction at a distance between molecules, the dimensions of which can be neglected due to their small size. The basic gas laws and the Mendeleev-Clapeyron equation are closely interrelated - all laws follow from the equation. They describe isoprocesses in systems, that is, processes as a result of which one of the main parameters remains unchanged (isochoric process - volume does not change, isothermal - constant temperature, isobaric - temperature and volume change at constant pressure). The Clapeyron-Mendeleev law is worth examining in more detail.

Ideal gas equation of state

The Clapeyron-Mendeleev law expresses the relationship between pressure, volume, temperature, and the amount of substance of an ideal gas. It is also possible to express the relationship only between the basic parameters, that is, absolute temperature, molar volume and pressure. The essence does not change, since the molar volume is equal to the ratio of the volume to the amount of substance.

Mendeleev-Clapeyron law: formula

The equation of state of an ideal gas is written as the product of pressure and molar volume, equated to the product of the universal gas constant and absolute temperature. The universal gas constant is a coefficient of proportionality, a constant (unchangeable value) expressing the work of expansion of a mole in the process of increasing the temperature value by 1 Kelvin under conditions of an isobaric process. Its value is (approximately) 8.314 J/(mol*K). If we express the molar volume, we get an equation of the form: р*V=(m/М)*R*Т. Or it can be put in the form: p=nkT, where n is the concentration of atoms, k is Boltzmann’s constant (R/NA).

Problem solving

The Mendeleev-Clapeyron law and solving problems with its help greatly facilitate the calculation part when designing equipment. When solving problems, the law is applied in two cases: one state of the gas and its mass are given, and if the value of the gas mass is unknown, the fact of its change is known. It must be taken into account that in the case of multicomponent systems (mixtures of gases), an equation of state is written for each component, i.e., for each gas separately. Dalton's law is used to establish the relationship between the pressure of the mixture and the pressures of the components. It is also worth remembering that for each state of the gas it is described by a separate equation, then the already obtained system of equations is solved. And finally, you must always remember that in the case of the equation of state of an ideal gas, temperature is an absolute value; its value is necessarily taken in Kelvin. If in the conditions of the problem the temperature is measured in degrees Celsius or in any other way, then it is necessary to convert to degrees Kelvin.