Isotherms at low temperature ( T< T k ) to one pressure value, for example, p 1 corresponds to three volume values V 1 , V 2 and V 3, and at T > Tk— one volume value. At the critical point, all three volumes (three roots) coincide and are equal V k.

Consider the isotherm at T< T k in Fig. 9.12.

Rice. 9.12 Fig. 9.13

In sections 1-3 and 5-7 with a decrease in volume Vm pressure p increases. In section 3-5, compression of the substance leads to a decrease in pressure; practice shows that such states do not occur in nature. The presence of section 3-5 means that with a gradual change in volume, the substance cannot remain all the time in the form of a homogeneous medium; At some point, an abrupt change in state and the decomposition of the substance into two phases must occur. Thus, the true isotherm looks like a broken line 7-6-2-1. Part 7-6 corresponds to the gaseous state, and part 2-1 corresponds to the liquid state. In states corresponding to the horizontal section of isotherm 6-2, equilibrium is observed between the liquid and gaseous phases of the substance.

If through extreme points draw a line across the horizontal sections of the family of isotherms, you will get a bell-shaped curve (Fig. 9.13), limiting the region of two-phase states of matter. This curve and the critical isotherm divide the diagram p,Vm under the isotherm into three regions: under the bell-shaped curve there is a region of two-phase states (liquid and saturated steam), to the left of it is the region of the liquid state, and to the right is the region pair. Steam- a substance in a gaseous state at a temperature below critical. Saturated steam- vapor in equilibrium with its liquid.

Problems for chapters 8, 9

1. Consider a model of an ideal gas enclosed in a vessel. Overestimated or underestimated compared to real gas (for given V And T) values: a) internal energy; b) gas pressure on the wall of the vessel?

2. The internal energy of some gas is 55 MJ, and the fraction of energy rotational movement accounts for 22 MJ. How many atoms are there in a molecule of this gas?

3. The molecules of which of the listed gases that make up air in an equilibrium state have the highest average arithmetic speed? 1)N 2 ; 2) O 2; 3) H 2; 4) CO 2 .

4. Some gas with a constant mass is transferred from one equilibrium state to another. Does the distribution of molecules by speed change: a) the position of the maximum of the Maxwell curve; b) area under this curve?

5. The volume of gas increases and the temperature decreases. How does pressure change? The mass is constant.

6. During adiabatic expansion of a gas, its volume changes from V 1 to V 2. Compare pressure ratios ( p 1 /p2), if the gas is: a) monatomic; b) diatomic.

7. A balloon with an elastic hermetic shell rises in the atmosphere. Temperature and air pressure decrease with altitude. Does the lifting force of a balloon depend on: a) air pressure; b) on temperature?

8. The figure shows adiabats for two gases H 2 and Ar. Indicate which graphs correspond to H 2. 1)I, III; 2)I, IV; 3)II, III; 4)II,IV.

9. Compare the work of gas expansion during an isothermal change in volume from 1 to 2 m 3 and from 2 to 4 m 3.

10. A gas, expanding, passes from the same state with the volume V 1 to volume V 2: a) isobaric; b) adiabatically; c) isothermal. In which processes does the gas do the least and most work?

11. Which of the following gases at room temperature has the highest specific heat capacity?

1) O 2; 2) H 2; 3) He; 4) Ne; 5) I 2.

12. How it changes internal energy gas in expansion processes: a) isobaric; b) in adiabatic?

13. An unknown gas is given. Is it possible to find out what kind of gas it is if given:

A) p, V, T, m; b) p, T, r; c) g, C V? Applicable to gas classical theory heat capacities.

14. Determine the signs of the molar heat capacity of the gas ( m=const, gas molecules are rigid) in a process for which T 2 V= const, if the gas is: a) monatomic; b) diatomic.

15. Let's move from the ideal gas model to a model that takes into account the forces of attraction between molecules. How do molar heat capacities change? C V And C p for given V And T?

16. Ideal gas containing N molecules expands at constant temperature. According to what law does the number of gas microstates increase? w? 1) w~V; 2) w~V N; 3) w~ln V; 4) the correct ratio is not given.

Critical phenomena

Isotherm at temperature T s plays a special role in the theory of the state of matter. Isotherm corresponding to the temperature below T s> behaves as already described: at a certain pressure, the gas condenses into a liquid, which can be distinguished by the presence of an interface. If compression is carried out at T s, then the surface separating the two phases does not appear, and the condensation point and the point of complete transition to liquid merge into one critical point of the gas. At temperatures above T s gas cannot be converted into liquid by any compression. The temperature, pressure and molar volume at the critical point are called the critical temperature T s, critical pressure r s and critical molar volume Vc substances. Collectively parameters R With, Vc, And T s are called the critical constants of a given gas (Table 10.2).

At T>T C the sample is a phase that completely occupies the volume of the container containing it, i.e. by definition is a gas. However, the density of this phase can be much greater than is typical for gases, so the name "supercritical fluid" is usually preferred. (supercritical fluid). When the points coincide T s And R s liquid and gas are indistinguishable.

Table 10.2

Critical constants and Boyle temperatures

At the critical point, the isothermal compressibility coefficient

equals infinity because

Therefore, near the critical point, the compressibility of the substance is so great that the acceleration of gravity leads to significant differences in density in the upper and lower parts of the vessel, reaching 10% in a column of substance only a few centimeters high. This makes it difficult to determine densities (specific volumes) and, accordingly, isotherms p - V near the critical point. At the same time, the critical temperature can be defined very precisely as the temperature at which the surface separating the gaseous and liquid phases disappears when heated and reappears when cooled. Knowing the critical temperature, it is possible to determine the critical density (and, accordingly, the critical molar volume), using the empirical rule of rectilinear diameter (Calete Mathias rule), according to which the average density of liquid and saturated vapor is linear function temperatures:

(10.2)

Where A And IN - constant quantities for a given substance. By extrapolating the straight line of average density to the critical temperature, the critical density can be determined. The high compressibility of matter near the critical point leads to an increase in spontaneous density fluctuations, which are accompanied by anomalous light scattering. This phenomenon is called critical opalescence.

Van der Waals equation

The equation of state and transport phenomena in real gases and liquids are closely related to the forces acting between molecules. Molecular statistical theory linking general properties with intermolecular forces, is now well developed for rarefied gases and to a lesser extent for dense gases and liquids. At the same time, measuring macroscopic properties makes it possible, in principle, to determine the law according to which forces act between molecules. Moreover, if the type of interaction is determined, then it becomes possible to obtain an equation of state or transfer coefficients for real gases.

For ideal gases, the equation of state is

This relationship is absolutely accurate in the case when the gas is very rarefied or its temperature is relatively high. However, already at atmospheric pressure and temperature, deviations from this law for real gas become noticeable.

Many attempts have been made to take into account deviations of the properties of real gases from the properties of an ideal gas by introducing various corrections into the equation of state of an ideal gas. Due to its simplicity and physical clarity, the most widely used equation is the van der Waals equation (1873).

Van der Waals made the first attempt to describe these deviations by obtaining equations of state for a real gas. Indeed, if the equation of state of an ideal gas pV = RT apply to real gases, then, firstly, by the volume that can vary up to a point, it is necessary to understand the volume of intermolecular space, since only this volume, like the volume of an ideal gas, can decrease to zero with an unlimited increase in pressure.

The first correction in the equation of state of an ideal gas considers the intrinsic volume occupied by the molecules of a real gas. In Dupre's equation (1864)

(10.3)

constant b takes into account the intrinsic molar volume of the molecules.

As the temperature decreases, intermolecular interaction in real gases leads to condensation (formation of liquid). Intermolecular attraction is equivalent to the existence of some internal pressure in the gas (sometimes called static pressure). Initially, the quantity was taken into account in general form in the Girn equation (1865)

J. D. Van der Waals in 1873 gave a functional interpretation of internal pressure. According to the van der Waals model, the attractive forces between molecules (van der Waals forces) are inversely proportional to the sixth power of the distance between them or the second power of the volume occupied by the gas. It is also believed that the forces of attraction are added to external pressure. Taking these considerations into account, the equation of state of an ideal gas is transformed into the van der Waals equation:

(10.5)

or for 1 mole

(10.6)

Van der Waals constant values a and b, which depend on the nature of the gas, but do not depend on temperature, are given in table. 10.3.

Equation (10.6) can be rewritten to express pressure explicitly

(10.7)

or volume

(10.8)

Table 10.3

Van der Waals constants for various gases

Equation (10.8) contains volume to the third power and therefore has three real roots, or one real and two imaginary.

At high temperatures, equation (10.8) has one real root, and as the temperature increases, the curves calculated from the van der Waals equation approach hyperbolas corresponding to the ideal gas equation of state.

In Fig. 10.4 shows isotherms calculated using the van der Waals equation for carbon dioxide (values ​​of constants A And b taken from table. 10.3). The figure shows that at temperatures below critical (31.04°C), instead of horizontal straight lines corresponding to the equilibrium of liquid and vapor, wavy curves are obtained 1-2-3-4-5 with three real roots, of which only two, at the points 1 and 5, physically feasible. Third root (point 3) is not physically real because it is located on a section of the curve 2-3-4, contradicting the condition of stability of a thermodynamic system -

Rice. 10.4. Van der Waals isotherms for CO 2

Conditions at the sites 1-2 And 5-4 , which correspond to supercooled steam and superheated liquid, respectively, are unstable (metastable) and can only be partially realized in special conditions. So, carefully squeezing the steam above the point 1 (see Fig. 10.4), you can climb along the curve 1-2. This requires the absence of condensation centers, and primarily dust, in the pair. In this case, the steam turns out to be supersaturated, i.e. supercooled state. Conversely, the formation of liquid droplets in such vapor is facilitated, for example, by ions entering it. This property of supersaturated vapor is used in the famous Wilson chamber (1912), used to detect charged particles. A moving charged particle, entering a chamber containing supersaturated vapor and colliding with molecules, forms ions along its path, creating a foggy trail - a track that is recorded photographically.

According to Maxwell's rule (the Maxwell construction ), which has a theoretical justification, in order for the calculated curve to correspond to the experimental equilibrium isotherm, it is necessary instead of the curve 1-2-3-4-5 draw a horizontal line 1-5 so that the area 1-2-3-1 And 3-4-5-3 were equal. Then the ordinate of the line 1-5 will be equal to the saturated vapor pressure, and the abscissas of the points 1 and 5 - molar volumes of vapor and liquid at a given temperature.

As the temperature rises, all three roots move closer together, and at a critical temperature T s become equal. At the critical point, the van der Waals isotherm has an inflection point

with horizontal tangent

(10.9)

(10.10)

The joint solution of these equations gives

which makes it possible to determine the constants of the van der Waals equation from the critical parameters of the gas. Accordingly, according to the van der Waals equation, the critical compressibility factor Z c for all gases must be equal

From the table 10.2 it is obvious that although the value Z c for real gases it is approximately constant (0.27-0.30 for non-polar molecules), it is still noticeably less than that resulting from the van der Waals equation. For polar molecules, an even greater discrepancy is observed.

The fundamental significance of the van der Waals equation is determined by the following circumstances:

• 1) the equation was obtained from model concepts of the properties of real gases and liquids, and was not the result of an empirical selection of the function /(/?, V T), describing the properties of real gases;
• 2) the equation has long been considered as a certain general form of the equation of state of real gases, on the basis of which many other equations of state were constructed (see below);
• 3) using the van der Waals equation, for the first time it was possible to describe the phenomenon of the transition of gas into liquid and analyze critical phenomena. In this respect, the van der Waals equation has an advantage even over more accurate equations in virial form - see expressions (10.1), (10.2).

The reason for the insufficient accuracy of the van der Waals equation was the association of molecules in the gas phase, which cannot be described, taking into account the dependence of the parameters A And b on volume and temperature, without using additional constants. After 1873, Van der Waals himself proposed six more versions of his equation, the last of which dates back to 1911 and contains five empirical constants. Clausius proposed two modifications of equation (10.5), and both of them are associated with a complication of the form of the constant b. Boltzmann obtained three equations of this type by changing the expressions for the constant A. In total, more than a hundred similar equations are known, differing in the number of empirical constants, degree of accuracy and scope of applicability. It turned out that none of the equations of state containing less than five individual constants turned out to be accurate enough to describe real gases over a wide range p, V ", T, and all these equations turned out to be unsuitable in the field of gas condensation. From simple equations with two individual parameters, the Diterici and Berthelot equations give good results.

The ideal gas equation of state fairly well depicts the behavior of real gases at high temperatures and low pressures. However, when the temperature and pressure are such that the gas is close to condensation, then significant deviations from the laws of an ideal gas are observed.

Among a number of equations of state proposed to depict the behavior of real gases, the van der Waals equation is especially interesting due to its simplicity and due to the fact that it satisfactorily describes the behavior of many substances in a wide range of temperatures and pressures.

Van der Waals derived his equation from considerations based on kinetic theory, taking into account, as a first approximation, the size of the molecules and the forces of interaction between them. Its equation of state (written for one mole of substance) is:

where are constants depending on the characteristics of a given substance. When equation (99) turns into the ideal gas equation. The term describes the effect associated with the finite size of the molecules, and the term depicts the effect of molecular interaction forces.

In Fig. Figure 14 shows some isotherms calculated according to the van der Waals equation. Comparing these isotherms with the isotherms in Fig. 13, we see that their outlines have many similarities. In both cases, there is an inflection point on one isotherm. The isotherm containing the inflection point is the critical isotherm, and the inflection point itself is the critical point. Isotherms at temperatures above critical behave similarly in both cases. However, isotherms below the critical temperature differ significantly. Van der Waals isotherms are continuous curves with a minimum and a maximum, while the isotherms in Fig. 13

have two “corner” points and are horizontal in the region where the van der Waals isotherms contain a maximum and a minimum.

The reason for the qualitatively different behavior of the two families of isotherms in the region indicated in Fig. 13, is that the points of the horizontal segment of the isotherms in Fig. 13 do not correspond to a homogeneous state, since in these areas the substance is divided into liquid and vapor parts.

If we compress unsaturated steam isothermally until we reach saturation pressure, and then continue to reduce the volume, then the condensation of part of the steam is not accompanied by a further increase in pressure, which corresponds to the horizontal isotherms in Fig. 13. However, if the steam is compressed very carefully and kept free of dust particles, a pressure much higher than the saturation pressure at the time of condensation can be achieved. When this situation occurs, the steam becomes overheated. But the overheated state is unstable (labile). As a result of any even slight disturbance of the state, condensation can occur, and the system will go into a stable (stable) state, characterized by the presence of liquid and vapor parts.

Unstable states are important for our discussion, since they illustrate the possibility of the existence of homogeneous states in the range of parameter values ​​that are characteristic of saturated vapor above a liquid. Let us assume that these unstable states are depicted by a section of the van der Waals isotherm in Fig. 15. The horizontal section of a continuous isotherm shows stable states of liquid - vapor. If it were possible to realize all unstable states on the van der Waals isotherm, then they would resemble a continuous isothermal process from steam, shown by the isotherm section, to liquid, depicted by the section. If the van der Waals isotherm is known, then it is possible to determine what the pressure of saturated vapor at a given temperature, or, in geometric language, how high above the axis should a horizontal segment be drawn that corresponds to the state of liquid - vapor. Let us prove that this distance must be such that the areas and are equal. To prove this, we first show that the work done

system during a reversible isothermal cycle is always zero. From equation (16) it follows that the work done during the cycle is equal to the heat absorbed by the system. But for a reversible cycle, equality (66) remains in force, and since our cycle is isothermal, it can be taken out from under the integral sign in (66). Equation (66) shows that all the heat absorbed and, therefore, all the work performed during the cycle is equal to zero.

Now consider a reversible isothermal cycle (Fig. 15).

The work done during the cycle must go to zero.

The section is traversed clockwise, so the corresponding area is positive, and the section is traversed counterclockwise, and the corresponding area is negative. Since the entire area of ​​the cycle is zero, the absolute values ​​of the areas of the two cycles must be equal, which is what needed to be proven.

The following objection might arise to the above proof: since the area of ​​an isothermal cycle is obviously not zero, it is not true that the work done during a reversible isothermal cycle is always zero. The answer to this objection is that the cycle is not reversible.

To see this, note that a point on the diagram represents two different states, depending on whether it is considered as a point on the van der Waals isotherm or as a point on the liquid-vapor isotherm. The volume and pressure depicted by the dot are the same in both cases, but on the van der Waals isotherm D represents an unstable homogeneous (uniform) state, and on the liquid-vapor isotherm a stable inhomogeneous (inhomogeneous) state formed from liquid and gaseous parts. When we complete a cycle, we pass from a state on the van der Waals isotherm to a state on the liquid-vapor isotherm. Since the state on the liquid-vapor isotherm is more stable than on the van der Waals isotherm, this path is irreversible - it could not be spontaneously carried out in the opposite direction. Thus, the entire cycle is irreversible and therefore the area of ​​the cycle should not be zero.

The critical values ​​of a substance can be expressed through constants that are included in the van der Waals equation.

The van der Waals equation (99), when given, is an equation of third order with respect to Therefore, generally speaking, there are three different roots of V (for fixed values ​​However, the critical isotherm has a horizontal inflection point at i.e. at a third-order curve - critical isotherm - touches the horizontal line. It follows that the cubic equation for V, which is obtained if we put in has a triple root. This equation can be written in the form

Since the triple root of the above equation, the left-hand side must have the form Comparing, we find

Solving these three equations for we get

These equations express the critical values ​​in terms of

It is useful to note that if volume, pressure and temperature are used as units, then the van der Waals equation has the same form for all substances.

and using equalities (100), from (99) we obtain:

Since this equation contains only numerical constants, it is the same for all substances. States various substances, which are determined by the same quantities are called corresponding states, and (101) is often called the “van der Waals equation for corresponding states.”

In section 14 it was shown that if a substance obeys the equation of state of an ideal gas, then it can be deduced thermodynamically that its energy is determined only by temperature and does not depend on volume. This result is only true for

One of the first real gas equations. Proposed in 1873. physicist J. D. van der Waals. For a mole of gas with volume V at temperature T and pressure p, it has the form:

(p+a/V2)(V-b)=RT,

V. u. yavl. approximate and quantitatively determines the properties of real gases only in the region of high T and low r. However, qualitatively it allows us to describe the behavior of gas at high p, gas condensation and criticality. state.

The figure shows isotherms calculated according to V. at. At low T, all three roots of V. y. are real, and higher than critical. temperature (Tk) there is only one valid one left. root. This means that at T>TK the substance can only be in one (gaseous) state, and at T the pressure is saturated. a pair of RNP and volumes from Vzh to Vr.

Diagram of the state of the substance in coordinates p - V: T1, T2, T3, Tk - isotherms calculated according to the van der Waals equation; K - critical dot. The dKe line (spinodal) outlines the region of unstable states.

At lower p (beyond the region where the simultaneous existence of gas and liquid is possible) characterizes the properties of the gas. Left, almost vertical. part of the isotherm reflects small liquids. Sections ad and ec (and similar sections of other isotherms) belong respectively. to superheated liquid and supercooled steam (metastable states). Section de is physically unfeasible, since here V increases with increasing p. The set of points a, a", a" and c, c", c", . . . defines a curve called binodal, the edge outlines the region of coexistence of gas and liquid. In critical at point K the parameters Tk, pk and Vk have values ​​characteristic of a given island. However, if in V. u. enter relates. values ​​T/Tk, p/pk and V/VK, then we can obtain the so-called. given by V. u., which is a phenomenon. universal.

Physical encyclopedic Dictionary. - M.: Soviet Encyclopedia. . 1983 .

Equation of state of a real gas. Proposed by J. D. van der Waals in 1873. For a gas containing N molecules, V. at. has the form:

Where V- volume, R - pressure, T- abs. gas temp, A And b- constants, taking into account the attraction and repulsion of molecules. Member called internal pressure, constant b equal to quadruple the volume of gas, if weakly attracting elastic spheres are taken as a model.

V. u. quantitatively determines the properties of real gases only in a small range T And R - in the area of ​​relatively high T and low R, because A And b are temperature functions. However, V. u. qualitatively correctly describes the behavior of gas and liquid even at high R, as well as features of the phase transition between them. At low pressures and relatively high temperatures, it passes into the ideal gas state level ( Clapeyron equation), and at high pressures and low temp-pax takes into account the low compressibility of liquids. V. u. describes, in addition, the critical and metastable states of the system - par.

In Fig. given in coordinates p - V isotherms calculated using V. equation, which is cubic relative to V. There are 3 possible cases of solving a V. equation: 1) all three roots are real and equal to each other; this case corresponds to critical. state (isotherm T cr; 2) all three roots are real and distinct - the so-called. subcritical state (isotherms at T cr ); 3) two imaginary roots that do not have physical roots. meaning, one root is real; this case corresponds to supercritical. state (isotherms at T>T cr ). Isotherms at T/T kr qualitatively describe the behavior of real gases. At subcritical temp-pax T<T the behavior of the gas is described by the isotherm-isobar of saturated vapor - a straight line on the diagram p - V, eg. straight ac(p n.n. =const), not S-shaped curve adec, corresponding V. at.

Geom. the place of the initial and final points of “equilibrium” a and c of the stable and metastable phases (determined from the condition of equality of the shaded areas) is called. binodal (curve aKc). Curve connecting extreme points of type d And e, called spinodal (curve dKe). The region enclosed between the binodal and the spinodal is the region of the unstable, metastable state of the system. Thus, sections of isotherms like ad And EU refer to the metastable equilibrium of, respectively, a superheated liquid and a liquid+ system, as well as a liquid+gas system and a supercooled gas. Plot dbe has no physical meaning, because in this area, with growth R increases and V, which is impossible.

At sufficiently low temp-pax the area adb goes lower R=0. In this case, having a physical meaning plot ad will fall into the negative area. pressure, which corresponds to the unstable state of the stretched liquid.

State diagram of matter in coordinates p-V: T 1 T 3< T cr< T 4< Т 5 ,-isotherms calculated according to the V. equation; TO - critical point, lines aKs - binodal, dKe- spinodal; 1 - liquid + gas region; 2 And 3 - areas of metastable state of systems: superheated liquid and liquid + steam, supercooled steam and liquid + steam. Shaded areas adb And beс are equal.

With the help of V. u. you can get critical. options R cr, V kp And T kp. At the point TO Van der Waals isotherms have both a maximum and an inflection point, i.e. The solution to the van der Waals system of equations and the two above has the form:

Despite the fact that constant b has a fitting nature, the molecular sizes obtained using the expression are in good agreement with those obtained by other methods.

V.u., in which the introduced relates. quantities T/T cr, R/R cr, T/T cr, called given level of state; it has wider application than V. at. If in V. u. expand the pressure into degrees of density and compare with virial decomposition, then constant A And b can be expressed in terms of virial coefficients.

Lit. see under Art. Gas. Yu. N. Lyubitov

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1988 .

See what the "VAN DER WAALS EQUATION" is in other dictionaries:

VAN DER WAALS EQUATION, an equation of state (see EQUATION OF STATE), describing the properties of a real gas (see REAL GAS). Proposed by J.D. Van der Waals (see VAN DER WAALS Johannes Diederick) in 1873. Widely used for... ... encyclopedic Dictionary

Proposed by J. D. van der Waals (1873), the equation of state of a real gas, taking into account the finite volume of molecules and the presence of intermolecular attractive forces; for one mole has the form: (p+a/V2)(V b) V = RT, where: p pressure, V molar volume, T... ... Big Encyclopedic Dictionary

One of the first equations of state of a real gas, proposed by the Dutch physicist J. D. van der Waals (1873): Here: p gas pressure; T is his temperature; V̅ volume of one mole of substance; R universal Gas... ... Great Soviet Encyclopedia

Level of state of a real gas. For pmol of gas having a volume V at t re Ti pressure p. has the form: where R is the gas constant, a and b are the van der Waals constants, characteristic of a given va. The term 2/V2 takes into account the attraction of gas molecules (decrease... ... Chemical encyclopedia

- [named after Goll. physics J. D. van der Waals (J. D. van der Waals; 1837 1923)] level of state of a real gas? where p pressure, V volume, T thermodynamic. rate pa, t mass of gas, M its molar mass, R gas constant, a and b constants depending... ... Big Encyclopedic Polytechnic Dictionary

The equation of state of a real gas proposed by J. D. van der Waals (1873), taking into account the finite volume of molecules and the presence of intermolecular forces of attraction; for one mole of gas has the form: (p + a/V2)(V – b) = RT, where p is pressure, V is volume... ... encyclopedic Dictionary

The equation of state of a real gas proposed by I. D. van der Waals (1873), taking into account the finite volume of molecules and the presence of intermolecular forces of attraction; for and moles of gas has the form: (p + n2a/V2)(V nb) = nRT, where p pressure, V volume, T abs. temp ra, R… … Natural science. encyclopedic Dictionary

Equation of state Stat ... Wikipedia

Equation of state This article is part of the Thermodynamics series. Equation of state of an ideal gas Van der Waals equation Diterici equation Sections of thermodynamics Principles of thermodynamics Equations ... Wikipedia

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• Statistical theory of open systems. Volume II. Kinetic theory of plasma. Kinetic theory of second order phase transitions. Issue 91, Klimontovich Yu.L. , Volume 2 significantly expands the scope of application of the ideas and methods developed in Volume 1. The second volume consists of two parts. In 4.1, using the Landau damping phenomenon as an example, it is shown that due to... Category: Scientific and technical literature Series: Synergetics: from past to future Publisher:

The behavior of real gases differs to one degree or another from the behavior of an ideal gas described by the Mendeleev-Clapeyron equation. Deviations depend not only on how

gas - oxygen, nitrogen, etc. - we are dealing with, but also on the conditions in which the gas is located. The more rarefied the gas and the higher its temperature, the less noticeable these deviations are. Therefore, the applicability of the ideal gas model to any real gas is determined not only by the properties of the gas itself, such as the size and mass of its molecules and the interactions between them, but also by the conditions in which the gas is located.

About models of real gas. The interaction between the molecules of a real gas is so complex that it is impossible to obtain an equation of state that would quantitatively correctly describe the behavior of a real gas over the entire range of possible changes in its temperature and density. However, it is possible to write an approximate equation of state for a real gas, taking into account the main qualitative features of intermolecular interaction.

The ideal gas model did not take into account the interaction of molecules at a distance, nor the finite size of the molecules themselves. Let us recall that the interaction forces between molecules are of an electromagnetic nature, although in general the molecules are electrically neutral. At large distances, the forces of attraction between molecules predominate, and at small distances, the forces of repulsion predominate. The attractive forces between molecules quickly decrease with increasing distance between them, and therefore, at low density, the gas behaves as an ideal gas. But nevertheless, the forces of attraction are significant, since it is they that lead to the condensation of gas into liquid. The repulsion of molecules at short distances increases so quickly as they approach that, with good accuracy, molecules can be considered solid non-penetrating bodies and talk about their own volume and effective diameter. This volume sets a limit to the possible compression of the gas.

Model of hard balls. The above considerations allow us to propose the simplest microscopic model of a real gas - the model of hard spheres. In this model, molecules are considered as solid balls of radius that interact only when in contact with each other. In addition to the radius, the system is characterized by the mass of molecules (balls), their concentration and the average kinetic energy of thermal motion

To understand the properties of such a model, let’s find the corresponding dimensionless parameter, composed of Since time only enters, it cannot enter into the dimensionless parameter. But then mass cannot enter into it either. And from the remaining quantities it is possible to create only one independent dimensionless parameter, the physical meaning of which is the ratio of the ball’s own volume to the average

volume per molecule in a gas. The limiting case of an ideal gas corresponds to strong rarefaction, i.e. when the interaction energy of molecules should become negligible compared to the energy of thermal motion

In a highly compressed gas, the ratio of the average interaction energy of molecules to their kinetic energy should depend on temperature, since with decreasing temperature the deviation of a real gas from an ideal gas becomes more and more noticeable. However, in the hard-sphere model, this deviation is determined only by the parameter y and, therefore, does not depend on temperature. Therefore, a consistent microscopic model of a real gas must take into account the interaction of molecules with each other and at a distance, and not only during direct contact.

Van der Waals model. When starting to obtain an approximate equation of state for a real gas, we will assume that the interaction of molecules leads to only small corrections in the equation of state for an ideal gas. At sufficiently high temperatures and low gas densities, the required equation should lead to the same results as the Mendeleev-Clapeyron equation.

Accounting for molecular sizes. First of all, let us take into account the finite intrinsic sizes of the molecules. In fact, this leads to the fact that the volume provided to gas molecules for movement will be less than the volume of the vessel V. Therefore, in the equation of state of one mole of an ideal gas, we replace the volume V with where is a positive constant characteristic of a given gas, taking into account the volume occupied by the molecules:

From this equation it is clear that the volume of gas V cannot be made less than since the gas pressure increases indefinitely. Of course, one should not expect that for values ​​of V close to Equation (1) will correctly describe the behavior of the gas, because in its meaning the constant is a small correction:

Accounting for interactions between molecules. Let us now take into account the attraction between molecules that occurs at large distances. The attraction should lead to a decrease in the pressure exerted by the gas on the walls of the vessel, since each molecule located near the wall will be acted upon by the remaining gas molecules by a force directed into the vessel. Therefore, the pressure on the walls will be less than the value given by expression (1) by a certain amount

What can the value depend on? It can be expected that, to a rough approximation, the force acting on each molecule from all the others will be proportional to the number of surrounding molecules, i.e., the density of the gas. Consequently, the correction to the momentum transmitted by an individual molecule upon impact with the wall is proportional to the density of the gas. The correction to the momentum transmitted when all molecules hit the wall (i.e., to the pressure) will be proportional to the square of the gas density or, which is the same thing, inversely proportional to the square of the volume. Therefore, the expression for can be written in the form

where a is a positive constant characteristic of a given gas. Since as and is a small correction, the inequality must be satisfied. Substituting from formula (3) into equation (2), we obtain

Equation (4) is called the van der Waals equation. It approximately takes into account the behavior of a real gas due to intermolecular interactions and the intrinsic sizes of molecules. The van der Waals constants a and are determined experimentally: their values ​​for each gas are chosen in such a way that equation (4) best describes the behavior of a given gas.

Phenomenological nature of the van der Waals equation. The above considerations should not be considered as a strict derivation of the equation of state of a real gas. They represent an example of a phenomenological approach, in which the qualitative type of a pattern is established using guiding considerations, and quantitative characteristics - in this case constants - are found from comparison with experiment. Like any phenomenological relationship, the van der Waals equation in a certain region - at sufficiently high temperatures and low densities - gives a correct quantitative description of the properties of a real gas, while in the entire region of parameter changes it gives only a qualitative picture of the behavior of the gas.

In addition to the van der Waals equation, many other empirical equations of state for real gases have been proposed. Some of them give better agreement with experience due to the larger number of phenomenological constants included in them. However, when qualitatively studying the behavior of real gases, it is convenient to use the van der Waals equation due to its simplicity and clear physical meaning.

Van der Waals isotherms. To study the behavior of a gas that obeys the van der Waals equation, we consider the isotherms determined by this equation, i.e., the curves depending on V at given temperatures T. For this purpose, we rewrite equation (4) in the form

For fixed values, this is a third-degree equation with respect to V. A third-degree equation has either one or three real roots. Therefore, for given values ​​of pressure and temperature, equation (5) gives either one or three volume values. This means that on the -diagram the isotherm intersects the horizontal line either at one point or at three points.

In Fig. 85 shows isotherms corresponding to different temperatures

Rice. 85. Van der Waals gas isotherms on the -diagram

The isotherm corresponding to a sufficiently high temperature differs little from the isotherm of an ideal gas, and one of the most significant differences is that the value of volume V cannot be less at any pressure. At sufficiently low temperatures, isotherms contain a wave-like section, so that the straight line crosses the isotherm three times . There is a certain temperature Tk such that the corresponding isotherm separates all monotonic isotherms lying above it from the “humpbacked” ones lying below. This temperature and the corresponding isotherm are called critical.

Andrews experimental isotherms. In order to find out the meaning of this seemingly very strange dependence, one should

turn to experience. In Fig. 86 shows the experimentally obtained isotherms of carbon dioxide. At high temperatures, these isotherms are very similar to the isotherms of an ideal gas.

Rice. 86. Experimental carbon dioxide isotherms

At low temperatures, the nature of isotherms is significantly different: they have a horizontal section in which the pressure remains constant, despite the change in volume.

Let us consider what physical processes correspond to different sections of the isotherms. When one mole of carbon dioxide is isothermally compressed in a section (Fig. 87), it behaves like an ideal gas: the pressure increases as the volume decreases.

Rice. 87. Comparison of the experimental isotherm with the van der Waals isotherm

With further compression, the pressure in the vessel remains unchanged, and liquid carbon dioxide appears in the vessel. This continues throughout the entire section until at point C all the gas condenses into liquid. The section corresponds to the equilibrium between liquid carbon dioxide and its saturated vapor. Further attempts to compress the liquid

accompanied by a sharp increase in pressure with the slightest decrease in volume.

Let us superimpose the theoretical van der Waals curve for the same temperature onto the graph of the experimentally obtained isotherm. The parameters can be selected so that in the section corresponding to the gaseous state, both curves practically coincide. This means that the van der Waals equation quite satisfactorily describes the properties of vapors that are far from saturation. The steep course of the theoretical isotherm in this section qualitatively correctly conveys the low compressibility of the liquid, although the experimental isotherm in this section is even steeper. But the wavy section of the theoretical isotherm is completely different from the corresponding plateau of the experimental curve.

Metastable states. However, it turns out that a number of states that fall within the “hump” region can be experimentally realized if certain conditions are met. Thus, in a space free from dust and charged particles, it is possible, through gradual compression, to obtain vapors at a pressure greater than the pressure of saturated vapors at a given temperature. Such pairs are called supersaturated, and their states are described quite well by a section of the theoretical isotherm (see Fig. 87). These states, when condensation centers appear - ions, dust particles, tiny droplets - become unstable, and the supersaturated steam instantly condenses into fog.

Supersaturated steam can be obtained not only by isothermal compression, but also by cooling saturated steam. Therefore, it is sometimes also called supercooled. This method of producing supersaturated steam is used in a cloud chamber, which was widely used at the dawn of nuclear physics for recording charged particles.

Liquid carbon dioxide, with gradual isothermal expansion, can remain in a liquid state and at a pressure less than the saturated vapor pressure at a given temperature. Such states, as already noted, are called superheated liquid, and they correspond to a section of the theoretical van der Waals isotherm. A superheated liquid can also be obtained by heating the liquid at constant pressure, if measures are taken to ensure that the liquid and the walls of the vessel do not contain dissolved gases, which, when heated, can be released to form bubbles and become nuclei of a new phase. When boiling centers appear, the superheated liquid boils instantly.

Thus, sections of the theoretical isotherm correspond to states of matter that, although unstable, can be realized experimentally. Such states are called metastable.

Absolutely unstable states. States of matter in the area of ​​Fig. 87) are absolutely unstable and cannot occur in nature at all, since they correspond to a decrease in pressure during compression. Let us imagine for a moment that such states are possible. Let a small part of the substance be accidentally compressed as a result of density fluctuations. Then the pressure in this place will decrease and become less than the ambient pressure. This will lead to further compression of the released substance, etc. - the system spontaneously exits such an unstable state.

If the instability of the states corresponding to the sections was due to the presence of centers of condensation and boiling and therefore was in principle removable, then the instability of the states in the section is associated with inevitable thermal fluctuations and is fundamentally irremovable. Therefore, such states of the theoretical isotherm are never realized.

The presence of an unrealizable section on the van der Waals isotherms at temperatures below critical means that with a gradual change in volume, the substance cannot remain homogeneous all the time: at some point, the substance must separate into two phases - liquid and gaseous.

In this regard, it is interesting to note that the empirical van der Waals equation, obtained for the purpose of introducing small corrections to the equation of state of an ideal gas, actually turned out to be effective in a much wider area. In fact, it indicated the existence of a critical temperature and the need for phase separation of matter at temperatures below the critical temperature. It also reflected the possibility of the existence of states of supersaturated steam and superheated liquid, qualitatively described the low compressibility of liquids, etc.

Critical state of matter.

Let us observe how the region of coexistence of two phases on the K-diagram changes as the temperature increases.

Rice. 88. Equilibrium boundaries of various phases of matter on the -diagram

The straight section of the experimental isotherm decreases with increasing temperature and, at the critical temperature, contracts to one point K in Fig. 86. If we connect the starting points of horizontal sections on all isotherms, as well as the points of the ends of these sections, we will get a certain curve (Fig. 88). This curve represents the boundary separating states in which a substance exists in two equilibrium states.

phases, from single-phase states. The pressure and volume corresponding to point K are called critical. The state of matter at point K is called the critical state.

At the critical point, three points merge into one, at which the horizontal section of the experimental isotherm intersects the van der Waals isotherm.

In a critical state, the difference between liquid and gas disappears, and there is no boundary between them. In this case, such concepts as surface tension and heat of vaporization lose their meaning.

The compressibility of a substance increases indefinitely as it approaches the critical state: the isotherm graph in the vicinity of this point goes horizontally and, therefore, a small change in the volume of the substance is not accompanied by a change in pressure. This leads to the existence of large density fluctuations in the critical state, which manifest themselves, for example, in the phenomenon of critical opalescence - strong light scattering by random inhomogeneities.

The boundaries of equilibrium between different phases of a substance are shown in the state diagram (Fig. 88). The boundary is purely arbitrary and corresponds to the historical division of the gaseous state into vapor and “true gas”, depending on whether the substance is at a temperature below or above the critical temperature. Unlike the boundary in this diagram, when crossing the boundary no qualitative changes occur in the substance.

What phenomena reveal deviations in the behavior of real gases from the predictions of the ideal gas model?

Why is the hard-sphere model unable to explain the fact that the deviation of the properties of a real gas from an ideal gas increases with decreasing temperature?

What physical meaning do the parameters a and appearing in the van der Waals equation have? What is their size? How are their values ​​determined for real gases?

What is the phenomenological nature of the van der Waals equation?

What is the reason for the qualitatively different appearance of van der Waals isotherms at different temperatures?

Describe the physical processes when the volume of the system changes, corresponding to different sections of the experimental isotherm,

How can metastable states corresponding to sections of the theoretical van der Waals isotherm be practically realized (Fig. 87)?

Why is the section of the theoretical isotherm (Fig. 87) fundamentally impossible to implement experimentally?

Explain how the van der Waals equation indicates the need for phase separation of a substance at temperatures below critical.

What is a critical condition? What properties does a substance have in a critical state?

Explain why, when crossing the border in Fig. 88, separating gas and vapor, no qualitative changes occur in the state of the substance.