Derivation of the van der Waals equation. Van der Waals equation. Critical constants and Boyle temperatures

The Clapeyron-Mendeleev equation (see § 40) describes the behavior of an ideal gas, the molecules of which can be considered as material points that do not interact with each other (see § 41). The molecules of a real gas, as we know, have a certain, albeit very small, size and are interconnected by cohesion forces, although also small. However, at low temperatures or at high pressures, when gas molecules are close to each other, it is no longer acceptable to neglect their sizes and adhesion forces. In these cases, the Clapeyron-Mendeleev equation, i.e., the equation of state of an ideal gas, turns out to be very inaccurate. To obtain the equation of state of a real gas, the Dutch physicist Van der Waals in 1873 introduced corrections to the Clapeyron-Mendeleev equation for the size of molecules and the action of cohesion forces between them. It was

done as follows. In the Clapeyron-Mendeleev equation for a mole of gas

where is the volume of gas, or, what is the same, the volume of the vessel provided for the movement of molecules. In a real gas, part of this volume is occupied by the molecules themselves. Therefore, the actual free volume in which molecules of a real gas can move will be less and equal. Substituting this value instead into formula (1), we obtain

The volume occupied by the molecules themselves is greater than the sum of the own volumes of these molecules, since even with the densest packing between the molecules there would remain “useless” gaps inaccessible to the movement of molecules (Fig. 121). In reality, these gaps will be even larger, since repulsive forces will not allow such a dense packing of molecules. Calculations show that the volume occupied by the molecules themselves of a mole of gas is approximately equal to four times the intrinsic volume of these molecules:

where is the intrinsic volume of the molecule, Avogadro’s constant.

The pressure in formula (1), exerted on an ideal gas by the walls of the vessel, is external. The action of attractive forces between the molecules of a real gas causes additional compression of the gas, thereby creating additional internal pressure similar to the internal pressure of the liquid (see § 59). Therefore, the actual pressure of the real gas will be greater and equal. Substituting this value instead into formula (2), we obtain

It is easy to establish that the internal pressure should be approximately proportional to the square of the gas density. Indeed, let us divide the gas into two parts by an imaginary plane (Fig. 122) and consider the layers of gas adjacent to this plane. It is obvious that the force of mutual attraction of these layers is proportional to the number of molecules in each of them, that is, proportional to the square of the number of gas molecules. But the number of molecules is proportional to the density of the gas. Therefore, the force of attraction of the layers, and therefore the internal pressure, is proportional to the square of the density: Since density is inversely proportional to the volume, the internal pressure is inversely proportional to the square of the volume:

where a is the proportionality coefficient. Substituting the expression into formula (3), we get

This is the equation of state of a real gas or the van der Waals equation for a mole of gas. Transforming this equation as was done with the Clapeyron-Mendeleev equation (see § 40), we obtain the van der Waals equation for any mass of gas

where V is the volume of the gas mass, the molar mass of the gas.

At low pressures and high temperatures the volume becomes large; therefore, i.e., the corrections in the van der Waals equation become negligibly small and it turns into the Clapeyron-Mendeleev equation.

The values ​​are almost constant for each gas. For example, for nitrogen they are determined experimentally; it is necessary to write the van der Waals equation for two gas states known from experience and solve a system of two equations for unknowns

Let's do some analysis of the van der Waals equation. For this purpose, first of all, we will compile tables of the dependence of pressure on gas volume at constant temperature for several temperature values. The results of such calculations are presented graphically in Fig. 123. The resulting curves - van der Waals isotherms - turn out to be quite peculiar: at low temperatures they have wavy sections (maxima and minima), at a certain temperature the isotherm has only an inflection point K, at high temperatures the van der Waals isotherms are similar on ideal gas isotherms (Boyle-Mariotte or Clapeyron-Mendeleev).

From a mathematical point of view, this nature of isotherms is explained very simply. If we bring the van der Waals equation to normal form, then it turns out to be a cubic equation for volume

A cubic equation can have either three real roots or one real root and two imaginary roots. Obviously, the first case corresponds to isotherms at low temperatures (three gas volume values ​​correspond to one pressure value, and the second case corresponds to isotherms at high temperatures (one volume value corresponds to one pressure value

For real gases, the results of the ideal gas theory should be used with great caution. In many cases it is necessary to move to more realistic models. One of a large number of such models can be Van der Waals gas. This model takes into account the intrinsic volume of molecules and the interactions between them. Unlike the Mendeleev-Clapeyron equation pV=RT, valid for an ideal gas, the van der Waals gas equation contains two new parameters A And b, not included in the equation for an ideal gas and taking into account intermolecular interactions (parameter A) and real (non-zero) intrinsic volume (parameter b) molecules. It is assumed that taking into account the interaction between molecules in the equation of state of an ideal gas affects the pressure value p, and taking into account their volume will lead to a decrease in the space free for the movement of molecules - the volume V, occupied by gas. According to van der Waals, the equation of state of one mole of such a gas is written as:

Where Mind- molar volume of quantity ( a/Um) And b describe deviations of a gas from ideality.

Magnitude a/V^, corresponding in dimension to pressure, describes the interaction of molecules with each other at large (compared to the size of the molecules themselves) distances and represents the so-called “internal pressure” of the gas, additional to the external one r. Constant Kommersant in expression (4.162) takes into account the total volume of all gas molecules (equal to four times the volume of all gas molecules).

Rice. 4.24. Towards the definition of a constant b in the van der Waals equation

Indeed, using the example of two molecules (Fig. 4.24), one can be convinced that molecules (like absolutely rigid balls) cannot approach each other at a distance less than 2 G between their centers

those. the area of ​​space “excluded” from the total volume occupied by the gas in the vessel, which is accounted for by two molecules, has a volume

In terms of one molecule this is

its quadruple volume.

That's why (V M - b) is the volume of the vessel available for the movement of molecules. For arbitrary volume V and masses T gas with molar mass M equation (4.162) has the form

Rice. 4.25.

where v = t/m is the number of moles of gas, and a"= v 2 a And b"= v b- van der Waals constants (corrections).

The expression for the internal gas pressure in (4.162) is written as a/Vj, for the following reason. As was said in subsection 1.4.4, the potential energy of interaction between molecules is, to a first approximation, well described by the Lennard-Jones potential (see Fig. 1.32). At relatively large distances, this potential can be represented as the dependence U ~ g~ b, Where G- distance between molecules. Because strength F interactions between molecules is related to potential energy U How F--grad U(r), That F~-g 7. The number of molecules in the volume of a sphere of radius r is proportional to r 3, therefore the total interaction force between molecules is proportional it 4 , and the additional “pressure” (force divided by area proportional to g 2) proportionally g b(or ~ 1/F 2). At small values G strong repulsion between molecules appears, which is indirectly taken into account

coefficient b.

The van der Waals equation (4.162) can be rewritten as a polynomial (virial) expansion in powers Mind(or U):

Relatively V M this equation is cubic, so at a given temperature T must have either one real root or three (further, assuming that we are still dealing with one mole of gas, we will omit the index M V V M so as not to clutter the formulas).

In Figure 4.25 in coordinates p(V) at different temperatures T The isotherms that are obtained as solutions to equation (4.163) are given.

As the analysis of this equation shows, there is such a value of the parameter T-Г* (critical temperature), which qualitatively separates the different types of its solutions. At T > T k curves p(V) monotonically decrease with growth V, which corresponds to the presence of one real solution (one intersection of the straight line p = const with isotherm p(V))- each pressure value r matches only one volume value V. In other words, when T > T k gas behaves approximately as ideal (there is no exact correspondence and it is obtained only when T -> oo, when the energy of interaction between molecules compared to their kinetic energy can be neglected). At low temperatures, when T to one value r corresponds to three values V, and the shape of the isotherms changes fundamentally. At G = T k The van der Waals isotherm has one singular point (one solution). This point corresponds to /^ (critical pressure) and V K(critical volume). This point corresponds to a state of matter called critical, and, as experiments show, in this state the substance is neither a gas nor a liquid (an intermediate state).

Experimental obtaining of real isotherms can be carried out using a simple device, the diagram of which is shown in Fig. 4.26. The device is a cylinder with a movable piston and a pressure gauge r. Volume measurement V produced by the position of the piston. The substance in the cylinder is maintained at a certain temperature T(located in the thermostat).

Rice. 4.26.

By changing its volume (lowering or raising the piston) and measuring the pressure, an isotherm is obtained p(V).

It turns out that the isotherms obtained in this way (solid lines in Fig. 4.25) differ markedly from the theoretical ones (dash-dotted line). At T = T and larger V a decrease in volume leads to an increase in pressure according to the calculated curve to the point N(dash-dotted isotherm in Fig. 4.25). After this decrease V does not lead to further growth r. In other words, point N corresponds to the beginning of condensation, i.e. the transition of a substance from a vapor state to a liquid state. When the volume decreases from a point N to the point M the pressure remains constant, only the ratio between the amounts of liquid and gaseous substances in the cylinder changes. The pressure corresponds to the equilibrium between vapor and liquid and is called saturated steam pressure(marked in Fig. 4.25 as p„. p). At the point M all the matter in the cylinder is liquid. With a further decrease in volume, the isotherms rise sharply, which corresponds to a sharp decrease in the compressibility of the liquid compared to vapor.

When the temperature in the system increases, i.e. when moving from one isotherm to another, the length of the segment MN decreases (A/UU"at T 2 > T), and at T=T K it contracts to a point. Envelope of all segments of the type MN forms a bell-shaped curve (binodal) - dotted curve MKN in Fig. 4.25, separating the two-phase region (under the binodal bell) from the single-phase region - vapor or liquid. At T>T k No increase in pressure can turn a gaseous substance into a liquid. This criterion can be used to make a conditional distinction between gas and steam: when T substance can exist both in the form of vapor and in the form of liquid, but at T > Because no amount of pressure can convert a gas into a liquid.

In carefully designed experiments one can observe the so-called metastable states, characterized by areas MO And NL on the van der Waals isotherm at T= T(dash-dotted curve in Fig. 4.25). These states correspond to supercooled steam (section MO) and superheated liquid (section NL). Supercooled steam - This is a state of matter when, according to its parameters, it should be in a liquid state, but in its properties continues to follow gaseous behavior - for example, it tends to expand with increasing volume. And vice versa, superheated liquid - this state of a substance when, according to its parameters, it should be a vapor, but according to its properties it remains a liquid. Both of these states are metastable (i.e., unstable): with a small external influence, the substance transforms into a stable single-phase state. Plot OL(defined mathematically from the van der Waals equation) corresponds to a negative compression coefficient (as the volume increases, the pressure also increases!), it is not realized in experiments under any conditions.

Constants A And b are considered independent of temperature and are, generally speaking, different for different gases. It is possible, however, to modify the van der Waals equation so that any gases satisfy it if their states are described by equation (4.162). To do this, let’s find the connection between the constants A And b and critical parameters: r k, V K n T k. From (4.162) for moles of real gas we obtain 1:

Let us now use the properties of the critical point. At this point the magnitude yr/dV And tfp/dV 2 are equal to zero, so this point is an inflection point. From this follows a system of three equations:

1 Index M when volume moles of gas are omitted to simplify notation. Here and below the constants A And b are still reduced to one mole of gas.

These equations are valid for the critical point. Their solution is relative/>*, U k, Guess:

and, accordingly,

From the last relation in this group of formulas, in particular, it follows that for real gases the constant R turns out to be individual (for each gas with its own set of rk, U k, T k it is its own), and only for an ideal or real gas far from the critical temperature (at T » T k) it can be assumed to be equal to the universal gas constant R = k b N A . The physical meaning of this difference lies in the processes of cluster formation occurring in real gas systems in subcritical states.

Critical parameters and van der Waals constants for some gases are presented in table. 4.3.

Table 4.3

Critical parameters and van der Waals constants

If we now substitute these values ​​from (4.168) and (4.169) into equation (4.162) and express the pressure, volume and temperature in the so-called reduced (dimensionless) parameters l = r/r k, co = V/VK t = T/T to, then it (4.162) will be rewritten as:

This van der Waals equation in given parameters universal for all van der Waals gases (i.e. real gases obeying equation (4.162)).

Equation (4.170) allows us to formulate a law connecting the three given parameters - the law of corresponding states: if for any different gases two out of three coincide(l, so, t) given parameters, then the values ​​of the third parameter must also coincide. Such gases are said to be in corresponding states.

Writing the van der Waals equation in the form (4.170) also allows us to extend the concepts associated with it to the case of arbitrary gases that are no longer van der Waals. Equation (4.162), written as (4.164): p(V) = RT/(V-b)-a/V 2, resembles in form the expansion of the function p(Y) in order of powers V(up to the second term inclusive). If we consider (4.164) a first approximation, then the equation of state of any gas can be represented in a universal form:

where are the coefficients A„(T) are called virial coefficients.

With an infinite number of terms in this expansion, it can accurately describe the state of any gas. Odds A„(T) are functions of temperature. Different models are used in different processes, and to calculate them, it is theoretically estimated how many terms of this expansion must be used in cases of different types of gases to obtain the desired accuracy of the result. Of course, all models of real gases depend on the chosen type of intermolecular interaction adopted when considering a specific problem.

• Proposed in 1873 by the Dutch physicist J.D. van der Waals.

The Clapeyron-Mendeleev equation follows from the molecular kinetic theory under the assumption that the gas is ideal. If we want to describe the behavior of real systems, we must take into account the interaction of molecules with each other. Accurately accounting for intermolecular forces is an extremely difficult task. Therefore, several modifications of the ideal gas equation of state have been proposed that could take into account the main features of real systems. The most successful attempt was van der Waals equation, upon obtaining which amendments were made to the equation of state of the ideal gas

In the van der Waals approach, firstly, it is taken into account that molecules have finite sizes. If we denote the intrinsic volume of all molecules in a mole of a substance by the letter b, then there remains free volume for the movement of molecules

and it is he who should appear in the equation of state. Secondly, it is taken into account that a molecule approaching the wall of a vessel “feels” the attraction of other molecules, which was balanced when the molecule was inside the vessel. Additional force directed into the vessel is equivalent to additional pressure p i, (it is called the “internal” gas pressure). Therefore, instead of pressure r gas onto the walls of the vessel, the equation of state must contain the sum р+р i .

How does internal pressure depend? p i from system parameters? The force acting on each molecule is proportional to the concentration n molecules in the system. The number of molecules approaching the wall is also proportional n, and therefore the internal pressure is proportional to the square of the particle concentration:

Denoting the proportionality coefficient with the letter A, we come to van der Waals equation

For one mole of a substance, this equation simplifies:

Additional information

http://eqworld.ipmnet.ru/ru/library/physics/thermodynamics.htm - J. de Boer Introduction to molecular physics and thermodynamics, Ed. IL, 1962 - pp. 38–47, part I, § 6, pp. b, c. - the van der Waals equation is discussed and the experimentally obtained intermolecular potential interaction energies for helium, hydrogen, argon and carbon dioxide are given;

http://www.plib.ru/library/book/14222.html - Yavorsky B.M., Detlaf A.A. Handbook of Physics, Science, 1977 - pp. 246–248 - detailed information on intermolecular attractive forces in van der Waals gas.

Let us consider the form of van der Waals gas isotherms on ( p,V) - diagram (Fig. 2.14). They are described by the function

At sufficiently high temperatures and large volumes, the introduced corrections can be neglected, and the appearance of the isotherms will turn out to be normal. As the temperature decreases, the appearance of the isotherms becomes increasingly distorted and at a certain critical temperature value T s this isotherm acquires an inflection point ( critical point) with coordinates ( p s, V c), in which the first and second derivatives of pressure with respect to volume are equal to zero. With a further decrease in temperature, the inflection point turns into the minimum and maximum of the function p(V).

Rice. 2.14. Van der Waals gas isotherms

Let us first find the values ​​of the parameters corresponding to the critical point. We take the first and second derivatives of function (2.37) and equate them to zero:

Solving this pair of equations will give us the critical values Tc And V c . Finding the value from the first equation

we substitute it into the second equation, which then follows

We first obtain the value of the molar critical volume

Substituting it into equation (2.39), we find the critical temperature

Finally, substituting the found values T s, V c into equation (2.37), we find the critical pressure

These critical values ​​were obtained for one mole of substance. To find them for an arbitrary number of moles, we note that when passing from equation (2.36) to (2.35) it is necessary to carry out a scaling transformation

Carrying out the same transformation in the formulas for the critical values ​​of thermodynamic parameters, we make sure that the critical temperature and pressure do not change, and the volume is transformed naturally:

The values ​​of critical parameters are taken from experimental data. Note that the gas constant R can also be expressed in terms of critical parameters:

For each real gas, one should calculate its own individual gas constant R, which will differ from the universal gas constant N A k B ideal gas. This should not be surprising, given the phenomenological approximate nature of the van der Waals equation. The values ​​of the critical parameters of some substances and their gas constant are given in table. 2.

Table 2.

Critical parameters of some gases

As already indicated in § 60, for real gases it is necessary to take into account the sizes of molecules and their interaction with each other, therefore the ideal gas model and the Clapeyron-Mendeleev equation (42.4) pV m = RT(for a mole of gas), which describes an ideal gas, are unsuitable for real gases.

Taking into account the intrinsic volume of molecules and the forces of intermolecular interaction, the Dutch physicist J. van der Waals (1837-1923) derived the equations of state of a real gas. Van der Waals introduced two amendments to the Clapeyron-Mendeleev equation.

1. Taking into account the intrinsic volume of molecules. The presence of repulsive forces that counteract the penetration of other molecules into the volume occupied by a molecule means that the actual free volume in which molecules of a real gas can move will not V m , a V m - b, Where b- the volume occupied by the molecules themselves. Volume b equal to four times the intrinsic volume of the molecules. If, for example, there are two molecules in a vessel, then the center of any of them cannot approach the center of the other molecule at a distance less than the diameter d molecules. This means that a spherical volume of radius is inaccessible to the centers of both molecules d, i.e., a volume equal to eight volumes of a molecule, and per one molecule - quadruple the volume of a molecule.

2. Taking into account the attraction of molecules. The action of gas attraction forces leads to the appearance of additional pressure on the gas, called internal pressure. According to van der Waals calculations, internal pressure is inversely proportional to the square of the molar volume, i.e.

p" = a/V 2 m, (61.1)

where a is the van der Waals constant, characterizing the forces of intermolecular attraction, V m - molar volume.

Introducing these corrections, we get van der Waals equationfor mole of gas(equation of state of real gases):

(p+a/V 2 m )(V m -b)=RT.(61.2)

For an arbitrary amount of substance v gas (v=t/M) taking into account that V = vV m , the van der Waals equation takes the form

where are the corrections a and b- constant values ​​for each gas, determined experimentally (van der Waals equations are written for two gas states known from experience and solved relative to A And b).

When deriving the van der Waals equation, a number of simplifications were made, so it is also very approximate, although it agrees better (especially for slightly compressed gases) with experience than the equation of state of an ideal gas.

The van der Waals equation is not the only equation that describes real gases. There are other equations, some of which even more accurately describe real gases, but are not considered due to their complexity.

§ 62. Van der Waals isotherms and their analysis

To study the behavior of a real gas, consider van der Waals isotherms- dependence curves r from V m for given T, determined by the van der Waals equation (61.2) for begging gas These curves (considered for four different temperatures; Fig. 89) have a rather peculiar character. At high temperatures (T>T k), the isotherm of a real gas differs from the isotherm of an ideal gas only by some distortion of its shape, remaining a monotonically decreasing curve. At some temperature T To there is only one inflection point on the isotherm TO. This isotherm is called critical, its corresponding temperature T To - critical temperature. The critical isotherm has only one inflection point TO, called critical point; at this point the tangent to it is parallel to the x-axis. Corresponding to this point volumeV To and pressurer To also called critical. State with critical parameters (p k, V To , T To ) called critical condition. At low temperatures (T<Т To ) Isotherms have a wave-like section, first going down monotonically, then going up monotonically, and then going down monotonically again.

To explain the nature of the isotherms, let us transform the van der Waals equation (61.2) to the form

pV 3 m -(RT+pb) V 2 m+a V m-ab=0.

Equation (62.1) for given r And T is an equation of the third degree with respect to V m; therefore, it can have either three real roots, or one real and two imaginary roots, and only real positive roots have a physical meaning. Therefore, the first case corresponds to isotherms at low temperatures (three values ​​of gas volume V 1 , V 2 And V 3 correspond (we omit the “t” symbol for simplicity) to one pressure value r 1 ), the second case is isotherms at high temperatures.

Considering different sections of the isotherm at T<Т To (Fig.90), we see that in areas 1 -3 And 5-7 when volume decreases V m pressure r increases, which is natural. On the site 3-5 compression of a substance leads to a decrease in pressure; practice shows that such states do not occur in nature. Availability of land 3-5 means that with a gradual change in volume, a substance cannot remain in the form of a homogeneous medium all the time; At some point, an abrupt change in state and the decomposition of the substance into two phases must occur. Thus, the true isotherm will look like a broken line 7- 6-2-1. Part 7- 6 corresponds to the gaseous state, and part 2-1 - liquid. In states corresponding to horizontal

new section of the isotherm 6-2, equilibrium of the liquid and gaseous phases of the substance is observed. A substance in a gaseous state at a temperature below critical is called ferry, and vapor that is in equilibrium with its liquid is called saturated.

These conclusions, following from the analysis of the van der Waals equation, were confirmed by the experiments of the Irish scientist T. Andrews (1813-1885), who studied the isothermal compression of carbon dioxide. The difference between experimental (Andrews) and theoretical (Van der Waals) isotherms is that the transformation of gas into liquid in the first case corresponds to horizontal sections, and in the second - wavy ones.

To find the critical parameters, we substitute their values ​​into equation (62.1) and write

p To V 3 -(RT To +p To b)V 2 +aV-ab= 0

(we omit the “t” symbol for simplicity). Since at the critical point all three roots coincide and are equal V To , the equation is reduced to the form

p To (V-V To ) 3 = 0,

p To V 3 -3p To V To V 2 +3p To V 2 To V-p To V To = 0.

Since equations (62.2) and (62.3) are identical, the coefficients in them for the unknown corresponding powers must also be equal. Therefore we can write

pkV 3 k =ab, 3p k V 2 k =a, 3p To V To =RT To +p To b. Solving the resulting equations, we find: V к = 3b, r k = a/(27b 2), T To =8a/(27Rb).

If a line is drawn through the extreme points of the horizontal sections of the family of isotherms, then a bell-shaped curve will be obtained (Fig. 91), limiting the region of two-phase states of matter. This curve and the critical isotherm divide

diagram p,V m under the isotherm into three regions: under the bell-shaped curve there is a region of two-phase states (liquid and saturated vapor), to the left of it is the region of the liquid state, and to the right is the region of vapor. Steam differs from other gaseous states in that during isothermal compression it undergoes a liquefaction process. A gas at a temperature above the critical temperature cannot be converted into a liquid at any pressure.

Comparing the van der Waals isotherm with the Andrews isotherm (upper curve in Fig. 92), we see that the latter has a straight section 2-6, corresponding to two-phase states of matter. True, under certain conditions, states depicted by sections of the van der Waals isotherm can be realized 5-6 And 2-3. These unstable states are called metastable. Plot 2-3 depicts superheated liquid 5-6 - supersaturated steam Both phases have limited stability

At sufficiently low temperatures, the isotherm crosses the V m axis, moving into the region of negative pressures (lower curve in Fig. 92). A substance under negative pressure is in a state of tension. Under certain conditions, such states are also realized. Plot 8 -9 on the lower isotherm corresponds superheated liquid, plot 9 - 10 - stretched liquid.

The gas laws discussed in the previous sections are strictly satisfied only for ideal gases, which do not condense when cooled down to absolute zero temperature.

The properties of most gases are close to those of an ideal gas when they are at temperatures sufficiently far from the condensation point, that is, when there is no interaction between molecules and when the intrinsic volume of the gas molecules is small compared to the volume of the gas.

Near the condensation point (at high pressure and low temperature), the properties of gases differ significantly from the properties of an ideal gas. In these cases we talk about real gases.

Equation of state for 1 mole of ideal gas ( Vm- molar volume) changes in the case of real gases.

For real gases, it is necessary to take into account the intrinsic volume of the molecules. The presence of repulsive forces that counteract the penetration of other molecules into the volume occupied by a molecule means that the actual free volume in which molecules of a real gas can move will not Vm, A Vm - b, b- the volume occupied by the molecules themselves. Volume b equal to four times the intrinsic volume of the molecules.

The action of attractive gas forces leads to the appearance of additional pressure on the gas, called internal pressure. According to van der Waals calculations, internal pressure is inversely proportional to the square of the molar volume, i.e.

Where a- van der Waals constant, characterizing the forces of intermolecular attraction.

By introducing corrections into the equation for an ideal gas, we obtain the van der Waals equation for 1 mole of gas

Considering that , we obtain the equation for an arbitrary amount of substance:

(9.46)

Van der Waals amendments ( a And b) are constant quantities for each gas. To determine them, equations are written for two gas states known from experience and solved for a And b.

Equation (9.45) can be written as:

For given p And T is an equation of the third degree with respect to Vm therefore, it can have either three real roots, or one real and two imaginary roots, and only real positive roots have a physical meaning.

Van der Waals isotherms are curves of the dependence of p on V m at given T, determined by the van der Waals equation for a mole of gas.

At some temperature Tk - critical temperature- on the isotherm (Fig. 9.11) there is only one inflection point (at this point the tangent to it is parallel to the abscissa axis). Dot K - critical point, corresponding to this point the volume V k and pressure p k also called critical. Isotherm at Tk called critical isotherm.

At high temperature ( T > Tk) the isotherm of a real gas differs from the isotherm of an ideal gas only by some distortion of its shape, remaining a monotonically decreasing curve. At low temperature ( T ) isotherms have a wave-like section, first going down monotonically, then going up monotonically and then going down monotonically again.