Van der Waals equation online. Van der Waals equation

Van der Waals equation:

where are the constant corrections A And b depend on the nature of the gas.

Amendment b takes into account the volume inaccessible to the movement of molecules due to the finite volume of the molecules themselves and the presence of interaction between them. The value of b is approximately four times the volume of the molecules themselves.

Amendment A takes into account the forces of mutual attraction. Assuming that the internal pressure of a gas varies proportionally to the square of the density or inversely proportional to the square of the specific volume of the gas, van der Waals took it equal to a/J 2, where a is the proportionality coefficient.

Expanding the brackets on the left side:

Multiplying the equality by J 2 and dividing by r:

The resulting equation has three roots, i.e. for given parameters p and T, there are three values of the variable J, which turn the equation into an identity.

Let us consider isotherms constructed using the van der Waals equation in the р–J coordinate system.

First case occurs at high temperatures, when isotherms have the form of hyperbolic curves (line 1-2). Each pressure corresponds to a certain specific volume (pressure p a corresponds to specific volume J a). In this case, the body is in a gaseous state at any pressure.

Second case occurs at relatively low temperatures, when the isotherms have two bends (line 3-4).

In this case, between points e and f there is a region in which each pressure corresponds to three values of the specific volume (pressure p a corresponds to the specific volumes J b, J c and J d), which are three real and different roots of the van der equation Vaals.

Section 3-b corresponds to isothermal compression of the body, located in a gaseous state, and at the point b it is already beginning to turn into a liquid state.

Dot d corresponds to the state of the body when it has already completely turned into liquid, according to which section d-4 represents isothermal compression of the liquid.

Dot With corresponds to an intermediate two-phase state of the body. Plot b-f curve corresponds to the unstable state of steam, and section d-e– unstable state of the liquid.

Regarding section e-f, then it has no physical meaning at all, since in reality, during isothermal compression, the body passes from a gaseous to a liquid state at constant pressure, i.e. along the horizontal line b-d.

Third case takes place at a certain temperature for each body, when points b and d, approaching with increasing temperature, merge into one point k, at which the corresponding isotherm is inflected, and the tangent to it at this point has a horizontal direction.

Dot k is called the critical point, above which it is impossible to achieve the transition of a gas into a liquid state by means of isothermal compression, and the corresponding parameters p cr, J cr and T cr are called critical parameters.

Analytically, the conditions for the critical state of a body are expressed by the equations

The first of them shows that the critical isotherm at point k has a horizontal tangent, the second - that the isotherm has an inflection at point k.

Using these equations together with the equation of state, it is possible to determine the values of the critical parameters of the gas state.

The critical parameters are defined as follows.

Let's transform the van der Waals equation:

Let's differentiate:

We define the second derivative:

Dividing the first equation by the second

and therefore ,

where

The van der Waals equation can be represented in dimensionless form with substitution.

The most well-known equation of state of real gases, taking into account the intrinsic volume of gas molecules and their interaction, is equation (1873) by the Dutch physicist I.D. van der Waals(1837–1923). Let us briefly consider the derivation of this equation.

The finite volume (size) of molecules increases the pressure of a real gas compared to the IG, because The transfer of momentum to the walls through the space of the vessel occurs faster than with point molecules due to the fact that they travel a shorter distance between collisions. Only (repulsive forces) paired collisions of molecules are taken into account - the collision of two molecules when the others do not act on them. The likelihood and influence of simultaneous triples, quadruples, etc. collisions are neglected. When calculating pressure, we can assume that one molecule remains motionless, while the other moves with twice the speed. kinetic energy. During a collision, the centers of molecules can approach each other to a distance less than d is the diameter of the molecule, therefore we can consider a stationary molecule surrounded fencing sphere radius d, and the moving molecule is point-like. If we apply this approximation to a gas from N molecules, then half the molecules N/2 will be at rest (surrounded by enclosure spheres), and the other half can be considered as a gas from N 1=N/ 2with temperature T 1 =2T. This gas would have access to the volume of the vessel V except volume b all areas of fencing N/ 2 resting molecules, i.e. V–b. Then, according to equation (9.12), the pressure exerted by these molecules on the wall of the vessel has the form

or for one mole of gas .

Obviously, the volume b approximately equal to four times the volume of all gas molecules (Fig. 13.2). Let us now take into account the action of attractive forces between gas molecules. When a molecule is inside a substance (gas), the attractive forces from other molecules on all sides are approximately compensated. If the molecule is in the surface layer, then an uncompensated attractive force appears F, directed from the surface into the gas. Under the influence of these forces, the molecule may not reach the wall of the vessel at all, but will be reflected from the surface layer of the substance. The action of attractive forces creates additional - internal or molecular pressure P i ~N cl F, Where N sl– number of molecules in the near-surface (near-wall) layer. Quantities N sl And F directly proportional to the density and inversely proportional to the volume of the gas. For one mole of gas P i =a/V m 2 and the real gas pressure is , Where R– IG pressure. For leaky gases, corrections for the forces of repulsion and attraction can be introduced independently, then, generalizing, we obtain

(13.2)

or for an arbitrary amount of substance, taking into account V=nV m:

. (13.3)

Equation (13.3) – van der Waals equation, a And b– constants, van der Waals corrections.

Equation (13.2), considered as an equation for determining the volume given T And R, there is an equation of the third degree, in transformed form it has the form

. (13.4)

Since an equation of the third degree with real coefficients can have either one real root and two complex conjugates, or three real roots, then on the plane PV straight, parallel to the axis V, can intersect the isotherm either at three points or at one. Constructing the van der Waals isotherm from points leads to the family of curves shown in Fig. 13.3 (theoretically Van der Waals, experimentally T. Andrews (1813–1885) for CO 2).

The left, steeply falling branch corresponds to a small change in volume with a change in pressure, which is characteristic of the liquid state of the substance. The right sloping branch corresponds to a significant change in volume with a change in pressure, which corresponds to the gaseous state of the substance.

The transition from liquid to gaseous state and back occurs not along the van der Waals isotherm, but along the isobar AE, which is also an isotherm of a real gas. At the same time, the areas of the figures ABC And CDE equal ( Maxwell's rule). Isotherm points A And E depict two-phase states of matter, and between them two phases exist simultaneously. The closer the representing point G To A, the more liquid in the system, the closer to E- the more steam. If we denote the maximum volume of a mole of liquid and the minimum volume of vapor in a system at temperature T through V 1 and V 2, respectively, and the volume of the two-phase region at the point G through V 0, That , Where X– mole fraction of liquid in the state G; from here, knowing the volume V 0, we can also find the share x liquids. Sites AB And DE Van der Waals isotherms depict metastable states of matter: supercooled liquid and supersaturated steam, which can exist under certain conditions (with a very slow quasi-equilibrium process and careful preparation, for example, removing all contaminants from the volume of the heated liquid and from the walls of the vessel, since the boiling process begins more easily on foreign particles - inclusions). Plot ВD corresponds to absolutely unstable (increasing pressure with increasing volume) states of matter and is not realized under any conditions. At sufficiently low temperatures the area ABC may fall below the axis O.V., which is adequate to the negative pressure corresponding to the state of the stretched liquid (due to the action of surface tension forces).

With increasing temperatures, the area of humps and troughs on the van der Waals isotherm decreases at temperature T k– critical temperature – turns into an inflection point with a horizontal tangent. For this point, equation (13.4) has three identical roots and takes the form . The critical parameters of a given gas are determined by the formulas

gas plasma kinetics thermodynamic

In a gas, interactions between molecules are weak. As it intensifies, the properties of the gas deviate ever closer from the properties of ideal gases, and, in the end, turns into a concentrated state - a liquid. In a liquid, the interaction between molecules is high and, therefore, the properties of the liquid depend on the specific type of liquid. Therefore it is impossible to install any general formulas, which would quantitatively describe the properties of the liquid. It is possible, however, to find some interpolation formula, which qualitatively describes the transition between liquid and gas. This formula should give correct results in two extreme cases. For rarefied gases it should transform into the ideal gas formulas. As density increases, it must take into account the limited compressibility of substances. To obtain such a formula, we will study in more detail the deviation from ideality at high temperatures. We will consider a monatomic gas. For the same reasons, the formulas will be applicable to polyatomic gases. The previously described nature of the interaction of gas atoms allows us to determine the form of the first terms of the expansion of B(T) relative to the inverse power of T, and we will assume that the ratio U 0 /kT is small<< 1.

Bearing in mind that U 12 is a function only of the distance r between atoms, we have. Dividing the domain of integration over dr into two parts, we write:

But for values of r from 0 to 2r 0, the potential energy of U 12 is very high. Therefore, in the first integral we can neglect the term exp(-U 12 /kT) compared to unity. Then the integral becomes equal to the positive value b = 16рr 0 3 /3 (if for a monatomic gas we consider r as the radius of the atom, then b is its quadruple volume). In the second integral everywhere |U 12 |/kT< U 0 /kT << 1. Поэтому можно разложить подынтегральное выражение по степеням U 12 /kT, ограничиваясь первым неисчезающим членом. Тогда второй интеграл становится равным

where a is a positive constant. Thus, we find that

Finding the free energy of a gas

Let's substitute into this expression

which we received earlier from the statistical sum for ideal gas. Then we get

When deriving the formula for the free energy of a gas, we assume that the gas, although not rarefied enough to be considered ideal, nevertheless has a sufficiently large volume (so that triple, etc. interactions could be neglected), i.e. the distance between molecules is much greater than their size. It can be said that the volume V of the gas is, in any case, significantly larger than Nb. That's why

Hence

In this form, this formula satisfies the conditions stated above, because at large V it turns into the formula for the free energy of an ideal gas, and at small V it reveals the impossibility of infinite gas compression (at V< Nb аргумент логарифма становится отрицательным). Зная свободную энергию, можно определить давление газа:

This is the desired equation of state of a real gas - the van der Waals equation. It is only one of many possible interpolation formulas. Jan van der Waals derived this equation in 1873 (Nobel Prize 1910).

Entropy of real gas from (*):

Energy E = F + TS

From this it is clear that the heat capacity of the van der Waals gas will coincide with the heat capacity of an ideal gas (depending only on T) and can be constant. The heat capacity C p, as is easy to see, depends not only on T, but also on V and therefore cannot be reduced to a constant. The second term in E corresponds to the gas interaction energy. It is negative, because gravity prevails.

Reduced equation of state.

Let us write the van der Waals equation for one mole of gas:

The P(V) dependences at constant temperature are called van der Waals isotherms. Among the various isotherms, there is one that corresponds to a critical state, mathematically characterized by an inflection point. Equating the first and second derivatives to zero.

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 any 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 special 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 pressure saturated steam (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 gaseous substance It can no longer be turned into 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 liquid state, but in its properties continues to follow gaseous behavior - it tends, for example, 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 Mendeleev-Clapeyron equation is the equation of state of an ideal gas and quite accurately describes the behavior of real gases at low density, i.e. sufficiently low pressure and high temperature ( ).

As the temperature decreases and the pressure increases, the density of the gas increases, and the distance between its molecules decreases, so neglect their volume and interaction

Rice. 23 by action we cannot.

The forces of mutual attraction between the molecules are directed into the gas, i.e., towards the greatest environment of the peripheral molecules (Fig. 23).

The action of these forces is similar to the presence of some additional pressure on the gas, called internal.

Due to the fact that gas molecules occupy finite sizes, they occupy a total volume V/. Therefore, the volume provided for molecules to move will be less by the amount V". Thus, to describe the state of real gases it is necessary to make two corrections:

A) on additional pressure caused by the interaction of molecules;

b) to reduce the volume, due to taking into account the size of the molecules themselves.

Let us take the equation of state of an ideal gas as a basis and, making appropriate amendments to it, obtain the equation of state of a real gas. For one mole of gas we have

The amendments introduced were first calculated and proposed by Van der Waals (Gol.)

Where A And V– van der Waals constants.

The van der Waals equation for one mole of real gas has the form:

. (26)