Equation of state

For equilibrium thermodynamic system there is a functional relationship between state parameters, which is called equation of state. Experience shows that the specific volume, temperature and pressure of the simplest systems, which are gases, vapors or liquids, are related thermal equation state of the species.

The equation of state can be given another form:

These equations show that of the three main parameters that determine the state of the system, any two are independent.

To solve problems using thermodynamic methods, it is absolutely necessary to know the equation of state. However, it cannot be obtained within the framework of thermodynamics and must be found either experimentally or by methods of statistical physics. Specific view equation of state depends on individual properties substances.

Equation of state of ideal gases

From equations (1.1) and (1.2) it follows that .

Consider 1 kg of gas. Considering what it contains N molecules and, therefore, we get: .

Constant value Nk, per 1 kg of gas is denoted by the letter R and call gas constant. That's why

Or . (1.3)

The resulting relationship is the Clapeyron equation.

Multiplying (1.3) by M, we obtain the equation of state for an arbitrary gas mass M:

The Clapeyron equation can be given a universal form if we relate the gas constant to 1 kmol of gas, i.e., to the amount of gas whose mass in kilograms is numerically equal to the molecular mass μ. Putting in (1.4) M=μ and V=V μ, We obtain the Clapeyron-Mendeleev equation for one mole:

Here is the volume of a kilomole of gas, and is the universal gas constant.

In accordance with Avogadro's law (1811), the volume of 1 kmol, the same under the same conditions for all ideal gases, at normal physical conditions is equal to 22.4136 m 3, therefore

The gas constant of 1 kg of gas is .

Equation of state of real gases

In real gases V The difference from ideal ones is that the forces of intermolecular interactions are significant (attractive forces when the molecules are at a considerable distance, and repulsive forces when they are sufficiently close to each other) and the own volume of the molecules cannot be neglected.

The presence of intermolecular repulsive forces leads to the fact that molecules can approach each other only up to a certain minimum distance. Therefore, we can assume that the volume free for the movement of molecules will be equal to , Where b- the smallest volume to which a gas can be compressed. In accordance with this, the free path of molecules decreases and the number of impacts on the wall per unit time, and therefore the pressure increases compared to an ideal gas in the ratio , i.e.

Attractive forces act in the same direction as external pressure and result in molecular (or internal) pressure. The force of molecular attraction of any two small parts of a gas is proportional to the product of the number of molecules in each of these parts, i.e., the square of the density, therefore the molecular pressure is inversely proportional to the square of the specific volume of the gas: rmol= a/ v 2 where A - proportionality coefficient depending on the nature of the gas.

From this we obtain the van der Waals equation (1873):

At large specific volumes and relatively low pressures of real gas, the van der Waals equation practically degenerates into an equation of state ideal gas Clapeyron, because the size a/v 2

(compared with p) And b(compared with v) become negligibly small.

The van der Waals equation qualitatively describes the properties of a real gas quite well, but the results of numerical calculations do not always agree with experimental data. In a number of cases, these deviations are explained by the tendency of real gas molecules to associate in separate groups, consisting of two, three or more molecules. The association occurs due to the asymmetry of the external electric field of the molecules. The resulting complexes behave like independent unstable particles. During collisions, they disintegrate, then unite again with other molecules, etc. As the temperature increases, the concentration of complexes with a large number molecules decreases rapidly, and the proportion of single molecules increases. Polar molecules of water vapor exhibit a greater tendency to associate.

All parameters, including temperature, depend on each other. This dependence is expressed by equations like

F(X 1 ,X 2 ,...,x 1 ,x 2 ,...,T) = 0,

where X 1, X 2,... are generalized forces, x 1, x 2,... are generalized coordinates, and T is temperature. Equations that establish the relationship between parameters are called equations of state.

Equations of state are given for simple systems, mainly for gases. For liquids and solids, which are generally assumed to be incompressible, practically no equations of state have been proposed.

By the middle of the twentieth century. a significant number of equations of state for gases were known. However, the development of science has taken such a path that almost all of them have not found application. The only equation of state that continues to be widely used in thermodynamics is the equation of state of an ideal gas.

Ideal gas is a gas whose properties are similar to that of a low-molecular-weight substance at very low pressure and a relatively high temperature (quite far from the condensation temperature).

For an ideal gas:

Boyle's law - Mariotta(at a constant temperature, the product of gas pressure and its volume remains constant for given quantity substances)

Gay-Lussac's law(at constant pressure the ratio of gas volume to temperature remains constant)

Charles's law(at constant volume the ratio of gas pressure to temperature remains constant)

S. Carnot combined the above relations into a single equation of the type

B. Clapeyron gave this equation a form close to the modern one:

The volume V included in the equation of state of an ideal gas refers to one mole of the substance. It is also called molar volume.

The generally accepted name for the constant R is the universal gas constant (very rarely you can find the name “Clapeyron’s constant” ). Its value is

R=8.31431J/mol TO.

Approaching a real gas to an ideal one means achieving such large distances between molecules that their own volume and the possibility of interaction can be completely neglected, i.e. the existence of forces of attraction or repulsion between them.

Van der Waals proposed an equation that takes these factors into account in the following form:

where a and b are constants determined for each gas separately. The remaining quantities included in the van der Waals equation have the same meaning as in the Clapeyron equation.

The possibility of the existence of an equation of state means that to describe the state of the system, not all parameters can be specified, but their number is less by one, since one of them can be determined (at least hypothetically) from the equation of state. For example, to describe the state of an ideal gas, it is enough to indicate only one of the following pairs: pressure and temperature, pressure and volume, volume and temperature.

Volume, pressure and temperature are sometimes called external parameters of the system.

If simultaneous changes in volume, pressure and temperature are allowed, then the system has two independent external parameters.

The system, located in a thermostat (a device that ensures constant temperature) or a manostat (a device that ensures constant pressure), has one independent external parameter.

State parameters are related to each other. The relation that defines this connection is called the equation of state of this body. In the simplest case equilibrium state a body is determined by the values ​​of those parameters: pressure p, volume V and temperature, the mass of the body (system) is usually considered known. Analytically, the relationship between these parameters is expressed as a function F:

Equation (1) is called the equation of state. This is a law that describes the nature of the change in the properties of a substance when changing external conditions.

What is an ideal gas

Particularly simple, but very informative is the equation of state of the so-called ideal gas.

Definition

An ideal gas is one in which the interaction of molecules with each other can be neglected.

Rarefied gases are considered ideal. Helium and hydrogen are especially close in their behavior to ideal gases. An ideal gas is a simplified mathematical model real gas: molecules are considered to move chaotically, and collisions between molecules and impacts of molecules on the walls of the vessel --- elastic, such that do not lead to energy losses in the system. This simplified model is very convenient, since it does not require taking into account the interaction forces between gas molecules. Most real gases do not differ in their behavior from an ideal gas under conditions when the total volume of molecules is negligible compared to the volume of the container (i.e., when atmospheric pressure and room temperature), which allows the use of the ideal gas equation of state in complex calculations.

The equation of state of an ideal gas can be written in several forms (2), (3), (5):

Equation (2) -- Mendeleev -- Clayperon equation, where m is gas mass, $\mu $ -- molar mass gas, $R=8.31\ \frac(J)(mol\cdot K)$ is the universal gas constant, $\nu \ $ is the number of moles of the substance.

where N is the number of gas molecules in mass m, $k=1.38\cdot 10^(-23)\frac(J)(K)$, Boltzmann constant, which determines the “fraction” of the gas constant per molecule and

$N_A=6.02\cdot 10^(23)mol^(-1)$ -- Avogadro's constant.

If we divide both sides in (4) by V, we get the following form writing the equation of state of an ideal gas:

where $n=\frac(N)(V)$ is the number of particles per unit volume or particle concentration.

What is real gas

Let us now turn to more complex systems- to non-ideal gases and liquids.

Definition

A real gas is a gas that has noticeable interaction forces between its molecules.

In non-ideal, dense gases, the interaction of molecules is strong and must be taken into account. It turns out that the interaction of molecules complicates the physical picture so much that the exact equation of state of a nonideal gas cannot be written in a simple form. In this case, they resort to approximate formulas found semi-empirically. The most successful such formula is the van der Waals equation.

The interaction of molecules has complex nature. Comparatively long distances There are attractive forces between molecules. As the distance decreases, the attractive forces first increase, but then decrease and turn into repulsive forces. The attraction and repulsion of molecules can be considered and taken into account separately. Van der Waals equation describing the state of one mole of a real gas:

\[\left(p+\frac(a)(V^2_(\mu ))\right)\left(V_(\mu )-b\right)=RT\ \left(6\right),\]

where $\frac(a)(V^2_(\mu ))$ is the internal pressure caused by the forces of attraction between molecules, b is the correction for the intrinsic volume of molecules, which takes into account the action of repulsive forces between molecules, and

where d is the diameter of the molecule,

the value a is calculated by the formula:

where $W_p\left(r\right)\ $- potential energy attraction between two molecules.

As the volume increases, the role of corrections in equation (6) becomes less significant. And in the limit, equation (6) turns into equation (2). This is consistent with the fact that as the density decreases, real gases approach ideal gases in their properties.

The advantage of the van der Waals equation is the fact that it is very high densities also approximately describes the properties of the liquid, in particular its poor compressibility. Therefore, there is reason to believe that the van der Waals equation will also reflect the transition from liquid to gas (or from gas to liquid).

Figure 1 shows the van der Waals isotherm for some constant value temperature T, constructed from the corresponding equation.

In the area of ​​the “convolution” (the CM section), the isotherm crosses the isobar three times. In the section [$V_1$, $V_2$], the pressure increases with increasing volume.

Such dependence is impossible. This may mean that something unusual is happening to the substance in this area. What exactly this is cannot be seen from the van der Waals equation. It is necessary to turn to experience. Experience shows that in the region of the “convolution” on the isotherm in a state of equilibrium, the substance is stratified into two phases: liquid and gaseous. Both phases coexist simultaneously and are in phase equilibrium. The processes of liquid evaporation and gas condensation occur in phase equilibrium. They flow with such intensity that they completely compensate each other: the amount of liquid and gas remains unchanged over time. A gas that is in phase equilibrium with its liquid is called saturated vapor. If there is no phase equilibrium, there is no compensation for evaporation and condensation, then the gas is called unsaturated vapor. How does the isotherm behave in the region of the two-phase state of matter (in the region of the “convolution” of the van der Waals isotherm)? Experience shows that in this region, when the volume changes, the pressure remains constant. The isotherm graph runs parallel to the V axis (Figure 2).

As the temperature increases, the area of ​​two-phase states on the isotherms narrows until it turns into a point (Fig. 2). This singular point K, in which the difference between liquid and vapor disappears. It is called critical point. The parameters corresponding to the critical state are called critical ( critical temperature, critical pressure, critical density substances).

Solution: From the Van Der Waals equation it follows that:

Let's convert the temperature to SI: T=t+273, According to the condition $T=173K, V = 0.1 l=10^(-4)m^3$

Let's carry out the calculation: $p=\frac(8.31\cdot 173)(\left(10-3.2\right)\cdot 10^(-5))-\frac(0.1358)(((10) ^(-4))^2)=75.61\cdot 10^5\left(Pa\right)$

For an ideal gas:

Let's carry out the calculation: $p_(id)=\frac(1\cdot 8.31\cdot 173)((10)^(-4))=143\cdot 10^5\ \left(Pa\right)\left( 2.3\right)$

Answer: $p\approx 0.53p_(id)$

At constant mass system parameters p, V, t can change due to external influences(mechanical and thermal). If the system is homogeneous in its physical properties and it doesn't happen chemical reactions, then, as experience shows, when changing one of its parameters in general case changes occur and others. Thus, based on experiments, it can be argued that the parameters of a homogeneous system (at constant mass) must be functionally connected:

Equation (3.1) is called the thermal equation of state of the system or simply the equation of state. Finding this equation explicitly is one of the main problems of molecular physics. At the same time, thermodynamically, using general laws, it is impossible to find the form of this equation. It is only possible by studying individual characteristics of certain systems, select dependencies (3.1) that will have the meaning of empirical dependencies that approximately describe the behavior of systems in limited ranges of changes in temperature and pressure. In molecular

physics developed general method obtaining equations (3.1) based on taking into account intermolecular interactions, but in this way when considering specific systems there are big ones math difficulties. Using molecular kinetic methods, the equation of state for rarefied (ideal) gases was obtained, intermolecular interactions in which are negligible. Molecular physics also allows you to describe the properties quite well without being very strong compressed gases. But the question of the theoretical derivation of the equation of state for dense gases and liquids, despite the efforts of many scientists, currently remains unresolved.

A change in the state of a system associated with a change in its parameters is called a thermodynamic process. According to (3.1), the state of the body can be represented by a point in the coordinate system. In Figure 1.3, a two states of the system are depicted by points. The transition from state 1 to state 2 occurs as a result of a thermodynamic process as a sequence of a series of intermediate states replacing each other.

One can imagine such a transition from the initial state to the final state 2, in which each intermediate state will be equilibrium. Such processes are called equilibrium and are depicted in the coordinate system by a continuous line (Fig. 1.3,b). In laboratory-scale systems, equilibrium processes proceed infinitely slowly; only with such a course of the process can the pressure and temperature in changing objects at each moment of time be considered the same everywhere. Using the model shown in Figure 1.1, a similar process can be carried out either by removing or adding individual pellets, or by infinitely slowly changing the temperature of the thermostat in which there is a cylinder with heat-conducting walls.

If changes occur quickly enough in the system (in the model shown in Figure 1.1, the piston load changes abruptly by a finite amount), then inside its pressure and temperature are not the same different points, i.e. they are functions of coordinates. Such processes are called nonequilibrium, they

are realized through a sequence of nonequilibrium states, which cannot be displayed by any graph.

Equilibrium processes are idealized processes; their description is much simpler than nonequilibrium ones. The study of such processes is very important, since many of their characteristics are limiting for real processes occurring at finite speeds.

The curve (Fig. 1.3,b) can be projected onto a plane or. Therefore, in practice, a two-dimensional image of equilibrium processes is more often used (Fig. 1.4).

Examples Equation of state can be used for gases Clapeyron equation for ideal gas p u = RT, Where R – gas constant, u – volume 1 begging gas;

D. N. Zubarev.

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