State Options .
1. - absolute pressure
2. 
- specific volume
3. 
Temperature
4. Density
F (p, v, t) = 0.
process .
equilibrium process
Reversible process -
thermodynamic process
p-v, p-T process curve
– equation of the form 
.
State equation for a simple body - 
.
Ideal gas
PV=nRT
real gas 
Question 3. Thermodynamic work, P-V coordinates.
Thermodynamic work: 
, where is the generalized force, is the coordinate.
Specific work: 
, 
, where is the mass.
If 
And 
, then there is a process of expansion, the work is positive.
- If 
And 
, then the compression process is negative.
- With a small change in volume, the pressure practically does not change.
Full thermodynamic work: 
.
1. In case 
, That .
, then the work is divided into two parts: 
, where - effective work, - irreversible losses, while 
- the heat of internal heat transfer, that is, irreversible losses are converted into heat.
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Question 4. Potential work, P-V coordinates, work distribution.
Potential Job is the work caused by the change in pressure. 
- If 
And 
- If 
And 
, then the compression process is in progress.
- With a small change in pressure, the volume almost does not change.
The total potential work can be found by the formula: 
.
1. In case 
, That .
2. If the process equation is given - 
, That 
.
Where is the work
transferred to external systems.
, with E is the speed of the body, dz is the change in the height of the center of gravity of the body in the gravitational field.
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Question 16. Isobaric process of changing the state of a simple body. Process equation, P-V representation, relation between parameters, work and heat transfer, change of state functions.
If 
, then the expansion process is in progress.
isobaric process.
Because 
, That 
.
For an ideal gas:
First law of thermodynamics: .
For an ideal gas: 
And 
Question 63. Throttling. Joule-Thomson effect. Basic concepts
Throttling- the process of movement of matter through a sudden narrowing. The reasons for the occurrence of local resistance during the movement of the flow of the working fluid through the channels can be locking, regulating and measuring devices; turns, narrowing, pollution of channels, etc.
Joule-Thomson effect- change in the temperature of the substance during adiabatic throttling.
Rice. 1.7. Throttling process in h-s diagram
Distinguish differential And integral choke - effects. Value of differential choke – effect is determined from the relation
, Where –
Joule-Thomson coefficient, [K/Pa].
Integral choke effect: 
.
The Joule-Thomson coefficient is derived from the mathematical expressions of the first law of thermodynamics and the second law of thermostatics
1. If the throttle-effect is positive ( D h > 0), then the temperature of the working fluid decreases ( dT<0 );
2. If the choke-effect is negative ( D h< 0
), then the temperature of the working fluid rises ( dT>0);
3. If the choke-effect is zero ( D h = 0), then the temperature of the working fluid does not change. The state of the gas or liquid that the condition corresponds to D h = 0, is called point of inversions.
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Two stroke diesel
Workflow in two-stroke diesel basically proceeds in the same way as in a two-stroke carburetor engine, and differs only in that the cylinder is purged with clean air. At the end of it, the air remaining in the cylinder is compressed. At the end of compression, fuel is injected through the nozzle into the combustion chamber and ignites.
The working process in a two-stroke diesel engine proceeds as follows.
First beat. When the piston moves up from n. m. t. to v. m.t., first, the purge ends, and then the end of the release. On the indicator diagram, the purge is shown by the line b "- a" and the outlet - a "- a.
After the exhaust port is closed by the piston, air is compressed in the cylinder. The compression line on the indicator diagram is shown by curve a-c. At this time, a vacuum is created under the piston in the crank chamber, under the action of which the automatic valve opens, and clean air is sucked into the crank chamber. As the piston begins to move downwards, due to the decrease in volume under the piston, the air pressure in the crank chamber increases and the valve closes.
Second beat. The piston moves from m. t. to n. m. t. Fuel injection and combustion begin before the end of compression and end after the piston passes through. m. t. At the end of combustion, expansion occurs. The flow of the expansion process on the indicator diagram is shown by the curve r-b.
The remaining processes, exhaust and purge proceed in the same way as in a carbureted two-stroke engine.
Question 2. State parameters and equations of state.
State Options- physical quantities characterizing the internal state of the thermodynamic system. The state parameters of a thermodynamic system are divided into two classes: intensive (do not depend on the mass of the system) and extensive (proportional to the mass).
Thermodynamic state parameters called intensive parameters characterizing the state of the system. The simplest parameters:
1. - absolute pressure - numerically equal to the force F acting per unit area f of the surface of the body ┴ to the last, [Pa \u003d N / m 2]
2. - specific volume
is the volume per unit mass of a substance.
3. Temperature
is the only state function of a thermodynamic system that determines the direction of spontaneous heat transfer between bodies.
4. Density substance is called the ratio of the mass of a body to its volume
The connection between the parameters characterizing the state of a simple body is called the equation of state F (p, v, t) = 0.
The change in the state of the system is called process .
equilibrium process is a continuous sequence of equilibrium states of the system.
Reversible process - an equilibrium process that allows the return of this system from the final state to the initial state by the reverse process.
thermodynamic process considered to be a reversible equilibrium process.
Equilibrium processes can be depicted graphically on state diagrams p-v, p-T etc. The line depicting the change in parameters in the process is called process curve. Each point of the process curve characterizes the equilibrium state of the system.
Thermodynamic process equation
– equation of the form .
State equation for a simple body - .
Ideal gas- a set of material points (molecules or atoms) that are in chaotic motion. These points are considered as absolutely elastic bodies, having no volume and not interacting with each other. The equation of state for an ideal gas is the Mendeleev-Clapeyron equation:
PV=nRT, where P – pressure, [Pa]; V is the volume of the system [m 3]; n is the amount of substance, [mol]; T – thermodynamic temperature, [K]; R is the universal gas constant.
real gas- a gas whose molecules interact with each other and occupy a certain volume. The equation of state of a real gas is the generalized Mendeleev-Clapeyron equation: , where Z r = Z r (p,T) is the gas compressibility factor; m is the mass; M is the molar mass.
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With a constant mass, the parameters of the system p, V, t can change due to external influences (mechanical and thermal). If the system is homogeneous in its physical properties and no chemical reactions occur in it, then, as experience shows, when one of its parameters changes, in the general case, changes also occur in others. Thus, based on experiments, it can be argued that the parameters of a homogeneous system (at a constant mass) should be functionally related:
Equation (3.1) is called the thermal equation of state of the system or simply the equation of state. Finding this equation in explicit form 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 the individual characteristics of certain systems, to select dependences (3.1) that will have the meaning of empirical dependences that approximately describe the behavior of systems in limited ranges of temperature and pressure changes. In molecular
Physics has developed a general method for obtaining equations (3.1) on the basis of taking into account intermolecular interactions, but on this path, when considering specific systems, great mathematical difficulties are encountered. Molecular-kinetic methods have been used to obtain the equation of state for rarefied (ideal) gases, in which intermolecular interactions are negligibly small. Molecular physics also makes it possible to describe quite well the properties of gases that are not very strongly compressed. But the question of the theoretical derivation of the equation of state for dense gases and liquids, despite the efforts of many scientists, remains unresolved at present.
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 displayed by a point in the coordinate system. Figure 1.3, a shows two states of the system with points. The transition from state 1 to state 2 is carried out as a result of a thermodynamic process as a sequence of a number of successive intermediate states.
It is possible to imagine such a transition from the initial state to the final state 2, in which each intermediate state will be in equilibrium. Such processes are called equilibrium and in the coordinate system are depicted 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, pressure and temperature in changing objects can be considered the same everywhere at every moment of time. Using the model shown in Figure 1.1, a similar process can be carried out both by removing or adding individual pellets, and by infinitely slow changes in the temperature of the thermostat, in which there is a cylinder with heat-conducting walls.
If changes occur in the system quickly enough (in the model shown in Figure 1.1, the piston load changes by a finite amount in a jump), then inside its pressure and temperature are not the same at different points, i.e., they are functions of the coordinates. Such processes are called non-equilibrium, they
Since the equation of state pV = nRT is simple and reflects with reasonable accuracy the behavior of many gases over a wide range of environmental conditions, it is very useful. But, of course, it is not universal. It is obvious that not a single substance in the liquid and solid state obeys this equation. There are no such condensed substances, the volume of which would be halved when the pressure is doubled. Even gases under high compression or near their dew point show noticeable deviations from the indicated behavior. Many other more complex equations of state have been proposed. Some of them are highly accurate in a limited area of changing external conditions. Some are applicable to special classes of substances. There are equations that apply to a wider class of substances under more widely differing environmental conditions, but they are not very precise. Here we will not waste time on a detailed consideration of such equations of state, but nevertheless we will give some idea of them.
Let us assume that the gas molecules are absolutely elastic hard balls, so small that their total volume can be neglected in comparison with the volume occupied by the gas. Let us also assume that there are no attractive or repulsive forces between the molecules and that they move completely randomly, colliding randomly with each other and with the walls of the vessel. If elementary classical mechanics is applied to this gas model, then we obtain the relation pV = RT without resorting to any generalizations of experimental data such as the laws of Boyle - Mariotte and Charles - Gay-Luss. In other words, the gas that we have called "ideal" behaves as a gas consisting of very small hard balls interacting with each other only at the moment of collision should behave. The pressure exerted by such a gas on any surface is simply equal to the average value of the momentum transferred per unit time by the molecules to the unit surface when colliding with it. When a molecule of mass m hits a surface with a velocity component perpendicular to the surface, and is reflected with a velocity component, then the resulting momentum transferred to the surface, according to the laws of mechanics, is equal to These speeds are quite high (several hundred meters per second for air under normal conditions), so the collision time is very short and the momentum transfer occurs almost instantly. But the collisions are so numerous (on the order of 1023 per 1 cm2 per 1 s in air at atmospheric pressure) that when measured by any instrument, the pressure is absolutely constant in time and continuous.
Indeed, most direct measurements and observations show that gases are a continuous medium. The conclusion that they must consist of a large number of individual molecules is purely speculative.
We know from experience that real gases do not follow the rules of behavior predicted by the ideal model just described. At sufficiently low temperatures and sufficiently high pressures, any gas condenses into a liquid or solid state, which, compared with a gas, can be considered incompressible. Thus, the total volume of the molecules is not always negligible compared to the volume of the vessel. It is also clear that attractive forces exist between molecules, which at sufficiently low temperatures can bind molecules, leading to the formation of a condensed form of matter. These considerations suggest that one way to obtain an equation of state that is more general than the equation of state for an ideal gas is to take into account the finite volume of real molecules and the forces of attraction between them.
Accounting for the molecular volume is not difficult, at least at a qualitative level. Let us simply assume that the free volume available for the movement of molecules is less than the total volume of the gas V by a value of 6, which is related to the size of the molecules and is sometimes called the bound volume. So we have to replace V in the ideal gas equation with (V - b); then we get
This relationship is sometimes called the Clausius equation of state, after the German physicist Rudolf Clausius, who played a major role in the development of thermodynamics. We will learn more about his work in the next chapter. Note that equation (5) is written for 1 mol of gas. For n mol, you need to write p(V-nb) = nRT.
It is somewhat more difficult to take into account the forces of attraction between molecules. A molecule located in the center of the gas volume, that is, far from the walls of the vessel, will "see" the same number of molecules in all directions. Therefore, the attractive forces are the same in all directions and balance each other, so that no net force arises. When a molecule approaches the vessel wall, it "sees" more molecules behind it than in front of it. As a result, there is an attractive force directed towards the center of the vessel. The movement of the molecule is somewhat restrained, and it hits the vessel wall less strongly than in the absence of attractive forces.
Since the pressure of a gas is due to the transfer of momentum by molecules colliding with the walls of the vessel (or with any other surface located inside the gas), the pressure created by the attracted molecules is somewhat less than the pressure created by the same molecules in the absence of attraction. It turns out that the decrease in pressure is proportional to the square of the density of the gas. Therefore we can write
where p is the density in moles per unit volume, is the pressure created by an ideal gas of non-attractive molecules, and a is a proportionality coefficient characterizing the magnitude of the forces of attraction between molecules of a given type. Recall that , where n is the number of moles. Then relation (b) can be rewritten for 1 mol of gas in a slightly different form:
where a has a characteristic value for a given type of gas. The right side of equation (7) is the "corrected" pressure of an ideal gas, which needs to replace p in the equation. If we take into account both corrections, one due to volume in accordance with (b) and the other due to attractive forces according to (7), we get for 1 mol of gas
This equation was first proposed by the Dutch physicist D. Van der Waals in 1873. For n mol, it takes the form
The van der Waals equation takes into account in a simple and clear form two effects that cause deviations in the behavior of real gases from the ideal. Obviously, the surface representing the van der Waals equation of state in the space p, V, Tu cannot be as simple as the surface corresponding to an ideal gas. A part of such a surface for specific values of a and b is shown in Fig. 3.7. Isotherms are shown as solid lines. The isotherms corresponding to temperatures above the temperature of which the so-called critical isotherm corresponds do not have minima and inflections and look similar to the ideal gas isotherms shown in Fig. 3.6. At temperatures below the isotherm, they have maxima and minima. At sufficiently low temperatures, there is a region in which the pressure becomes negative, as indicated by the portions of the isotherms depicted by the dashed lines. These humps and dips, as well as the area of negative pressures, do not correspond to physical effects, but simply reflect the shortcomings of the van der Waals equation, its inability to describe the true equilibrium behavior of real substances.
Rice. 3.7. Surface p - V - T for a gas obeying the van der Waals equation.
In fact, in real gases at temperatures below and at sufficiently high pressure, attractive forces between molecules lead to the condensation of the gas into a liquid or solid state. Thus, the anomalous region of peaks and dips on isotherms in the region of negative pressure, which is predicted by the van der Waals equation, in real substances corresponds to the region of the mixed phase, in which vapor and liquid or solid state coexist. Rice. 3.8 illustrates this situation. Such "discontinuous" behavior cannot be described at all by any relatively simple and "continuous" equation.
Despite its shortcomings, the van der Waals equation is useful for describing corrections to the ideal gas equation. The values of a and b for various gases are determined from experimental data, some typical examples are given in Table. 3.2. Unfortunately, for any particular gas, there are no single values of a and b that would provide an accurate description of the relationship between p, V and T over a wide range using the van der Waals equation.
Table 3.2. Characteristic values of the van der Waals constants
However, the values given in the table give us some qualitative information about the expected amount of deviation from the behavior of an ideal gas.
It is instructive to consider a specific example and compare the results obtained with the ideal gas equation, the Clausius equation, and the van der Waals equation with measured data. Consider 1 mol of water vapor in a volume of 1384 cm3 at a temperature of 500 K. Remembering that (mol K) and using the values from Table. 3.2, we get
a) from the equation of state of an ideal gas:
b) from the Clausius equation of state: atm;
c) from the van der Waals equation of state:
d) from experimental data:
For these specific conditions, the ideal gas law gives an overestimated value of pressure by about 14%, the equation
Rice. 3.8. A surface for a substance that contracts when cooled. A surface like this cannot be described by a single equation of state and must be built on the basis of experimental data.
Clausius gives an even larger error, about 16%, and the van der Waals equation overestimates the pressure by about 5%. Interestingly, the Clausius equation gives a larger error than the ideal gas equation. The reason is that the correction for the finite volume of the molecules increases the pressure, while the term for attraction decreases it. Thus, these corrections partially offset each other. The ideal gas law, which does not take into account either correction, gives a pressure value closer to the actual value than the Clausius equation, which takes into account only its increase due to a decrease in free volume. At very high densities, the correction for the volume of molecules becomes much more significant and the Clausius equation turns out to be more accurate than the ideal gas equation.
Generally speaking, for real substances we do not know the explicit relationship between p, V, T, and n. For most solids and liquids, there are not even rough approximations. Nevertheless, we are firmly convinced that such a ratio exists for every substance and that the substance obeys it.
A piece of aluminum will occupy a certain volume, always exactly the same, if the temperature and pressure are at the given values. We write this general statement in mathematical form:
This entry asserts the existence of some functional relationship between p, V, T and n, which can be expressed by an equation. (If all the terms of such an equation are moved to the left, the right side will obviously be equal to zero.) Such an expression is called an implicit equation of state. It means the existence of some relationship between the variables. It also says that we do not know what this ratio is, but the substance "knows" it! Rice. 3.8 allows us to imagine how complex an equation must be that would describe real matter in a wide range of variables. This figure shows the surface for a real substance that shrinks when it freezes (this is how almost all substances behave, except water). We are not adept enough to predict, by calculation, how much volume a substance will take up given arbitrarily given values of p, T, and n, but we are absolutely sure that the substance "knows" how much volume it will take up. This confidence is always confirmed by experimental verification. A substance always behaves in a unique way.
EQUATION OF STATE - an equation that relates pressure R, volume V and abs. temp-ru T physically homogeneous system in a state of thermodynamic equilibrium: f(p, V, T) = 0. This equation is called. thermal U. s., in contrast to the caloric U. s., which determines the internal. energy U systems as f-tion to-l. two of the three parameters p, v, t. Thermal W. s. allows you to express pressure in terms of volume and temperature, p=p(V, T), and determine the elementary work for an infinitesimal expansion of the system . W. s. is a necessary addition to thermodynamic. laws that make it possible to apply them to real substances. It cannot be derived using laws alone, but is determined from experience or calculated theoretically on the basis of ideas about the structure of matter by statistical methods. physics. From first law of thermodynamics follows only the existence of caloric. U.S., and from second law of thermodynamics- the relationship between caloric and thermal U. with .:
Where A And b- constants depending on the nature of the gas and taking into account the influence of the forces of intermolecular attraction and the finiteness of the volume of molecules; virial U. s. for a non-ideal gas:
Where B (T), C (T), ...- 2nd, 3rd, etc. virial coefficients, depending on the forces of intermolecular interaction. Virial U. s. makes it possible to explain many experimental results based on simple models intermolecular interaction in gases. Also offered are diff. empirical At. pages, based on eksperim. data on the heat capacity and compressibility of gases. W. s. non-ideal gases indicate the existence of critical. points (with parameters p To, V K , T j), in which the gaseous and liquid phases become identical. If U. s. represent in the form of a reduced U.S., that is, in dimensionless variables r / r k, V/V K , T/ T to, then at not too low temp-pax this equation changes little for decomp. substances (law of corresponding states),
For liquids, due to the difficulty of taking into account all the features of intermolecular interaction, it has not yet been possible to obtain a general theoretical ultrasonic coefficient. The van der Waals equation and its modifications, although they are used for qualities, assess the behavior of liquids, but in essence it is not applicable below critical. points where coexistence of liquid and gaseous phases is possible. The ultrasonic density, which describes well the properties of a number of simple liquids, can be obtained from approximate theories of liquids. Knowing the probability distribution of the mutual arrangement of molecules (pair correlation functions; see. Liquid), it is possible in principle to calculate the W. s. liquids, but this problem is complex and has not been completely solved even with the help of a computer.
For receiving U. page. solids use the theory vibrations of the crystal lattice, but the universal U. s. for solids not received.
For (photon gas) W. with. determined
For an equilibrium thermodynamic system, there is a functional relationship between the state parameters, which is called equation costanding. Experience shows that the specific volume, temperature and pressure of the simplest systems, which are gases, vapors or liquids, are related termic equation view states.
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 by 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. The specific form of the equation of state depends on the individual properties of the substance.
Equation of state of ideal hacall
Equations (1.1) and (1.2) imply that 
.
Consider 1 kg of gas. Considering that it contains N molecules and therefore 
, we get: 
.
Constant value Nk, referred to 1 kg of gas, denoted by the letter R and call gas constantlyNoah. That's why
, or 
.
(1.3)
The resulting relation is the Clapeyron equation.
Multiplying (1.3) by M, we obtain the equation of state for an arbitrary mass of gas M: 
.
(1.4)
The Clapeyron equation can be given a universal form if we refer the gas constant to 1 kmole of gas, that is, 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 for one mole the Clapeyron - Mendeleev equation:
.
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 kmole, which is the same under the same conditions for all ideal gases, under normal physical conditions is 22.4136 m 3, therefore
The gas constant of 1 kg of gas is 
.
Equation of state of real hacall
In real gases V the difference from the ideal is significant forces of intermolecular interactions (attractive forces when the molecules are at a considerable distance, and repulsive forces when they are sufficiently close to each other) and the intrinsic 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
is the smallest volume to which a gas can be compressed. In accordance with this, the mean free path of molecules decreases and the number of impacts on the wall per unit time, and, consequently, the pressure increases compared to an ideal gas in relation to
,
i.e.
.
Attractive forces act in the same direction as the external pressure and give rise to 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, that is, to the square of the density, so the molecular pressure is inversely proportional to the square of the specific volume of the gas: Rthey say= a/ v 2 , where A - coefficient of proportionality, 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 a real gas, the van der Waals equation practically degenerates into the Clapeyron equation of state for an ideal gas, because the quantity a/v 2
(compared with p) And b (compared with v) become negligible.
Qualitatively, the van der Waals equation 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 into 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 break up, then recombine with other molecules, etc. As the temperature rises, the concentration of complexes with a large number of molecules decreases rapidly, and the proportion of single molecules increases. Polar water vapor molecules exhibit a greater tendency to association.
