The equation of state of an ideal gas is written as: Ideal gas. Equation of state of an ideal gas. Isoprocesses. The pressure remains constant
The state of gases is characterized by pressure R, temperature 7, and volume V. The relationship between these quantities is determined by the laws of the gas state.
Oil and natural gases have significant deviations from the laws of ideal gases due to the interaction between molecules, which occurs when real gases are compressed. The degree of deviation of the compressibility of real gases from ideal ones is characterized by the compressibility coefficient z, which shows the ratio of the volume of a real gas to the volume of an ideal gas under the same conditions.
In a reservoir, hydrocarbon gases can be found in a variety of conditions. With an increase in pressure from O to 3-4 MPa, the volume of gases decreases. In this case, the hydrocarbon gas molecules come closer together and the attractive forces between them help the external forces that compress the gas. When a hydrocarbon gas is highly compressed, the intermolecular distances are so small that repulsive forces begin to resist further reduction in volume and the compressibility of the gas decreases.
In practice, the state of real hydrocarbon gases at various temperatures and pressures can be described based on the Clapeyron equation:
P-V=z-m-R-T (2.9)
Where R - pressure gz. Pa; V" - volume occupied by gas at a given pressure, m 3 ; T - gas mass, kg; R- gas constant, J/(kg-K); T- temperature, K; G - compressibility factor.
The compressibility coefficient is determined from graphs constructed from experimental data.
State of hydrocarbon gas-liquid systems with changes in pressure and temperature.
When oil and gas move in the formation, wellbore, collection and treatment systems, pressure and temperature change, which causes a change in the phase state of hydrocarbons - a transition from liquid to gaseous state and vice versa. Since oil and gas consist of a large number of components with different properties, under certain conditions some of these components can be in the liquid phase, and the other in the vapor (gas) phase. It is obvious that the patterns of movement of a single-phase system in the formation and wellbore are significantly different from the patterns of multiphase movement. Conditions for long-distance transport of oil and gas and subsequent processing require the separation of easily evaporating components from the liquid condensed fraction. Therefore, the choice of field development technology and in-field oil and gas treatment system is largely related to the study of the phase state of hydrocarbons under changing thermodynamic conditions.
Phase transformations of hydrocarbon systems are illustrated by phase diagrams showing the relationship between pressure, temperature and specific volume of a substance.
In Fig. 2.2, A The state diagram of pure gas (ethane) is shown. The solid lines in the diagram show the relationship between pressure and the specific volume of a substance at constant temperatures. The lines passing through the area bounded by the dotted curve have three characteristic sections. If we consider one of the lines of the high pressure region, then at first the increase in pressure is accompanied by a slight increase in the specific volume of the substance, which is compressible and in this region is in a liquid state.
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Rice. 2.2. Pure gas phase diagram
At a certain pressure, the isotherm breaks sharply and looks like a horizontal line. At constant pressure there is a continuous increase in the volume of the substance. In this area, the liquid evaporates and enters the vapor phase. Evaporation ends at the point of the second break of the isotherm, after which the change in volume is accompanied by an almost proportional decrease in pressure. In this region, all matter is in gaseous form.
state (in the vapor phase). The dotted line connecting the break points of the isotherms limits the region of transition of a substance from a liquid to a vapor state or vice versa (in the direction of decreasing specific volumes). This region corresponds to the conditions under which a substance is simultaneously in two states, liquid and gaseous (region of a two-phase state of a substance). The dotted line located to the left of point C is called vaporization point curve. The coordinates of the points on this line are the pressure and temperature at which the substance begins to boil. To the right of point C lies a dotted line called condensation point curve or dew points. It shows at what pressures and temperatures vapor condensation begins - the transition of a substance into a liquid state. Point C, lying at the top of the two-phase region, is called critical point. At the pressure and temperature corresponding to this point, the properties of the vapor and liquid phases are the same. In addition, for a pure substance, the critical point determines the highest values of pressure and temperature at which the substance can simultaneously be in a two-phase state. When considering an isotherm that does not cross the two-phase region, it is clear that the properties of the substance change continuously and the transition of the substance from the liquid to the gaseous state or vice versa occurs without passing through the two-phase state.
In Fig. 2.2, b The state diagram of ethane is shown, rearranged in pressure-temperature coordinates. Since a pure substance passes from one phase state to another at constant pressure, the curves of the evaporation and condensation points in this diagram coincide and end with the critical point C. The resulting line delimits the regions of liquid and vaporous substances. A substance can be in a two-phase state only at pressures and temperatures corresponding to the coordinates of this line.
« Physics - 10th grade"
This chapter will discuss the implications that can be drawn from the concept of temperature and other macroscopic parameters. The basic equation of the molecular kinetic theory of gases has brought us very close to establishing connections between these parameters.
We examined in detail the behavior of an ideal gas from the point of view of molecular kinetic theory. The dependence of gas pressure on the concentration of its molecules and temperature was determined (see formula (9.17)).
Based on this dependence, it is possible to obtain an equation connecting all three macroscopic parameters p, V and T, characterizing the state of an ideal gas of a given mass.
Formula (9.17) can only be used up to a pressure of the order of 10 atm.
The equation relating three macroscopic parameters p, V and T is called ideal gas equation of state.
Let us substitute the expression for the concentration of gas molecules into the equation p = nkT. Taking into account formula (8.8), the gas concentration can be written as follows:
where N A is Avogadro's constant, m is the mass of the gas, M is its molar mass. After substituting formula (10.1) into expression (9.17) we will have
The product of Boltzmann's constant k and Avogadro's constant N A is called the universal (molar) gas constant and is denoted by the letter R:
R = kN A = 1.38 10 -23 J/K 6.02 10 23 1/mol = 8.31 J/(mol K). (10.3)
Substituting the universal gas constant R into equation (10.2) instead of kN A, we obtain the equation of state of an ideal gas of arbitrary mass
The only quantity in this equation that depends on the type of gas is its molar mass.
The equation of state implies a relationship between the pressure, volume and temperature of an ideal gas, which can be in any two states.
If index 1 denotes the parameters related to the first state, and index 2 denotes the parameters related to the second state, then according to equation (10.4) for a gas of a given mass
The right-hand sides of these equations are the same, therefore, their left-hand sides must also be equal:
It is known that one mole of any gas under normal conditions (p 0 = 1 atm = 1.013 10 5 Pa, t = 0 °C or T = 273 K) occupies a volume of 22.4 liters. For one mole of gas, according to relation (10.5), we write:
We have obtained the value of the universal gas constant R.
Thus, for one mole of any gas
The equation of state in the form (10.4) was first obtained by the great Russian scientist D.I. Mendeleev. He is called Mendeleev-Clapeyron equation.
The equation of state in the form (10.5) is called Clapeyron equation and is one of the forms of writing the equation of state.
B. Clapeyron worked in Russia for 10 years as a professor at the Institute of Railways. Returning to France, he participated in the construction of many railways and drew up many projects for the construction of bridges and roads.
His name is included in the list of the greatest scientists of France, placed on the first floor of the Eiffel Tower.
The equation of state does not need to be derived every time, it must be remembered. It would be nice to remember the value of the universal gas constant:
R = 8.31 J/(mol K).
So far we have talked about the pressure of an ideal gas. But in nature and in technology, we very often deal with a mixture of several gases, which under certain conditions can be considered ideal.
The most important example of a mixture of gases is air, which is a mixture of nitrogen, oxygen, argon, carbon dioxide and other gases. What is the pressure of the gas mixture?
Dalton's law is valid for a mixture of gases.
Dalton's law
The pressure of a mixture of chemically non-interacting gases is equal to the sum of their partial pressures
p = p 1 + p 2 + ... + p i + ... .
where p i is the partial pressure of the i-th component of the mixture.
The molecular kinetic concepts developed above and the equations obtained on their basis make it possible to find those relationships that connect the quantities that determine the state of the gas. These quantities are: the pressure under which the gas is located, its temperature and the volume V occupied by a certain mass of gas. These are called state parameters.
The three quantities listed are not independent. Each of them is a function of the other two. The equation connecting all three quantities - pressure, volume and temperature of a gas for a given mass is called the equation of state and can be generally written as follows:
This means that the state of a gas is determined by only two parameters (for example, pressure and volume, pressure and temperature, or, finally, volume and temperature), the third parameter is uniquely determined by the other two. If the equation of state is known explicitly, then any parameter can be calculated by knowing the other two.
To study various processes in gases (and not only in gases), it is convenient to use a graphical representation of the equation of state in the form of curves of the dependence of one of the parameters on another at a given constant third. For example, at a given constant temperature, the dependence of gas pressure on its volume
has the form shown in Fig. 4, where different curves correspond to different temperature values: the higher the temperature, the higher the curve lies on the graph. The state of the gas on such a diagram is represented by a dot. The curve of the dependence of one parameter on another shows a change in state, called a process in a gas. For example, the curves in Fig. 4 depict the process of expansion or compression of a gas at a given constant temperature.
In the future, we will widely use such graphs when studying various processes in molecular systems.
For ideal gases, the equation of state can be easily obtained from the basic equations of kinetic theory (2.4) and (3.1).
In fact, substituting into equation (2.4) instead of the average kinetic energy of molecules its expression from equation (3.1), we obtain:
If volume V contains particles, then substituting this expression into (4.1), we have:
This equation, which includes all three parameters of state, is the equation of state of ideal gases.
However, it is useful to transform it so that, instead of the number of particles inaccessible to direct measurement, it includes an easily measurable mass of gas. For such a transformation, we will use the concept of a gram molecule, or mol. Let us recall that a mole of a substance is a quantity of it whose mass, expressed in grams, is equal to the relative molecular mass of the substance (sometimes called molecular weight). This unique unit of quantity of a substance is remarkable, as is known, in that a mole of any substance contains the same number of molecules. In fact, if we denote the relative masses of two substances by and and the masses of the molecules of these substances, then we can write such obvious equalities;
where is the number of particles in a mole of these substances. Since from the very definition of relative mass it follows that
dividing the first of equalities (4.3) by the second, we obtain that a mole of any substance contains the same number of molecules.
The number of particles in a mole, the same for all substances, is called Avogadro's number. We will denote it by We can thus define the mole as a unit of a special quantity - the amount of a substance:
1 mole is an amount of substance containing a number of molecules or other particles (for example, atoms, if the substance is made of atoms) equal to Avogadro's number.
If we divide the number of molecules in a given mass of gas by Avogadro's number, then we get the number of moles in this mass of gas. But the same value can be obtained by dividing the mass of a gas by its relative mass so that
Let's substitute this expression for into formula (4.2). Then the equation of state will take the form:
This equation includes two universal constants: Avogadro’s number and Boltzmann’s constant. Knowing one of them, for example Boltzmann’s constant, the other (Avogadro’s number) can be determined by simple experiments using equation (4.4) itself. To do this, you should take some gas with a known relative mass, fill it with a vessel of known volume V, measure the pressure of this gas and its temperature and determine its mass by weighing the empty (evacuated) vessel and the vessel filled with gas. Avogadro's number turned out to be equal to moles.
1. An ideal gas is a gas in which there are no intermolecular interaction forces. With a sufficient degree of accuracy, gases can be considered ideal in cases where their states are considered that are far from the regions of phase transformations.
2. The following laws are valid for ideal gases:
a) Boyle’s Law - Mapuomma: at constant temperature and mass, the product of the numerical values of pressure and volume of a gas is constant:
pV = const
Graphically, this law in PV coordinates is depicted by a line called an isotherm (Fig. 1).
b) Gay-Lussac's law: at constant pressure, the volume of a given mass of gas is directly proportional to its absolute temperature:
V = V0(1 + at)
where V is the volume of gas at temperature t, °C; V0 is its volume at 0°C. The quantity a is called the temperature coefficient of volumetric expansion. For all gases a = (1/273°С-1). Hence,
V = V0(1 +(1/273)t)
Graphically, the dependence of volume on temperature is depicted by a straight line - an isobar (Fig. 2). At very low temperatures (close to -273°C), Gay-Lussac's law is not satisfied, so the solid line on the graph is replaced by a dotted line.
c) Charles’s law: at constant volume, the pressure of a given mass of gas is directly proportional to its absolute temperature:
p = p0(1+gt)
where p0 is the gas pressure at temperature t = 273.15 K.
The value g is called the temperature coefficient of pressure. Its value does not depend on the nature of the gas; for all gases = 1/273 °C-1. Thus,
p = p0(1 +(1/273)t)
The graphical dependence of pressure on temperature is depicted by a straight line - an isochore (Fig. 3).
d) Avogadro's law: at the same pressures and the same temperatures and equal volumes of different ideal gases, the same number of molecules is contained; or, what is the same: at the same pressures and the same temperatures, the gram molecules of different ideal gases occupy the same volumes.
So, for example, under normal conditions (t = 0°C and p = 1 atm = 760 mm Hg), gram molecules of all ideal gases occupy a volume Vm = 22.414 liters. The number of molecules located in 1 cm3 of an ideal gas at under normal conditions, is called the Loschmidt number; it is equal to 2.687*1019> 1/cm3
3. The equation of state of an ideal gas has the form:
pVm = RT
where p, Vm and T are the pressure, molar volume and absolute temperature of the gas, and R is the universal gas constant, numerically equal to the work done by 1 mole of an ideal gas when heated isobarically by one degree:
R = 8.31*103 J/(kmol*deg)
For an arbitrary mass M of gas, the volume will be V = (M/m)*Vm and the equation of state has the form:
pV = (M/m)RT
This equation is called the Mendeleev-Clapeyron equation.
4. From the Mendeleev-Clapeyron equation it follows that the number n0 of molecules contained in a unit volume of an ideal gas is equal to
n0 = NA/Vm = p*NA /(R*T) = p/(kT)
where k = R/NA = 1/38*1023 J/deg - Boltzmann's constant, NA - Avogadro's number.
Gas pressure arises as a result of collisions of molecules with the walls of a vessel (and on a body placed in a gas), in which there are randomly moving gas molecules. The more frequent the blows, the stronger they are - the higher the pressure. If the mass and volume of a gas are constant, then its pressure in a closed vessel depends entirely on temperature. Pressure also depends on the speed of forward moving gas molecules. The unit of pressure is pascal p(Pa) . Gas pressure is measured with a pressure gauge (liquid, metal and electric).
Ideal gas is a model of real gas. A gas in a vessel is taken to be an ideal gas when a molecule flying from wall to wall of the vessel does not experience collisions with other molecules. More precisely, an ideal gas is a gas in which the interaction between its molecules is negligible ⇒ E to >> E r.
Basic MKT equation relates macroscopic parameters (pressure p , volume V , temperature T , weight m ) gas system with microscopic parameters (mass of molecules, average speed of their movement):
Where n - concentration, 1/m 3; m — molecular mass, kg; - root mean square speed of molecules, m/s.
Ideal gas equation of state- a formula establishing the relationship between pressure, volume and absolute temperature ideal gas, characterizing the state of a given gas system. 
—Mendeleev-Clapeyron equation (for an arbitrary mass of gas). R = 8.31 J/mol K
— universal gas constant. pV
=
RT
– (for 1 mole).
It is often necessary to investigate a situation when the state of a gas changes while its quantity remains unchanged ( m=const ) and in the absence of chemical reactions ( M=const ). This means that the amount of substance ν=const . Then:
For a constant mass of an ideal gas, the ratio of the product of pressure and volume to the absolute temperature in a given state is a constant value: — Clapeyron equation.
Thermodynamic process (or simply process) is a change in the state of a gas over time. During the thermodynamic process, the values of macroscopic parameters change - pressure, volume and temperature. Of particular interest are isoprocesses - thermodynamic processes in which the value of one of the macroscopic parameters remains unchanged. Fixing each of the three parameters in turn, we get t Three types of isoprocesses.
The last equation is called the unified gas law. It makes laws of Boyle - Mariotte, Charles and Gay-Lussac. These laws are called laws for isoprocesses:
Isoprocesses - these are processes that occur at the same parameter or T-temperature, or V-volume, or p-pressure.
Isothermal process— - Boyle-Mariotte law (at a constant temperature and a given mass of gas, the product of pressure and volume is a constant value)
Isobaric process — 
- law