Ideal gas, laws and formulas.

The simplest object of study is an ideal gas. An ideal gas is a gas whose molecules are negligibly small and do not interact at a distance. And during collisions they interact like perfectly elastic balls. An ideal gas is an abstraction. But this concept is useful, as it simplifies engineering calculations of heat engines and the processes occurring in them.

The main parameters of a gas characterizing its state are volume, pressure, , and temperature, .

3. Atomic mass unit (a.u.m.).

The molecular masses are very small,
10 -27 kg. Therefore, to characterize the masses of atoms and molecules, quantities are used that are called the atomic mass unit of an element or molecule,

1a.u.m. = 1.67 10 -27 kg =
.

The masses of all atoms and molecules are measured in amu:

= 12 amu,
= 14 amu,
= 16 amu

Relative molecular (
) or atomic ( ) mass is the ratio of the mass of a molecule or atom to (1/12) the mass of a carbon atom
.

As can be seen from the definition
- dimensionless quantities. Unit of mass equal to (1/12) the mass of a carbon atom
called the atomic mass unit. (a.e.m.). Let us denote this unit (i.e. amu), expressed in kilograms, by
. Then the mass of the atom will be equal
, and the mass of the molecule is
.

A quantity of a substance that contains a number of particles (atoms or molecules) equal to the number of atoms in 0.012 kg of isotope
, is called a mole.

The number of particles contained in a mole of a substance is called Avogadro's number.
= 6.022 10 23 mol -1. The mass of a mole is called the molar mass.

(1)

In the case of carbon

= 1.66 10 -27 kg.

From (2) it follows that

= 0.001 kg/mol. (3)

Substituting (3) into (1), we have

= 0,001
kg/mol

=
g/mol.

Thus, the mass of a mole, expressed in grams, is numerically equal to the relative molecular mass.

= 12 amu
= 12 g/mol,

= 16 amu
= 16 g/mol,

= 32 a.m.u.

= 32 g/mol.

4. Properties of an ideal gas.

The dimensions of the molecules are about 1 A = 10 -10 m.

The pressure is equal to the force acting perpendicular to a unit area,
. Pressure in SI is measured in Pa (pascals). Pa = n/m2, 1 kg/cm2 = 1 atm = 9.8 10 4 Pa, 1 mm Hg. = 133 Pa.

5. Mendeleev-Clapeyron equation.

At low densities, gases obey the equation

Mendeleev-Clapeyron equation of state for an ideal gas, - number of moles, = 8.31 J/mol K. The equation can be given a different form if you enter the quantities

= 1.38 10 -23 J/K:

.

If
is the concentration of particles, then

.

If
, That

.

This expression is used in aerodynamics.

6. Basic equation of the kinetic theory of gases (Clausius equation).

The basic equation of molecular kinetic theory connects the parameters of the state of a gas with the characteristics of the movement of molecules.

To derive the equation, a statistical method is used, that is, knowing the characteristics of individual gas molecules
(concentration) can be found - gas pressure, characteristics of the entire gas.

To derive the equation, consider a monatomic ideal gas. Molecules move chaotically. The speeds of molecules are different. Let us assume that the number of mutual collisions between gas molecules is negligible compared to the number of impacts on the walls of the vessel; collisions of molecules with the walls of the vessel are absolutely elastic. Let us find the pressure on the walls of the vessel, assuming that the gas is in a cubic vessel with an edge . We look for pressure as the average result of impacts of gas molecules on the walls of the vessel.

1). According to Newton's third law, the wall receives momentum from each molecule

2). During the time
sites
reach only those molecules that are contained in the volume

3). The number of these molecules in volume
equals

.

4). The number of impacts on the platform is equal to
.

5). When molecules collide, they transfer momentum to the area

Considering that
- strength, and
- pressure,

we have for pressure

(1)

If the gas volume contains
molecules that move at speeds
, then we need to introduce the concept of root mean square speed using the formula

. (2)

Then expression (1) will take the form

=

Basic equation of the kinetic theory of gases.

This equation can be rearranged by noting that

.

.

On the other side

.

.

The average kinetic energy of the chaotic movement of molecules is directly proportional to temperature and does not depend on mass. At T=0
= 0, the movement of gas molecules stops and the pressure is zero.

Absolute temperature, T is a measure of average kinetic energy translational motion of ideal gas molecules. But this is true only at moderate temperatures, as long as there is no decay or ionization of molecules and atoms. If the number of particles in the system is small, then this is also incorrect, since it is impossible to introduce the concept of mean square velocity.

From
And
should

=.

DEFINITION: An ideal gas is a gas whose properties satisfy the following conditions:
a) collisions of molecules of such a gas occur as collisions of elastic balls, the dimensions of which are negligible;
b) from collision to collision, the molecules move uniformly and rectilinearly;
c) the forces of interaction between molecules are neglected.

Real gases at room temperature And normal pressure behave like ideal gases. Ideal gases can be considered gases such as helium and hydrogen, the properties of which even under ordinary conditions correspond to the laws of an ideal gas.

The state of a certain mass of ideal gas will be determined by the values ​​of three parameters: P, V, T. These values, characterizing the state of the gas, are called state parameters. These parameters are naturally related to each other, so a change in one of them entails a change in the other. This relationship can be analytically specified as a function:

A relationship that gives a connection between the parameters of a body is called equation of state. Therefore, this relationship is the equation of state of an ideal gas.

Let's consider some of the state parameters characterizing the state of the gas:

1) Pressure(P). In a gas, pressure arises as a result of the chaotic movement of molecules, as a result of which the molecules collide with each other and with the walls of the container. As a result of the impact of molecules on the wall of the vessel, a certain average force will act on the wall from the side of the molecules dF. Let us assume that the surface area dS, Then . Hence:

DEFINITION (mechanistic): Pressure- This physical quantity, numerically equal to strength, acting per unit area of ​​the surface normal to it.

If the force is uniformly distributed over the surface, then . In the SI system, pressure is measured in 1Pa=1N/m2.

2) Temperature(T).

DEFINITION (provisional): Temperature body is a thermodynamic quantity that characterizes the state of thermodynamic equilibrium of a macroscopic system.

The temperature is the same for all parts of an isolated system in a state of thermodynamic equilibrium. That is, if the contacting bodies are in a state thermal equilibrium, i.e. do not exchange energy through heat transfer, then these bodies are assigned the same temperature. If, when thermal contact is established between bodies, one of them transfers energy to the other through heat transfer, then the first body is assigned high temperature than the second.

Any of the body properties (temperature signature) that depends on temperature can be used to quantify (measure) temperature.

For example: if we choose volume as a temperature indicator and assume that volume changes linearly with temperature, then choosing the melting temperature of ice as “0”, and the boiling temperature of water as 100°, we obtain a temperature scale called the Celsius scale. According to which the state in which a thermodynamic body has a volume V should be assigned a temperature:

To unambiguously determine the temperature scale, it is necessary to agree, in addition to the calibration method, also on the choice of a thermometric body (i.e., the body that is selected for measurement) and the temperature characteristic.

Known two temperature scales:

1) t– empirical or practical temperature scale (°C). (We will talk about the choice of a thermometric body and a temperature characteristic for this scale later).

2) T– thermodynamic or absolute scale (°K). This scale does not depend on the properties of the thermodynamic body (but this will be discussed later).

Temperature T, measured on an absolute scale, is related to temperature t on a practical scale by the relation

T = t + 273,15.

The unit of absolute temperature is called Kelvin. Temperature on a practical scale is measured in degrees. Celsius (°C). Deg values. Kelvin and deg. Celsius are the same. A temperature equal to 0°K is called absolute zero, it corresponds to t=-273.15°C

DEFINITION

Ideal gas- this is the most simple model system consisting of large quantity particles.

It is a gas that consists of material points that have a finite mass but no volume. These particles cannot interact at a distance. Collisions of ideal gas particles are described using the laws of absolutely elastic collision of spheres. It should be noted that this refers to the laws of collisions of balls, since point particles experience only head-on collisions, which cannot change the direction of velocities at different angles.

An ideal gas exists only in theory. IN real life it cannot exist in principle, since point molecules and the absence of their interaction at a distance is analogous to their existence outside space, that is, their non-existence. The closest in their properties to the ideal gas model are gases at low pressure (rarefied gases) and (or) high temperature. The ideal gas model is suitable for studying methods for studying multiparticle systems and becoming familiar with relevant concepts.

In the intervals between collisions, the molecules of an ideal gas move in straight lines. The laws of collisions and impacts on the walls of vessels containing gas are known. Consequently, if you know the positions and velocities of all particles of an ideal gas at some point in time, then you can find their coordinates and velocities at any other point in time. This information most fully describes the state of the particle system. However, the number of particles is so large that the dynamic description of a system of many particles is unsuitable for theory and useless for practice. This means that to study systems of many particles, information must be generalized, and it is attributed not to individual particles, but to large aggregates of them.

Ideal gas pressure

Using the ideal gas model, it was possible to qualitatively and quantitatively explain the pressure of a gas on the walls of the vessel in which it is located. The gas exerts pressure on the walls of the vessel because its molecules interact with the walls as elastic bodies according to the laws of classical mechanics. Quantitatively, the pressure (p) of an ideal gas is equal to:

where is the average kinetic energy of the translational motion of gas molecules; - concentration of gas molecules (N - number of gas molecules in the vessel; V - volume of the vessel).

Ideal gas laws

Gases that strictly obey the Boyle-Mariotte and Gay-Lussac laws are called ideal.

Boyle's Law - Mariotte. For a constant mass (m) of an ideal gas at a constant temperature (T), the product of the pressure (p) of the gas and its volume (V) is a constant value for any state of the substance in question:

Gay-Lussac's law. For a constant mass of gas at a constant pressure, the following relation holds:

In the behavior of real gases, deviations from the Boyle-Mariotte and Gay-Lussac laws are observed, and these deviations are different for different gases.

For an ideal gas, Charles's law holds. Which says that for a constant mass of gas, at a constant volume, the ratio of gas pressure to temperature does not change:

To relate the parameters of an ideal gas, the equation of state is often used, which bears the names of two scientists Clapeyron and Mendeleev:

Where - molar mass gas; - universal gas constant.

Dalton's law. Mixture pressure ideal gases(p) is equal to the sum of the partial pressures () of the gases under consideration:

In this case, the equation of state for a mixture of ideal gases has the form (2), as if the gas were chemically homogeneous.

Examples of problem solving

EXAMPLE 1

Exercise	What processes in a constant mass of an ideal gas are represented by the graphs (Fig. 1)?
Solution	Let's consider the process depicted in graph number 1. We see that the product, according to the condition, the gas is ideal, the mass of the gas is constant, therefore, this is an isothermal process. Let's move on to the second graph. From the graph we can conclude that: where C is some constant value. Dividing the right and left sides of expression (1.1) we have: We got that the pressure is constant. Since , we have an isobaric process.
Answer	1- isothermal process. 2-isobaric process.

EXAMPLE 2

Exercise	How will the pressure of an ideal gas change in a process in which the mass of the gas is constant, the volume of the gas is increased, and the temperature is decreased?
Solution	We will take the Clapeyron-Mendeleev equation as the basis for solving the problem:

As is known, many substances in nature can be in three states of aggregation: solid, liquid And gaseous.

The doctrine of the properties of matter in various states of aggregation is based on ideas about the atomic-molecular structure of the material world. The molecular kinetic theory of the structure of matter (MKT) is based on three main principles:

• all substances are made up of tiny particles(molecules, atoms, elementary particles), between which there are gaps;
• particles are in continuous thermal motion;
• there are interaction forces between particles of matter (attraction and repulsion); the nature of these forces is electromagnetic.

Means, physical state substance depends on relative position molecules, the distance between them, the forces of interaction between them and the nature of their movement.

The interaction between particles of a substance is most pronounced in the solid state. The distance between molecules is approximately equal to their own sizes. This leads to a fairly strong interaction, which practically makes it impossible for the particles to move: they oscillate around a certain equilibrium position. They retain their shape and volume.

The properties of liquids are also explained by their structure. Particles of matter in liquids interact less intensely than in solids, and therefore can change their location abruptly - liquids do not retain their shape - they are fluid. Liquids retain volume.

A gas is a collection of molecules moving randomly in all directions independently of each other. Gases do not have their own shape, occupy the entire volume provided to them and are easily compressed.

There is another state of matter - plasma. Plasma is a partially or fully ionized gas in which the densities of positive and negative charges are almost equal. When heated strongly enough, any substance evaporates, turning into a gas. If you increase the temperature further, the process of thermal ionization will sharply intensify, i.e., gas molecules will begin to disintegrate into their constituent atoms, which then turn into ions.

Ideal gas model. Relationship between pressure and average kinetic energy.

To clarify the laws that govern the behavior of a substance in the gaseous state, an idealized model of real gases is considered - an ideal gas. This is a gas whose molecules are considered as material points, not interacting with each other at a distance, but interacting with each other and with the walls of the vessel during collisions.

Ideal gas – It is a gas in which the interaction between its molecules is negligible. (Ek>>Er)

An ideal gas is a model invented by scientists to understand the gases that we actually observe in nature. It cannot describe any gas. Not applicable when the gas is highly compressed, when the gas turns into a liquid state. Real gases behave like ideal gases when the average distance between molecules is many times larger than their sizes, i.e. at sufficiently large vacuums.

Properties of an ideal gas:

1. there is a lot of distance between molecules more sizes molecules;
2. gas molecules are very small and are elastic balls;
3. the forces of attraction tend to zero;
4. interactions between gas molecules occur only during collisions, and collisions are considered absolutely elastic;
5. the molecules of this gas move randomly;
6. movement of molecules according to Newton's laws.

State of some mass gaseous substance characterized by physical quantities dependent on each other, called state parameters. These include volumeV, pressurepand temperatureT.

Gas volume denoted by V. Volume gas always coincides with the volume of the container it occupies. SI unit of volume m 3.

Pressure– physical quantity equal to the ratio of forceF, acting on a surface element perpendicular to it, to the areaSthis element.

p = F/ S SI unit of pressure pascal[Pa]

Until now, non-systemic units of pressure are used:

technical atmosphere 1 at = 9.81-104 Pa;

physical atmosphere 1 atm = 1.013-105 Pa;

millimeters of mercury 1 mmHg Art. = 133 Pa;

1 atm = = 760 mm Hg. Art. = 1013 hPa.

How does gas pressure arise? Each gas molecule, hitting the wall of the vessel in which it is located, acts on the wall with a certain force for a short period of time. As a result of random impacts on the wall, the force exerted by all molecules per unit area of ​​the wall changes rapidly with time relative to a certain (average) value.

Gas pressureoccurs as a result of random impacts of molecules on the walls of the vessel containing the gas.

Using the ideal gas model, we can calculate gas pressure on the vessel wall.

During the interaction of a molecule with the wall of a vessel, forces arise between them that obey Newton’s third law. As a result, the projection υ x the molecular speed perpendicular to the wall changes its sign to the opposite, and the projection υ y the speed parallel to the wall remains unchanged.

Devices that measure pressure are called pressure gauges. Pressure gauges record the time-average pressure force per unit area of ​​its sensitive element (membrane) or other pressure receiver.

Liquid pressure gauges:

1. open – for measuring small pressures above atmospheric
2. closed - for measuring small pressures below atmospheric, i.e. small vacuum

Metal pressure gauge– for measuring high pressures.

Its main part is a curved tube A, the open end of which is soldered to tube B, through which gas flows, and the closed end is connected to the arrow. Gas enters through the tap and tube B into tube A and unbends it. The free end of the tube, moving, sets the transmission mechanism and the pointer in motion. The scale is graduated in pressure units.

Basic equation of the molecular kinetic theory of an ideal gas.

Basic MKT equation: the pressure of an ideal gas is proportional to the product of the mass of the molecule, the concentration of the molecules and the mean square of the speed of the molecules

p= 1/3m 0· n·v 2

m 0 - mass of one gas molecule;

n = N/V – number of molecules per unit volume, or concentration of molecules;

v 2 - root mean square speed of movement of molecules.

Since the average kinetic energy of translational motion of molecules is E = m 0 *v 2 /2, then multiplying the basic MKT equation by 2, we obtain p = 2/3 n (m 0 v 2)/2 = 2/3 E n

p = 2/3 E n

Gas pressure is equal to 2/3 of the average kinetic energy of translational motion of the molecules contained in a unit volume of gas.

Since m 0 n = m 0 N/V = m/V = ρ, where ρ is the gas density, we have p= 1/3· ρ·v 2

United gas law.

Macroscopic quantities that unambiguously characterize the state of a gas are calledthermodynamic parameters of gas.

The most important thermodynamic parameters of a gas are itsvolumeV, pressure p and temperature T.

Any change in the state of a gas is calledthermodynamic process.

In any thermodynamic process, the gas parameters that determine its state change.

The relationship between the values ​​of certain parameters at the beginning and end of the process is calledgas law.

The gas law expressing the relationship between all three gas parameters is calledunited gas law.

p = nkT

Ratio p = nkT relating the pressure of a gas to its temperature and concentration of molecules was obtained for a model of an ideal gas, the molecules of which interact with each other and with the walls of the vessel only during elastic collisions. This relationship can be written in another form, establishing a connection between the macroscopic parameters of a gas - volume V, pressure p, temperature T and the amount of substance ν. To do this you need to use the equalities

where n is the concentration of molecules, N is total number molecules, V – volume of gas

Then we get or

Since at a constant gas mass N remains unchanged, then Nk – constant number, Means

For a constant gas mass, the product of volume and pressure divided by absolute temperature gas, there is a value that is the same for all states of this mass of gas.

The equation establishing the relationship between pressure, volume and temperature of a gas was obtained in the middle of the 19th century by the French physicist B. Clapeyron and is often called Clayperon equation.

The Clayperon equation can be written in another form.

p = nkT,

considering that

Here N– number of molecules in the vessel, ν – amount of substance, N A is Avogadro’s constant, m– mass of gas in the vessel, M– molar mass of gas. As a result we get:

Product of Avogadro's constant N A byBoltzmann constantk is called universal (molar) gas constant and is designated by the letter R.

Its numerical value in SI R= 8.31 J/mol K

Ratio

called ideal gas equation of state.

In the form we received, it was first written down by D.I. Mendeleev. Therefore, the equation of state of the gas is called Clapeyron–Mendeleev equation.`

For one mole of any gas this relationship takes the form: pV=RT

Let's install physical meaning of the molar gas constant. Suppose that in a certain cylinder under the piston at temperature E there is 1 mole of gas, the volume of which is V. If the gas is heated isobarically (at constant pressure) by 1 K, then the piston will rise to a height Δh, and the volume of the gas will increase by ΔV.

Let's write the equation pV=RT for heated gas: p (V + ΔV) = R (T + 1)

and subtract from this equality the equation pV=RT, corresponding to the state of the gas before heating. We get pΔV = R

ΔV = SΔh, where S is the area of ​​the base of the cylinder. Let's substitute into the resulting equation:

pS = F – pressure force.

We obtain FΔh = R, and the product of the force and the movement of the piston FΔh = A is the work of moving the piston performed by this force against external forces during gas expansion.

Thus, R = A.

The universal (molar) gas constant is numerically equal to the work done by 1 mole of gas when it is heated isobarically by 1 K.

One of which is gas. Its constituent particles - molecules and atoms - are located at a great distance from each other. At the same time, they are in constant free movement. This property indicates that the interaction of particles occurs only at the moment of approach, sharply increasing the speed of the colliding molecules and their size. This distinguishes the gaseous state of a substance from solid and liquid.

The word “gas” itself translated from Greek sounds like “chaos”. This perfectly characterizes the movement of particles, which is actually random and chaotic. The gas does not form a specific surface; it fills the entire volume available to it. This state of matter is the most common in our Universe.

The laws that determine the properties and behavior of such a substance are easiest to formulate and consider using the example of a state in which molecules and atoms are low. It was called the “ideal gas”. In it, the distance between particles is greater than the radius of interaction of intermolecular forces.

So, an ideal gas is a theoretical model of matter in which there is almost no interaction between particles. The following conditions must exist for it:

Very small molecular sizes.

There is no interaction force between them.

Collisions occur like collisions of elastic balls.

A good example of such a state of matter is gases in which the pressure at low temperatures does not exceed atmospheric pressure by 100 times. They are considered discharged.

The very concept of “ideal gas” has enabled science to build a molecular kinetic theory, the conclusions of which are confirmed in many experiments. According to this doctrine, ideal gases are distinguished between classical and quantum.

The characteristics of the first are reflected in the laws of classical physics. The movement of particles in this gas does not depend on each other; the pressure exerted on the wall is equal to the sum of the impulses that, during a collision, are transmitted by individual molecules over a certain time. Their total energy is that of the individual particles. The work of an ideal gas in this case is calculated p = nkT. A striking example This is based on the laws derived by such physicists as Boyle-Marriott, Gay-Lussac, Charles.

If an ideal gas lowers its temperature or increases its particle density to a certain value, its wave properties increase. There is a transition to a quantum gas, in which atoms and molecules are comparable to the distance between them. There are two types of ideal gas:

The teaching of Bose and Einstein: particles of the same type have an integer spin.

Fermi and Dirac statistics: another type of molecules that have half-integer spin.

The difference between a classical ideal gas and a quantum one is that even at absolutely zero temperature the energy density and pressure differ from zero. They become larger as density increases. In this case, the particles have maximum (another name is boundary) energy. From this point of view, the theory of the structure of stars is considered: in those of them in which the density is higher than 1-10 kg/cm3, the law of electrons is clearly expressed. And where it exceeds 109 kg/cm3, the substance turns into neurons.

In metals, the use of the theory in which a classical ideal gas transforms into a quantum one makes it possible to explain most of the state of the substance: the denser the particles, the closer it is to the ideal.

With strongly expressed low temperatures various substances in liquid and solid states the collective motion of molecules can be considered as the work of an ideal gas represented by weak excitations. In such cases, the contribution to the energy of the body that the particles add is visible.