Ideal gas definition and formula. Examples of problem solving

The science of physics plays a significant role in the study of the world around us. Therefore, its concepts and laws begin to pass even at school. The properties of a substance are measured in different aspects. If we consider its state of aggregation, then there is a special technique. Ideal gas is a physical concept that allows us to evaluate the properties and characteristics of the material that makes up our entire world.

General definition

An ideal gas is a model in which the interaction between molecules is usually neglected. The process of interaction of particles of any substance with each other is quite complex.

When they fly close to each other and are at a very small distance, they strongly repel each other. But at a great distance, relatively small forces of attraction act between molecules. If the average distance at which they are from each other is large, this position of the substance is called a rarefied gas. The interaction of such particles manifests itself as rare impacts of molecules. This happens only when they fly close to each other. In an ideal gas, the interaction of molecules is not taken into account at all. An ideal gas has a very large number of molecules. Therefore, calculations occur only using the statistical method. Moreover, it should be noted that the particles of matter in this case are distributed evenly in space. This is the most common state of an ideal gas.

When can a gas be considered ideal?

There are several factors due to which a gas is called ideal. The first sign is the behavior of molecules as absolutely elastic bodies, there are no forces of attraction between them. In this case, the gas will be very discharged. The distance between the smallest constituents of matter will be much more sizes themselves. In this case, thermal equilibrium will be achieved instantly throughout the volume. In order to reach the position of an ideal gas in laboratory conditions, its real type is rarefied accordingly. Some substances in the gaseous state, even at room temperature and normal atmospheric pressure, practically do not differ from the ideal state.

Limits of application of the model

Real gas is considered depending on the tasks. If the researcher is tasked with determining the relationship between temperature, volume and pressure, then the ideal state of matter can be considered such a state of matter in which the gas exhibits high accuracy up to pressures measured in several tens of atmospheres. But in the case of studying a phase transition, for example, evaporation and condensation, the process of achieving equilibrium in a gas, the model under consideration cannot be applied even at very low pressure. The pressure of the gas on the wall of the test tube occurs when the molecules randomly hit the glass. When such shocks are frequent, the human body can pick up these changes as a continuous impact.

Ideal gas equation

Based on the main principles of molecular kinetic theory, the main equation of an ideal gas was derived.

The work of an ideal gas has the following expression: p \u003d 1 / 3 m 0 nv 2, where p is the pressure of an ideal gas, m 0 is the molecular weight, v 2 is the average value of the particle concentration, the square of the velocity of the molecules. If we designate the average indicator of the kinetic motion of particles of matter as Ek = m 0 n/ 2, then the equation will look like this: p = 2/3 nEk. Gas molecules, hitting the walls of the vessel, interact with them as elastic bodies according to the laws of mechanics. The momentum from such impacts is transferred to the walls of the vessel.

Temperature

Calculating only the pressure of the gas on the walls of the vessel, it is impossible to determine the average kinetic energy of its particles.

Moreover, this cannot be done either for an individual molecule or for their concentration. Therefore, to measure gas parameters, it is necessary to determine one more quantity. It is temperature, which is also related to the kinetic energy of molecules. Such an indicator is a scalar physical quantity. Temperature describes thermodynamic equilibrium. In this state, there is no change in parameters at the micro level. Temperature is measured as a deviation from zero. It characterizes the saturation of chaotic motion smallest particles gas. It is measured by the average value of their kinetic energy. This indicator is determined using thermometers in degrees of various marks. There is a thermodynamic absolute scale (Kelvin) and its empirical varieties. They differ in starting points.

The equation of position of an ideal gas, taking into account the temperature

The physicist Boltzmann states that the average kinetic energy of a particle is proportional to absolute indicator temperature. Ek \u003d 3 / 2 kT, where k \u003d 1.38 ∙ 10-23, T is temperature. The work of an ideal gas will be: Р = NkT/V, where N is the number of molecules, V is the volume of the vessel. If we add the concentration n = N/V to this indicator, then the above formula will look like this: p = nkT. These two equations are various forms records, but they relate pressure, volume, and temperature for an ideal gas. These calculations can be applied both to pure gases and their mixtures. In the latter version, n should be understood as the total number of molecules of substances, their total concentration, or the total number of moles in a substance.

Three gas laws

An ideal gas and its particular laws were discovered experimentally and only then confirmed theoretically.

The first particular law states that an ideal gas at constant mass and temperature will have an inversely proportional pressure to its volume. A process in which the temperature is constant is called isothermal. If the pressure is constant during the study, then the volume is proportional to the value of the absolute temperature. This law bears the name of Gay-Lussac. The isochoric process occurs at a constant volume. In this case, the pressure will be proportional to the absolute temperature. Its name is Charles' law. These are three particular laws of ideal gas behavior. They were confirmed only by mastering the knowledge of molecules.

Absolute measurement scale

AT absolute scale The unit of measurement is called Kelvin. It was chosen based on the popular Celsius scale. One Kelvin corresponds to one degree Celsius. But in the absolute scale, zero is taken as the value at which the pressure of an ideal gas at constant volume will be equal to zero.

This is the generally accepted system. This temperature value is called absolute zero. Having made the appropriate calculations, you can get the answer that the value of this indicator will be -273 degrees Celsius. This confirms that there is a relationship between the absolute and the Celsius scale. It can be expressed in the following equation: T = t + 237. It should be noted that it is impossible to reach absolute zero. Any cooling process is based on the evaporation of molecules from the surface of matter. Approaching absolute zero, the translational motion of particles slows down so much that evaporation stops almost completely. But from a purely theoretical point of view, if it were realistic to reach the point of absolute zero, then the speed of movement of molecules would decrease so much that it could be called absent altogether. The thermal motion of molecules would stop.

Having studied such a concept as an ideal gas, one can understand the principle of operation of any substance. By expanding knowledge in this area, one can understand the properties and behavior of any gaseous substance.

The main object of the molecular-kinetic theory of gases is the so-called "ideal gas". An ideal gas is a rarefied medium among many (very a large number) particles that do not interact with each other except through rare collisions. Each of the particles of the medium moves randomly and independently of the others. Each particle has the usual classical mechanics a set of physical parameters, such as: mass and speed. As well as derivatives of these quantities - energy and momentum. The particle sizes are considered to be negligibly small in relation to the other characteristic sizes of the considered physical system. More precisely, an ideal gas is characterized by the following properties, which follow directly from this definition:

• Since the particles practically do not interact with each other, their potential energy is negligibly small compared to their kinetic energy. This also applies to fundamental forces, like the forces of gravity, which are not included in the consideration.
• Particle collisions are assumed to be elastic, i.e. the same as collisions of absolutely rigid spheres, like billiard balls. When colliding with each other, the particles do not "stick" to each other. And this means that the time interval occupied by the collision process can be neglected.
• An ideal gas is considered together with a certain volume it occupies. The total volume of particles is assumed to be negligible compared to the volume they occupy.

Bottom line: we are talking about a very rarefied medium without resistance and any other external interactions, consisting of elastic particles of negligible size (molecules, atoms).

Macroscopic characteristics of an ideal gas

An ideal gas in a vessel, considered as a whole (that is, as a macroscopic object), has a certain set of macroscopic characteristics that do not depend on the behavior of its individual particles. These characteristics are derivatives of the average values ​​of the energies of individual particles of an ideal gas. These indicators include temperature and pressure ideal gas.

• Temperature ideal gas - is a measure of the average kinetic energy of the molecules of an ideal gas.
• Pressure ideal gas - is a measure of the average kinetic energy of impacts on a small, absolutely elastic area placed in a gas.

Already from the definition of temperature and pressure it should be clear that these parameters depend on each other. Indeed, if the walls of the vessel are allowed to expand freely, then the law of proportionality takes place: p~T, where p is pressure and T is temperature.

Laws of ideal gas behavior

Depending on the conditions imposed on the volume of the vessel, the pressure value or the temperature value, one can obtain various particular patterns of behavior of an ideal gas:

• Boyle-Mariotte law(temperature is assumed to be constant).
• Gay-Lussac's law(pressure is assumed to be constant).
• Charles' law(constant volume).

There are other ratios as well. The corresponding formulas can be seen in the picture below:

Annotation: traditional presentation of the topic, supplemented by a demonstration on a computer model.

Of the three aggregate states of matter, the simplest is the gaseous state. In gases, the forces acting between molecules are small and under certain conditions they can be neglected.

The gas is called perfect , if:

The size of molecules can be neglected, i.e. molecules can be counted material points;

We can neglect the forces of interaction between molecules (the potential energy of interaction of molecules is much less than their kinetic energy);

The collisions of molecules with each other and with the walls of the vessel can be considered absolutely elastic.

Real gases are close in properties to the ideal at:

Conditions close to normal conditions (t = 0 0 C, p = 1.03 10 5 Pa);

At high temperatures.

The laws that govern the behavior of ideal gases were discovered experimentally quite a long time ago. So, Boyle's law - Mariotte was established in the 17th century. We give the formulations of these laws.

Boyle's Law - Mariotte. Let the gas be under conditions where its temperature is kept constant (such conditions are called isothermal ). Then for a given mass of gas, the product of pressure and volume is a constant value:

This formula is called isotherm equation. Graphically, the dependence of p on V for different temperatures shown in the figure.

The property of a body to change pressure with a change in volume is called compressibility. If the change in volume occurs at T=const, then the compressibility is characterized by isothermal compressibility coefficient which is defined as the relative change in volume that causes a change in pressure per unit.

For an ideal gas, it is easy to calculate its value. From the isotherm equation we get:

The minus sign indicates that as the volume increases, the pressure decreases. Thus, the isothermal compressibility of an ideal gas is equal to the reciprocal of its pressure. With increasing pressure, it decreases, because. the greater the pressure, the less the gas has the ability to further compress.

Gay-Lussac law. Let the gas be under conditions where its pressure is maintained constant (such conditions are called isobaric ). They can be carried out by placing gas in a cylinder closed by a movable piston. Then a change in the temperature of the gas will move the piston and change the volume. The pressure of the gas will remain constant. In this case, for a given mass of gas, its volume will be proportional to the temperature:

where V 0 - volume at temperature t = 0 0 C, - volume expansion coefficient gases. It can be represented in a form similar to the compressibility factor:

Graphically, the dependence of V on T for various pressures is shown in the figure.

Moving from temperature in the Celsius scale to absolute temperature, Gay-Lussac's law can be written as:

Charles' law. If the gas is under conditions where its volume remains constant ( isochoric conditions), then for a given mass of gas, the pressure will be proportional to the temperature:

where p 0 - pressure at temperature t \u003d 0 0 C, - pressure coefficient. It shows the relative increase in gas pressure when it is heated by 10:

Charles' law can also be written as:

Avogadro's law: One mole of any ideal gas at the same temperature and pressure occupies the same volume. At normal conditions(t \u003d 0 0 C, p \u003d 1.03 10 5 Pa) this volume is equal to m -3 / mol.

The number of particles contained in 1 mole of various substances, called. Avogadro's constant :

It is easy to calculate the number n 0 particles in 1 m 3 under normal conditions:

This number is called Loschmidt number.

Dalton's law: the pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the gases included in it, i.e.

where - partial pressures- the pressure that the components of the mixture would exert if each of them occupied a volume equal to the volume of the mixture at the same temperature.

Equation of Clapeyron - Mendeleev. From the laws of an ideal gas, one can obtain equation of state , linking T, p and V of an ideal gas in a state of equilibrium. This equation was first obtained by the French physicist and engineer B. Clapeyron and Russian scientists D.I. Mendeleev, therefore bears their name.

Let some mass of gas occupies volume V 1 , has pressure p 1 and is at temperature T 1 . The same mass of gas in a different state is characterized by the parameters V 2 , p 2 , T 2 (see figure). The transition from state 1 to state 2 is carried out in the form of two processes: isothermal (1 - 1") and isochoric (1" - 2).

For these processes, one can write down the laws of Boyle - Mariotte and Gay - Lussac:

Eliminating p 1 " from the equations, we get

Since states 1 and 2 were chosen arbitrarily, the last equation can be written as:

This equation is called Clapeyron's equation , in which B is a constant, different for different masses of gases.

Mendeleev combined Clapeyron's equation with Avogadro's law. According to Avogadro's law, 1 mole of any ideal gas at the same p and T occupies the same volume V m, so the constant B will be the same for all gases. This common constant for all gases is denoted R and is called universal gas constant. Then

This equation is ideal gas equation of state , which is also called Clapeyron - Mendeleev equation .

The numerical value of the universal gas constant can be determined by substituting the values ​​of p, T and V m into the Clapeyron - Mendeleev equation under normal conditions:

The Clapeyron - Mendeleev equation can be written for any mass of gas. To do this, recall that the volume of a gas of mass m is related to the volume of one mole by the formula V \u003d (m / M) V m, where M is molar mass of gas. Then the Clapeyron - Mendeleev equation for a gas of mass m will look like:

where is the number of moles.

The equation of state for an ideal gas is often written in terms of Boltzmann's constant :

Based on this, the equation of state can be represented as

where is the concentration of molecules. From the last equation it can be seen that the pressure of an ideal gas is directly proportional to its temperature and concentration of molecules.

Small demo ideal gas laws. After pressing the button "Let's start" You will see the host's comments on what is happening on the screen (black color) and a description of the computer's actions after you press the button "Further" (Brown color). When the computer is "busy" (i.e., experience is in progress), this button is not active. Move on to the next frame only after understanding the result obtained in the current experiment. (If your perception does not match the host's comments, write!)

You can verify the validity of the ideal gas laws on the existing

Satisfying the following conditions:

1) the own volume of gas molecules is negligible compared to the volume of the vessel;

2) there are no interaction forces between gas molecules;

3) collisions of gas molecules with each other and with the walls of the vessel are absolutely elastic.

2. What parameters characterize the state of the gas? Give a molecular-kinetic interpretation of the parameters p, T.

The state of a given mass of gas m is characterized by the following parameters: pressure p, volume V, temperature T.

3. Write down the formula that relates the temperatures on the Kelvin scale and on the Celsius scale? What is the physical meaning of absolute zero?

The relationship between the thermodynamic temperature T and the temperature on the centigrade Celsius scale is T = t + 273.15. At absolute zero the energy of the molecules is zero.

4. Write down the equation of state for an ideal gas.

The equation of state of an ideal gas (sometimes the Clapeyron equation or the Clapeyron-Mendeleev equation) is a formula that establishes the relationship between pressure, molar volume and absolute temperature of an ideal gas. The equation looks like: , where p - pressure, Vμ - molar volume, T - absolute temperature, R - universal gas constant.

5. What process is called isothermal? Write down and formulate the Boyle-Mariotte law and draw a graph of pressure versus volume.

D For a given mass of gas at a constant temperature, the product of the gas pressure and its volume is a constant value, at . A process that takes place at a constant temperature is called isothermal.

6. What process is called isochoric? Write down and formulate Charles' law. Draw a graph of pressure versus temperature.

D The pressure of a given mass of gas at a constant volume varies linearly with temperature , at .

A process that takes place at a constant volume is called isochoric.

7. What process is called isobaric? Write down and formulate Gay-Lussac's law. Draw a plot of volume versus temperature.

O The volume of a given mass of gas at constant pressure varies linearly with temperature: , at . A process that takes place at constant pressure is called isobaric.

8. What process is called adiabatic? Write the Poisson equation and represent it graphically. (see Appendix No. 2)

BUT A diabatic process is one that does not exchange heat with environment, Consequently .

Work during adiabatic expansion is carried out due to the loss of internal energy.

Poisson's equation, where is the adiabatic exponent.

9. Write down and formulate the first law of thermodynamics. Give the concept of internal energy, work, amount of heat.

The amount of heat received by the system goes to changing its internal energy and doing work against external forces.

The change in the internal energy of the system during its transition from one state to another is equal to the sum of the work of external forces and the amount of heat transferred to the system and does not depend on the method by which this transition is carried out.

10. Write down the expression for the work of gas expansion. How to represent it graphically on the pV diagram.

11. Apply the first law of thermodynamics to all processes considered in this laboratory work and analyze its implications.
12. Define specific and molar heat capacities and write down the relationship between them.

The specific heat capacity of a substance is a value equal to the amount of heat required to heat 1 kg of a substance by 1 K.

C=cm.
13. Derive Mayer's equation. Which of the heat capacities C P or C V is greater and why?

Relationship between molar and heat capacities (Mayer's equations).

Relationship between specific heat capacities

The number of degrees of freedom in mechanics is the number of possible displacements of a mechanical system that are independent of each other. The number of degrees of freedom depends on the number of material particles that form the system, and the number and nature of the mechanical bonds imposed on the system. For a free particle, the number of degrees of freedom is 3, for a free solid body- 6, for a body with a fixed axis of rotation , the number of degrees of freedom is 1, etc. For any holonomic system (systems with geometric constraints), the number of degrees of freedom is equal to the number s of coordinates independent of each other that determine the position of the system, and is given by the equality 5 = 3n - k, where n

16. Draw and explain on the pV diagram successively all the processes that occur with gas.

17. What is the reason for the change in air temperature in the cylinder when air is pumped into the cylinder and when it is released from the cylinder?

18. Derive the calculation formula for determining the ratio of heat capacities γ.