If, as a result of heat exchange, a certain amount of heat is transferred to the body, then the internal energy of the body and its temperature change. Quantity of heat Q required to heat 1 kg of a substance by 1 K is called specific heat of a substance c.

where M Is the molar mass of the substance.

The heat capacity determined in this way is not unambiguous characteristic of the substance. According to the first law of thermodynamics, the change in the internal energy of a body depends not only on the amount of heat received, but also on the work done by the body. Depending on the conditions under which the heat transfer process was carried out, the body could perform various jobs... Therefore, the same amount of heat transferred to the body could cause various changes in its internal energy and, consequently, temperature.

This ambiguity in determining the heat capacity is characteristic only for a gaseous substance. When liquid and solid bodies are heated, their volume practically does not change, and the work of expansion turns out to be zero. Therefore, all the amount of heat received by the body is spent on changing its internal energy. Unlike liquids and solids, gas in the process of heat transfer can greatly change its volume and perform work. Therefore, the heat capacity of a gaseous substance depends on the nature of the thermodynamic process. Two values ​​of the heat capacity of gases are usually considered: CV – molar heat capacity in the isochoric process (V= const) and Cp – molar heat capacity in isobaric process (p= const).

In the process, at a constant volume, the gas does not perform work: A= 0. From the first law of thermodynamics for 1 mole of gas it follows

where Δ V- change in the volume of 1 mole of ideal gas when its temperature changes by Δ T... This implies:

where R Is a universal gas constant. At p= const

Molar heat capacity Cp gas in a process with constant pressure is always greater than the molar heat capacity CV in a constant volume process (Figure 3.10.1).

In particular, this ratio is included in the formula for the adiabatic process.

Between two isotherms with temperatures T 1 and T 2 in the diagram ( p, V) different transition paths are possible. Since for all such transitions, the temperature change Δ T = T 2 – T 1 is the same, therefore, the same is the change in Δ U internal energy. However, the work performed in this case A and the amount of heat obtained as a result of heat exchange Q will be different for different transition paths. It follows from this that the gas has an infinite number of heat capacities. Cp and CV- these are only particular (and very important for the theory of gases) values ​​of heat capacities.

Thermodynamic processes in which the heat capacity of a gas remains unchanged are called polytropic ... All isoprocesses are polytropic. In the case of an isothermal process Δ T= 0, therefore CT= ∞. In the adiabatic process, Δ Q= 0, therefore C hell = 0.

It should be noted that "heat capacity" as well as "amount of heat" are extremely unfortunate terms. They went to modern science as a legacy from theory caloric , which reigned in the XVIII century. This theory considered heat as a special weightless substance contained in bodies. It was believed that it could neither be created nor destroyed. The heating of bodies was explained by an increase, and cooling - by a decrease in the caloric contained inside them. The caloric theory is untenable. She cannot explain why the same change in the internal energy of a body can be obtained by transferring to it a different amount of heat, depending on the work that the body does. Therefore, the statement that "this body contains such and such a reserve of heat" is devoid of physical meaning.

In the molecular kinetic theory, the following relationship is established between the average kinetic energy translational motion molecules and absolute temperature T:

When the temperature changes by Δ T internal energy changes by the amount

This relationship is well confirmed in experiments with gases consisting of monoatomic molecules (helium, neon, argon). However, for diatomic (hydrogen, nitrogen) and polyatomic (carbon dioxide) gases, this ratio does not agree with experimental data. The reason for this discrepancy is that for di- and polyatomic molecules, the average kinetic energy should include the energy of not only the translational, but also the rotational motion of the molecules.

In fig. 3.10.2 shows a model of a diatomic molecule. The molecule can make five independent movements: three translational movements along the axes X, Y, Z and two rotations about the axes X and Y... Experience shows that rotation about an axis Z, on which the centers of both atoms lie, can be excited only at very high temperatures... At ordinary temperatures, rotation about the axis Z does not happen, just as a monatomic molecule does not rotate. Each independent movement is called degree of freedom... Thus, a monatomic molecule has 3 translational degrees of freedom, a "rigid" diatomic molecule has 5 degrees (3 translational and 2 rotational), and a polyatomic molecule has 6 degrees of freedom (3 translational and 3 rotational).

In classical statistical physics, the so-called theorem about even distribution energy by degrees of freedom :

If a system of molecules is in thermal equilibrium at a temperature T, then the average kinetic energy is uniformly distributed between all degrees of freedom and for each degree of freedom of the molecule it is equal to

It follows from this theorem that the molar heat capacities of the gas Cp and CV and their ratio γ can be written as

For gas consisting of diatomic molecules (i = 5)

The experimentally measured specific heats of many gases under normal conditions are in good agreement with the above expressions. However, in general, the classical theory of the heat capacity of gases cannot be considered completely satisfactory. There are many examples of significant discrepancies between theory and experiment. This is due to the fact that the classical theory is not able to fully take into account the energy associated with internal motions in the molecule.

The theorem on the uniform distribution of energy over the degrees of freedom can be applied to the thermal motion of particles in a solid. The atoms that make up the crystal lattice vibrate around equilibrium positions. The energy of these vibrations is the internal energy of a solid. Each atom in the crystal lattice can vibrate in three mutually perpendicular directions. Therefore, each atom has 3 vibrational degrees of freedom. With harmonic vibrations, the average kinetic energy is equal to the average potential energy. Therefore, in accordance with the uniform distribution theorem, for each vibrational degree of freedom there is an average energy kT, and for one atom - 3 kT... The internal energy of 1 mole of solid is equal to:

This ratio is called Dulong-Petit law ... For solids, there is practically no difference between Cp and CV due to negligible expansion or contraction work.

Experience shows that for many solids (chemical elements) the molar heat capacity at ordinary temperatures is really close to 3 R... However, at low temperatures, there are significant discrepancies between theory and experiment. This shows that the hypothesis of a uniform energy distribution over the degrees of freedom is an approximation. The experimentally observed dependence of heat capacity on temperature can be explained only on the basis of quantum concepts.

Internal energy and heat capacity of an ideal gas Average energy of one molecule Since an ideal gas molecule does not interact at a distance, the internal energy of a gas is equal to the sum of the internal energies of all molecules For 1 mole, where N = NA Internal energy of an arbitrary mass m Internal energy of an ideal gas depends only on from temperature

Heat capacity Heat capacity of a body is a value equal to the amount of heat that must be imparted to the body in order to increase its temperature by 1 degree to heat this body by one degree: if m = 1 kg

Specific heat (s) - the amount of heat required to heat a unit mass of a substance by one degree. [c] = For gases, it is convenient to use the molar heat capacity Cμ the amount of heat required to heat 1 mole of gas by 1 degree: Cμ = c different molar masses μ)

The heat capacity of a thermodynamic system depends on how the state of the system changes when heated. Of greatest interest is the heat capacity for cases when heating occurs under the condition V = Const (c. V) p = Const (cp).

V = Const (c. V) If the gas is heated at a constant volume, then all the supplied heat goes to heating the gas, that is, a change in its internal energy. No work is done on other bodies. d. QV = d. U (d. A = 0) Because for 1 mole T. o. CV does not depend on temperature, but depends only on the number of degrees of freedom i are equal, that is, on the number of atoms in a gas molecule.

p = Const (cp) If the gas is heated at constant pressure (CP) in a vessel with a piston, then the supplied heat is spent both on heating the gas and on performing work. Therefore, to increase T by 1 K, more heat is needed than in the case of V = Const Therefore, СР> СV

We write down the I beginning of TD for 1 mole of gas, we divide by d. T CV From the basic equation of the MKT we have: p. Vμ = RT / p So. the work that 1 mole of an ideal gas does when the temperature rises by 1 K is equal to the gas constant R. The ratio Cp / Cv is a constant value for each gas

The number of degrees of freedom manifested in specific heat depends on temperature. Rice. the qualitative dependence of the molar heat capacity СV on temperature for argon (Ar) and hydrogen (H 2) The MCT results are correct for certain temperature ranges, and each range has its own number of degrees of freedom.

Application of the first law of thermodynamics to isoprocesses Isoprocess is a process that takes place at a constant value of one of the main thermodynamic parameters - P, V or T. 1) isochoric process, in which the volume of the system remains constant (V = const). 2) isobaric process, in which the pressure exerted by the system on the surrounding bodies remains constant (p = const). 3) isothermal process in which the temperature of the system remains constant (T = const). 4) an adiabatic process, in which there is no heat exchange with the environment throughout the entire process (d. Q = 0; Q = 0)

An isothermal process is a process that occurs in a physical system at a constant temperature (T = const). In an ideal gas in an isothermal process, the product of pressure and volume is constant - Boyle's Mariotte law: Let's find the work of a gas in an isothermal process:

Using the formula U = c. VT, we get d. U = c. V d. T = 0 Consequently, the internal energy of the gas does not change during the isothermal process. Therefore, this means that in the isothermal process, all the heat imparted to the gas goes to work on the external bodies. Therefore, so that when the gas expands, its temperature does not decrease, it is necessary to supply an amount of heat to the gas equal to its work on external bodies.

Isochoric process is a process that occurs in a physical system at a constant volume (V = const). - Charles's law In an isochoric process, mechanical work with a gas does not occur.

Isochoric process: V = const 1. From the equation of state of an ideal gas 2. for two temperatures T 1 and T 2 3. it follows 4. whence 5. In process 1 6. In process 1 2 gas is heated 3 gas is cooled

Let the initial state of the gas correspond to the state under normal conditions T 0 = 0 ° C = 273.15 ° K, p0 = 1 atm, then for an arbitrary temperature T the pressure in the isochoric process is found from the equation Gas pressure is proportional to its temperature - Charles's Law Since d. A = pd. V = 0, then in the isochoric process, the gas does not work on external bodies. In this case, the heat transferred to the gas is equal to d. Q = d. A + d. U = d. U That is, during the isochoric process, all the heat transferred to the gas goes to increase its internal energy.

Isobaric process is a process that occurs in a physical system at constant pressure (P = const). const is Gay's law. Lussac

2) Isobaric process: p = const In the isobaric process, the gas does work The work is equal to the area under the straight line of the isobar. From the equation of state for an ideal gas, we obtain

Let us rewrite the last relation in the form This equality reveals the physical meaning of the gas constant R - it is equal to the work of 1 mole of an ideal gas performed by it when heated by 1 ° K under conditions of isobaric expansion. Let us take as the initial state - the state of an ideal gas under normal conditions (T 0, V 0), then the volume of gas V at an arbitrary temperature T in the isobaric process is equal to The volume of gas at constant pressure is proportional to its temperature - the Gay-Lussac law.

An adiabatic process is a process that occurs in a physical system without heat exchange with the environment (Q = 0). Poisson's equation. γ is the adiabatic index.

4) Adiabatic process: d. Q = 0 In an adiabatic process, there is no heat exchange between the gas and the environment. From the first law of thermodynamics, we obtain d. A = - d. Therefore, in the adiabatic process, the work of the gas on external bodies is performed due to the loss of its internal energy. Using d. U = c. Vd. T; d. A = pd. V we find pd. V = - c. V d. T On the other hand, it follows from the equation of state for an ideal gas that d (p. V) = pd. V + Vdp = Rd. T

Excluding d. T, we get pd. V = - c. V (pd. V + vdp) / R Whence Integrating, we find

The last formula can be rewritten as Consequently, this equation of the adiabatic process is the Poisson equation. Since> 1, then the pressure of the adiabat changes from the volume faster than that of the isotherm.

Using the equation of state for an ideal gas, we transform the Poisson equation to the form Means or During adiabatic expansion, an ideal gas is cooled, and when it is compressed, it heats up.

A polytropic process is a process that occurs at a constant heat capacity, cm = const. where cm is the molar heat capacity. where n is the polytropic exponent.

On the other hand, from the equation of state of an ideal gas Therefore, we can write Since c. P = c. V + R then

Entropy Adiabatic processes in thermodynamic systems can be equilibrium and nonequilibrium. To characterize the equilibrium adiabatic process, one can introduce a certain physical quantity that would remain constant throughout the entire process; it was called entropy S. Entropy is such a function of the state of the system, the elementary change of which during the equilibrium transition of the system from one state to another is equal to the received or given amount of heat divided by the temperature at which this process took place for an infinitely small change in the state of the system

Change in entropy in isoprocesses If the system makes an equilibrium transition from state 1 to state 2, then the change in entropy: Let's find the change in entropy in ideal gas processes. Since then

Or The change in the entropy S 1 2 of an ideal gas during its transition from state 1 to state 2 does not depend on the path of transition 1 2. isochoric process: isobaric process: p 1 = p 2 isothermal process: T 1 = T 2 adiabatic process:

Consequently, S = const, the adiabatic process is called in another way - isentropic process. In all cases when the system receives heat from the outside, then Q is positive, therefore, S 2> S 1 and the entropy of the system increases. If the system gives up heat, then Q has a negative sign and, therefore, S 2

Isoprocesses can be depicted graphically in coordinate systems, along the axes of which state parameters are plotted. pressure p - volume V temperature Т - volume V temperature Т - pressure p V 1 V 2 With adiabatic expansion, external work is performed only due to the internal energy of the gas, as a result of which the internal energy, and with it the gas temperature, decrease (Т 2

Convenience of the coordinate system p, V On the scale of the drawing, the external work is depicted by the area bounded by the process curve 1 - 2 and the ordinates of the initial and final states

Circular (closed) processes The totality of thermodynamic processes, as a result of which the system returns to its original state, is called a circular process (cycle). Forward cycle - work per cycle Reverse cycle - work per cycle

Heat engine A cyclically acting device that converts heat into work is called a heat engine or heat engine. Q 1 is the heat received by the RT from the heater, Q 2 is the heat transferred by the RT to the refrigerator, A is the useful work (the work done by the RT when transferring heat).

The cylinder contains gas - the working fluid (RT). The initial state of RT on the p (V) diagram is shown by point 1. The cylinder is connected to a heater, the RT heats up and expands. Therefore, positive work A 1 is performed, the cylinder goes to position 2 (state 2).

Process 1–2: - the first law of thermodynamics. Work A 1 is equal to the area under the curve 1 a 2. To return the piston of the cylinder to its original state 1, it is necessary to compress the working fluid, thus spending the work - A 2.

In order for the piston to do useful work, it is necessary to fulfill the condition: A 2

Let's add two equations and get: The working body performs a circular process 1 a 2 b 1 - a cycle. K. p. D.

The process of returning the working fluid to its original state occurs at a lower temperature. Consequently, a refrigerator is fundamentally necessary for the operation of a heat engine.

Cycle Carnot Nicola Leonard Sadi CARNO - a brilliant French officer of the engineering troops, in 1824 published the essay "Reflections on the driving force of fire and machines capable of developing this force." He introduced the concept of circular and reversible processes, the ideal cycle of heat engines, thereby laying the foundations of their theory. Came to the concept of the mechanical equivalent of heat.

Carnot deduced a theorem that now bears his name: of all periodically operating heat engines with the same temperatures of heaters and refrigerators, reversible machines have the highest efficiency. Moreover, the efficiency of reversible machines operating at the same temperatures of heaters and refrigerators are equal to each other and do not depend on the design of the machine. In this case, the efficiency is less than unity.

If T 2 = 0, then η = 1, which is impossible, since the absolute zero temperature does not exist. If T 1 = ∞, then η = 1, which is impossible, since infinite temperature is not attainable. Carnot cycle efficiency η

Karnot's theorems. 1. The efficiency η of a reversible ideal Carnot heat engine does not depend on the working substance. 2. The efficiency of an irreversible Carnot machine cannot be greater than the efficiency of a reversible Carnot machine.

Consider the internal energy of an ideal gas. In an ideal gas, there is no attraction between molecules. Therefore, their potential energy is zero. Then the internal energy of this gas will add up only from the kinetic energies of individual molecules. Let us first calculate the internal energy of one mole of gas. It is known that the number of molecules in one mole of a substance is equal to the Avogadro number N A. The average kinetic energy of a molecule is found by the formula. Therefore, the internal energy U one mole of an ideal gas is equal to:

(1)

because kN A = R- universal gas constant. Internal energy U arbitrary mass of gas M is equal to the internal energy of one mole, multiplied by the number of moles , equal to  = M /, where is the molar mass of the gas, i.e.

(2)

Thus, the internal energy of a given mass of an ideal gas depends only on temperature and does not depend on volume and pressure.

Quantity of heat

The internal energy of a thermodynamic system under the influence of a number of external factors can change, which, as can be seen from formula (2), can be judged by the change in the temperature of this system. For example, if a gas is quickly compressed, its temperature rises. When drilling metal, its heating is also observed. If two bodies with different temperatures are brought into contact, then the temperature of the colder body rises, and the warmer one decreases. In the first two cases, the internal energy changes due to the work of external forces, and in the latter, the kinetic energies of the molecules are exchanged, as a result of which the total kinetic energy of the molecules of the heated body decreases, and the less heated one increases. There is a transfer of energy from a hot body to a cold one without performing mechanical work... The process of transferring energy from one body to another without performing mechanical work is called heat transfer or heat transfer ... The transfer of energy between bodies with different temperatures is characterized by a quantity called the amount of warmth or warmth , i.e. quantity of heat - it energy transferred by heat exchange from one thermodynamic system to another due to the temperature difference between these systems.

The first law of thermodynamics

There is in nature energy conservation and transformation law , Whereby energy does not disappear and does not arise again, but only passes from one type to another... This law applies to thermal processes , i.e. processes associated with a change in the temperature of a thermodynamic system, as well as with a change in the aggregate state of matter, was called the first law of thermodynamics.

If a certain amount of heat is imparted to a thermodynamic system Q, i.e. some energy, then due to this energy, in the general case, there is a change in its internal energy U and the system, expanding, performs a certain mechanical work A... Obviously, according to the law of conservation of energy, the equality must be fulfilled:

(3)

those. the amount of heat imparted to the thermodynamic system is spent on changing its internal energy and on performing mechanical work by the system during its expansion. Relation (4) is called the first law of thermodynamics.

It is convenient to write the expression of the first law for a small change in the state of the system when an elementary amount of heat is imparted to it dQ and doing basic work by the system dA, i.e.

(4)

where dU- an elementary change in the internal energy of the system. Formula (4) is a record of the first law of thermodynamics in differential form.

Experience shows that the internal energy of an ideal gas depends only on temperature:

Here B is the coefficient of proportionality, which remains constant over a very wide range of temperatures.

The fact that the internal energy does not depend on the volume occupied by the gas indicates that the molecules of an ideal gas do not interact with each other most of the time. Indeed, if the molecules interacted with each other, the potential energy of interaction would enter into the internal energy, which would depend on the average distance between the molecules, i.e. on.

Note that the interaction should take place in collisions, that is, when molecules approach each other at a very small distance. However, such collisions in rarefied gas are rare. Each molecule spends most of its time in free flight.

The heat capacity of a body is a quantity equal to the amount of heat that must be imparted to the body in order to raise its temperature by one kelvin. If the message of the amount of heat to the body raises its temperature, then the heat capacity, by definition, is equal to

This value is measured in joules per kelvin (J / K).

The heat capacity of a mole of a substance, called the molar heat capacity, will be denoted by the capital letter C. It is measured in joules per mole-kelvin (J / (mol K)).

The heat capacity of a unit mass of a substance is called the specific heat. We will denote it lowercase letter With. Measured from in joules per kilogram-kelvin

There is a ratio between the molar and specific heat capacities of the same substance

( - molar mass).

The heat capacity value depends on the conditions under which the body is heated. Of greatest interest is the heat capacity for cases when heating occurs at constant volume or at constant pressure. In the first case, the heat capacity is called the heat capacity at constant volume (denoted), in the second - the heat capacity at constant pressure

If heating occurs at a constant volume, the body does not perform work on external bodies and, therefore, according to the first law of thermodynamics (see (83.4)), all heat goes to the increment of the body's internal energy:

From (87.4) it follows that the heat capacity of any body at constant volume is

This notation emphasizes the fact that when differentiating the expression for U with respect to T, the volume should be considered constant. In the case of an ideal gas, U depends only on T, so that expression (87.5) can be represented in the form

(to get the molar heat capacity, you need to take the internal energy of a mole of gas).

Expression (87.1) for one mole of gas has the form Differentiating it with respect to T, we obtain that Thus, the expression for the internal energy of one mole of an ideal gas can be represented in the form

where is the molar heat capacity of the gas at constant volume.

The internal energy of an arbitrary mass of gas will be equal to the internal energy of one mole multiplied by the number of moles of gas contained in the mass:

If the gas is heated at constant pressure, then the gas will expand, doing positive work on external bodies. Consequently, to increase the gas temperature by one kelvin, in this case, more heat will be needed than when heating at a constant volume - part of the heat will be spent on the gas to perform work. Therefore, the heat capacity at constant pressure must be greater than the heat capacity at constant volume.

Let us write the equation (84.4) of the first law of thermodynamics for a mole of gas:

In this expression, the index at indicates that heat is imparted to the gas under conditions where it is constant. Dividing (87.8) by we obtain an expression for the molar heat capacity of a gas at constant pressure:

The term is equal, as we have seen, to the molar heat capacity at constant volume. Therefore, formula (87.9) can be written as follows:

(87.10)

The value is the increment in the volume of a mole of gas with an increase in temperature by one kelvin, obtained in the case when it is constant. In accordance with the equation of state (86.3). Differentiating this expression with respect to T, setting p = const, we find