Heat capacity. Her types. Relationship between heat capacities. Mayer's law. Average and true heat capacities. Heat capacity of a mixture of gases. True and average heat capacity What is the difference between true and average heat capacity

Heat capacity is a function of state parameters - pressure and temperature, therefore, in technical thermodynamics, true and average heat capacities are distinguished.

The heat capacity of an ideal gas depends only on temperature and, by definition, can only be found in the temperature range. However, it can always be assumed that this interval is very small near some temperature value. Then we can say that the heat capacity is determined at a given temperature. This heat capacity is called true.

In the reference literature, the dependence of the true heat capacities with p And with v temperature is given in the form of tables and analytical dependencies. An analytical dependence (for example, for mass heat capacity) is usually represented as a polynomial:

Then the amount of heat supplied in the process in the temperature range [ t1,t2] is determined by the integral:

. (2)

In the study of thermodynamic processes, the average value of the heat capacity in the temperature range is often determined. It is the ratio of the amount of heat supplied in the process Q 12 to the final temperature difference:

Then, if the dependence of the true heat capacity on temperature is given, in accordance with (2):

.

Often in the reference literature, values ​​​​of the average heat capacities are given with p And with v for the temperature range from 0 before t about C. Like true ones, they are presented in the form of tables and functions:

(4)

When substituting the temperature value t this formula will be used to find the average heat capacity in the temperature range [ 0.t]. To find the average heat capacity in an arbitrary interval [ t1,t2], using dependence (4), it is necessary to find the amount of heat Q 12 applied to the system in this temperature range. Based on the rule known from mathematics, the integral in equation (2) can be divided into the following integrals:

.

, A .

After that, the desired value of the average heat capacity is found by formula (3).

Gas mixtures

In technology, not pure substances, but mixtures of various gases are more often used as working fluids. In this case, a gas mixture is understood as a mechanical mixture of pure substances, called mixture components that do not enter into chemical reactions with each other. An example of a gas mixture is air, the main components of which are oxygen and nitrogen. If the components of the mixture are ideal gases, then the mixture as a whole will also be considered an ideal gas.

When considering mixtures, it is assumed that:

Each gas that is part of the mixture is evenly distributed throughout the volume, that is, its volume is equal to the volume of the entire mixture;

Each of the components of the mixture has a temperature equal to the temperature of the mixture;

Each gas creates its own pressure on the walls of the vessel, called partial pressure.

Partial pressure, thus, is the pressure that a component of the mixture would have if it alone occupied the entire volume of the mixture at the same temperature. The sum of the partial pressures of each component is equal to the pressure of the mixture (Dalton's law):

.

Partial volume component V is the volume that this component would occupy at a pressure equal to the pressure of the mixture and a temperature equal to the temperature of the mixture. Obviously, the sum of the partial volumes is equal to the volume of the mixture (Amag's law):

.

When studying thermodynamic processes with gas mixtures, it is necessary to know a number of quantities characterizing them: gas constant, molar mass, density, heat capacity, etc. To find them, one must set mixture composition, which determines the quantitative content of each component included in the mixture. The composition of the gas mixture is usually given by massive, voluminous or molar shares.

Mass fraction mixture component g a value equal to the ratio of the mass of the component to the mass of the entire mixture is called:

Obviously, the mass of the mixture m is equal to the sum of the masses of all components:

,

and the sum of mass fractions:

Volume fraction mixture component r i is called the value equal to the ratio of the partial volume of the component to the volume of the mixture:

The equation for the volumetric composition of the mixture has the form:

and the sum of volume fractions:

Mole fraction mixture component x i called a value equal to the ratio of the number of moles of this component to the total number of moles of the mixture:

It's obvious that:

The composition of the mixture is given in fractions of a unit or in percent. The relationship between mole and volume fractions can be established by writing the Clapeyron-Mendeleev equation for a component of a mixture and for the entire mixture:

Dividing term by term the first equation by the second, we get:

Thus, for ideal gases, the volume and mole fractions turn out to be equal.

The relationship between mass and volume fractions is established by the relationships:

. (5)

From Avogadro's law it follows:

where μ is the molar mass of the mixture, which is called apparent. It can be found, in particular, through the volumetric composition of the mixture. Writing down the Clapeyron-Mendeleev equation for i-th mixture component in the form

and summing over all components, we get:

.

Comparing it with the equation of state for the mixture as a whole

we arrive at the obvious relation:

.

Once the molar mass of the mixture is found, the gas constant of the mixture can be determined in the usual way:

. (7)

These formulas are used in determining the true and average heat capacities of the mixture.

Heat capacity is the ratio of the amount of heat δQ received by a substance with an infinitesimal change in its state in any process to the change in temperature dT of the substance (symbol C, unit J / K):

С(T) = δQ/dT

The heat capacity of a unit of mass (kg, g) is called specific (unit J / (kg K) and J / (g K)), and the heat capacity of 1 mol of a substance is called molar heat capacity (unit J / (mol K)).

Distinguish the true heat capacity.

С = δQ/dT

Average heat capacity.

Ĉ \u003d Q / (T 2 - T 1)

The average and true heat capacities are related by the relation

The amount of heat absorbed by a body when its state changes depends not only on the initial and final states of the body (in particular, on temperature), but also on the transition conditions between these states. Consequently, its heat capacity also depends on the heating conditions of the body.

In an isothermal process (T = const):

C T = δQ T /dT = ±∞

In an adiabatic process (δQ = 0):

C Q = δQ/dT = 0

Heat capacity at constant volume, if the process is carried out at constant volume - isochoric heat capacity C V .

Heat capacity at constant pressure, if the process is carried out at constant pressure - isobaric heat capacity С Р.

For V = const (isochoric process):

C V = δQ V /dT = (ϭQ/ϭT) V = (ϭU/ϭT) V

δQ V = dU = C V dT

At P = const (isobaric process)%

C p = δQ p /dT = (ϭQ/ϭT) p = (ϭH/ϭT) p

The heat capacity at constant pressure C p is greater than the heat capacity at constant volume C V . When heated at constant pressure, part of the heat goes to the production of work of expansion, and part to increase the internal energy of the body; when heated at a constant volume, all the heat is spent on increasing the internal energy.

A connection between C p and C V for any systems that can only do expansion work. According to the first law of thermodynamics%

δQ = dU + PdV

Internal energy is a function of external parameters and temperature.

dU = (ϭU/ϭT) V dT + (ϭU/ϭV) T dV

δQ = (ϭU/ϭT) V dT + [(ϭU/ϭV) T + P] dV

δQ/dT = (ϭU/ϭT) V + [(ϭU/ϭV) T + P] (dV/dT)

The value of dV/dT (volume change with temperature) is the ratio of increments of independent variables, that is, the value is uncertain, unless you specify the nature of the process in which heat exchange occurs.

If the process is isochoric (V = const), then dV = 0, dV/dT = 0

δQ V /dT = C V = (ϭU/ϭT) V

If the process is isobaric (P = const).

δQ P /dT = C p = C V + [(ϭU/ϭV) T + P] (dV/dT) P

For any simple system, the following is true:

C p – C v = [(ϭU/ϭV) T + P] (dV/dT) P

Solidification and boiling point of the solution. Cryoscopy and ebullioscopy. Determination of the molecular weight of a solute.

crystallization temperature.

A solution, unlike a pure liquid, does not solidify entirely at a constant temperature; at a temperature called the temperature of the onset of crystallization, crystals of the solvent begin to precipitate, and as crystallization proceeds, the temperature of the solution decreases (therefore, the freezing point of a solution is always understood to be the temperature of the onset of crystallization). The freezing of solutions can be characterized by the magnitude of the decrease in the freezing point ΔT deputy, equal to the difference between the freezing point of a pure solvent T ° deputy and the temperature of the beginning of crystallization of the solution T deputy:

ΔT dep = T° dep - T dep

Solvent crystals are in equilibrium with the solution only when the saturation vapor pressure over the crystals and over the solution is the same. Since the vapor pressure of the solvent over the solution is always lower than over the pure solvent, the temperature that meets this condition will always be lower than the freezing point of the pure solvent. In this case, the decrease in the freezing point of the solution ΔT deputy does not depend on the nature of the solute and is determined only by the ratio of the number of particles of the solvent and the solute.

Lowering the freezing point of dilute solutions

The decrease in the freezing point of the solution ΔT deputy is directly proportional to the molar concentration of the solution:

ΔT deputy = Km

This equation is called Raoult's second law. The coefficient of proportionality K - the cryoscopic constant of the solvent - is determined by the nature of the solvent.

Boiling temperature.

The boiling point of solutions of a non-volatile substance is always higher than the boiling point of a pure solvent at the same pressure.

Any liquid - solvent or solution - boils at the temperature at which the saturated vapor pressure becomes equal to the external pressure.

Increasing the boiling point of dilute solutions

The increase in the boiling point of solutions of nonvolatile substances ΔTc = Tc - T°c is proportional to the decrease in saturated vapor pressure and, therefore, is directly proportional to the molar concentration of the solution. The coefficient of proportionality E is the ebullioscopic constant of the solvent, independent of the nature of the solute.

ΔT c \u003d Em

Raoult's second law. The decrease in the freezing point and the increase in the boiling point of a dilute solution of a non-volatile substance is directly proportional to the molar concentration of the solution and does not depend on the nature of the solute. This law is valid only for infinitely dilute solutions.

Ebullioscopy- a method for determining molecular weights by increasing the boiling point of a solution. The boiling point of a solution is the temperature at which the vapor pressure above it becomes equal to the external pressure.

If the solute is non-volatile, then the vapor above the solution consists of solvent molecules. Such a solution begins to boil at a higher temperature (T) compared to the boiling point of a pure solvent (T0). The difference between the boiling points of a solution and a pure solvent at a given constant pressure is called the rise in the boiling point of the solution. This value depends on the nature of the solvent and the concentration of the solute.

A liquid boils when the pressure of the saturated vapor above it is equal to the external pressure. When boiling, liquid solution and vapor are in equilibrium. If the solute is non-volatile, the increase in the boiling point of the solution obeys the equation:

∆ isp H 1 - enthalpy of evaporation of the solvent;

m 2 - the molality of the solution (the number of moles of a solute per 1 kg of solvent);

E is the ebullioscopic constant, equal to the increase in the boiling point of a 1-mol solution compared to the boiling point of a pure solvent. The value of E is determined by the properties of only the solvent, but not the solute.

Cryoscopy- a method for determining molecular weights by lowering the freezing point of a solution. When the solutions are cooled, they freeze. Freezing point - the temperature at which the first crystals of the solid phase are formed. If these crystals consist only of solvent molecules, then the freezing point of the solution (T) is always lower than the freezing point of the pure solvent (T pl). The difference between the freezing point of the solvent and the solution is called the drop in the freezing point of the solution.

The quantitative dependence of the freezing point decrease on the concentration of the solution is expressed by the following equation:

M 1 - molar mass of the solvent;

∆ pl H 1 - enthalpy of melting of the solvent;

m 2 - molality of the solution;

K is a cryoscopic constant, depending on the properties of the solvent alone, equal to the decrease in the freezing point of the solution with the molality of the substance dissolved in it equal to one.

The dependence of the saturation vapor pressure of the solvent on the temperature.

Lowering the freezing point and raising the boiling point of solutions, their osmotic pressure do not depend on the nature of the dissolved substances. Such properties are called colligative. These properties depend on the nature of the solvent and the concentration of the solute. As a rule, colligative properties appear when two phases are in equilibrium, one of which contains a solvent and a solute, and the second contains only a solvent.

is the amount of heat supplied to 1 kg of a substance when its temperature changes from T 1 to T 2 .

1.5.2. Heat capacity of gases

The heat capacity of gases depends on:

type of thermodynamic process (isochoric, isobaric, isothermal, etc.);

type of gas, i.e. on the number of atoms in the molecule;

gas state parameters (pressure, temperature, etc.).

A) Influence of the type of thermodynamic process on the heat capacity of a gas

The amount of heat required to heat the same amount of gas in the same temperature range depends on the type of thermodynamic process performed by the gas.

IN isochoric process (υ = const) heat is spent only on heating the gas by a value. The gas does not expand in a closed vessel (Fig. 1.2 A), so it doesn't do any work. The heat capacity of a gas in an isochoric process is denoted by the symbol With υ .

IN isobaric process (R= const), heat is spent not only on heating the gas by the same amount as in the isochoric process, but also on doing work when the piston is raised with an area of ​​\u200b\u200b(Fig. 1.2 b). The heat capacity of a gas in an isobaric process is denoted by the symbol With R .

Since, according to the condition, in both processes the value is the same, then in the isobaric process due to the work performed by the gas, the value. Therefore, in an isobaric process, the heat capacity With R With υ .

According to Mayer's formula for ideal gas

or . (1.6)

B) Influence of the type of gas on its heat capacity It is known from the molecular-kinetic theory of an ideal gas that

where is the number of translational and rotational degrees of freedom of motion of molecules of a given gas. Then

, A . (1.7)

A monatomic gas has three translational degrees of freedom for the movement of a molecule (Fig. 1.3 A), i.e. .

A diatomic gas has three translational degrees of freedom of motion and two degrees of freedom of rotational motion of the molecule (Fig. 1.3 b), i.e. . Similarly, it can be shown that for a triatomic gas.

Thus, the molar heat capacity of gases depends on the number of degrees of freedom of molecular motion, i.e. on the number of atoms in the molecule, and the specific heat also depends on the molecular weight, because the value of the gas constant depends on it, which is different for different gases.

C) Influence of gas state parameters on its heat capacity

The heat capacity of an ideal gas depends only on temperature and increases with increasing T.

Monatomic gases are an exception, because their heat capacity is practically independent of temperature.

The classical molecular-kinetic theory of gases makes it possible to fairly accurately determine the heat capacities of monatomic ideal gases in a wide range of temperatures and the heat capacities of many diatomic (and even triatomic) gases at low temperatures.

But at temperatures significantly different from 0 o C, the experimental values ​​of the heat capacity of two- and polyatomic gases turn out to be significantly different from those predicted by the molecular-kinetic theory.

On fig. 1.4 shows the dependence of the molar heat capacities of hydrogen and helium at constant volume With v from absolute temperature T over a wide range of changes. As can be seen, the values ​​of the heat capacity for a diatomic gas (and polyatomic gases) can depend significantly on temperature. This is explained by the fact that at low temperatures rotational degrees of freedom are not excited, and therefore the molar heat capacity of a diatomic (and polyatomic) gas is the same as that of a monatomic gas (hydrogen is the same as helium). At high temperatures, however, in two- and polyatomic gases, degrees of freedom are also excited, associated with vibrations of atoms in molecules, which leads to an additional increase in their heat capacity.

In heat engineering calculations, experimental values ​​of the heat capacity of gases are usually used, presented in the form of tables. In this case, the heat capacity determined in the experiment (at a given temperature) is called true heat capacity. And if in the experiment the amount of heat was measured q, which was spent on a significant increase in the temperature of 1 kg of gas from a certain temperature T 0 to temperature T, i.e. on  T = T T 0 , then the ratio

called middle heat capacity of the gas in a given temperature range.

Usually in reference tables, the values ​​\u200b\u200bof the average heat capacity are given at the value T 0 corresponding to zero degrees Celsius.

Heat capacity real gas depends, in addition to temperature, also on pressure due to the influence of intermolecular interaction forces.

Specific, molar and volumetric heat capacity. Although the heat that is part of the CCT equations can be theoretically represented as the sum of microworks performed by the collision of microparticles at the boundaries of the system without the occurrence of macroforces and macrodisplacements, in practice this method of calculating heat is of little use and historically heat was determined in proportion to the change in body temperature dT and some value C of the body characterizing the content of matter in the body and its ability to accumulate thermal motion (heat),

Q = C of the body dT. (2.36)

Value

C body = Q / dT; = 1 J / K, (2.37)

equal to the ratio of elemental heat Q, communicated to the body, to the change in body temperature dT, is called the (true) heat capacity of the body. The heat capacity of a body is numerically equal to the heat required to change the body temperature by one degree.

Since the temperature of the body also changes during the performance of work, then work, by analogy with heat (4.36), can also be determined through a change in body temperature (this method of calculating work has certain advantages when calculating it in polytropic processes):

W = C w dT. (2.38)

C w = dW/dT = pdV / dT, (2.39)

equal to the ratio of the work supplied (retracted) to the body to the change in body temperature, by analogy with the heat capacity can be called the "labor capacity of the body" The term "work capacity" is as arbitrary as the term "heat capacity". The term “heat capacity” (capacity for heat) - as a tribute to the real theory of heat (caloric) - was first introduced by Joseph Black (1728-1779) in the 60s of the 18th century. in his lectures (the lectures themselves were published only posthumously in 1803).

Specific heat capacity c (sometimes called mass, or specific mass heat capacity, which is obsolete) is the ratio of the heat capacity of a body to its mass:

c = body / m = dQ / (m dT) = dq / dT; [s] = 1 J /(kgK), (2.40)

where dq \u003d dQ / m - specific heat, J / kg.

The specific heat capacity is numerically equal to the heat that must be supplied to a substance of unit mass in order to change its temperature by one degree.

Molar heat capacity is the ratio of the heat capacity of a body to the amount of substance (molarity) of this body:

C m \u003d C body / m, \u003d 1 J / (molK). (2.41)

Volumetric heat capacity is the ratio of the heat capacity of a body to its volume, reduced to normal physical conditions (p 0 \u003d 101325 Pa \u003d 760 mm Hg; T 0 \u003d 273.15 K (0 o C)):

c" \u003d C body / V 0, \u003d 1 J / (m 3 K). (2.42)

In the case of an ideal gas, its volume under normal physical conditions is calculated from the equation of state (1.28)

V 0 = mRT 0 / p 0 . (2.43)

Molecular heat capacity is the ratio of the heat capacity of a body to the number of molecules of this body:

c m = C body / N; = 1 J / K. (2.44)

The relationship between different types of heat capacities is established by jointly solving relations (2.40) - (2.44) for heat capacities. The relationship between specific and molar heat capacities establishes the following relationship:

c \u003d C body / m \u003d C m. m / m \u003d C m / (m / m) \u003d C m / M, (2.45)

where M = m / m is the molar mass of the substance, kg / mol.

Since tabular values ​​for molar heat capacities are more often given, then relation (2.45) should be used to calculate the values ​​of specific heat capacities through molar heat capacities.

The relationship between volumetric and specific heat capacities is established by the relation

c" \u003d C body / V 0 \u003d cm / V 0 \u003d c 0, (2.46)

where 0 \u003d m / V 0 - gas density under normal physical conditions (for example, air density under normal conditions

0 \u003d p 0 / (RT 0) \u003d 101325 / (287273.15) \u003d 1.29 kg / m 3).

The relationship between volumetric and molar heat capacities is established by the relation

c" \u003d C body / V 0 \u003d C m m / V 0 \u003d C m / (V 0 / m) \u003d C m / V m0, (2.47)

where V 0 \u003d V 0 / m \u003d 22.4141 m 3 / kmol is the molar volume reduced to NFU.

In the future, when considering the general provisions for all types of heat capacities, we will consider the specific heat capacity as the initial one, which, for brevity, we will simply call the heat capacity, and the corresponding specific heat, simply the heat.

True and average heat capacity. The heat capacity of an ideal gas depends on the temperature c = c (T), while the heat capacity of a real gas also depends on the pressure c = c (T, p). On this basis, true and average heat capacity are distinguished. For gases with low pressure and high temperature, the dependence of heat capacity on pressure turns out to be negligible.

The true heat capacity corresponds to a certain body temperature (heat capacity at a point), since it is determined with an infinitely small change in body temperature dT

c = dq / dT. (2.48)

Often in thermal engineering calculations, the nonlinear dependence of the true heat capacity on temperature is replaced by a linear dependence close to it

c = b 0 + b 1 t = c 0 + bt, (2.49)

where c 0 \u003d b 0 - heat capacity at Celsius temperature t \u003d 0 o C.

Elementary specific heat can be determined from expression (4.48) for specific heat:

dq = c dT. (2.50)

Knowing the dependence of the true heat capacity on temperature c = c(t), it is possible to determine the heat supplied to the system in a finite temperature range by integrating expression (2.53) from the initial state 1 to the final state 2,

In accordance with the graphic representation of the integral, this heat corresponds to an area of ​​122 "1" under the curve c \u003d f (t) (Fig. 4.4).

Figure 2.4 - To the concept of true and average heat capacity

The area of ​​the curvilinear trapezoid 122 "1", corresponding to the heat q 1-2, can be replaced by the equivalent area of ​​the rectangle 1 "342" with the base ДT = T 2 - T 1 = t 2 - t 1 and height: .

Expression value

and will be the average heat capacity of the substance in the temperature range from t 1 to t 2.

If dependence (2.52) for the true heat capacity is substituted into expression (2.55) for the average heat capacity and integrated over temperature, we obtain

Co + b(t1 + t2) / 2 = , (2.53)

where t cp \u003d (t 1 + t 2) / 2 is the average Celsius temperature in the temperature range from t 1 to t 2.

Thus, in accordance with (2.56), the average heat capacity in the temperature range from t 1 to t 2 can be approximately determined as the true heat capacity, calculated from the average temperature t cp for a given temperature range.

For the average heat capacity in the temperature range from 0 ° C (t 1 \u003d 0) to t, dependence (2.56) takes the form

C o + (b / 2)t \u003d c o + b "t. (2.54)

When calculating the specific heat required to heat the gas from 0 ° C to t 1 and t 2, using such tables, where each temperature t corresponds to the average heat capacity, the following ratios are used:

q 0-1 = t 1 and q 0-2 = t 2

(in Fig. 4.4, these heats are depicted as the areas of figures 0511 "and 0522"), and to calculate the heat supplied in the temperature range from t 1 to t 2, the relation is used

q 1-2 = q 0-2 - q 0-1 = t 2 - t 1 = (t 2 - t 1).

From this expression, the average heat capacity of the gas is found in the temperature range from t 1 to t 2:

= = (t 2 - t 1) / (t 2 - t 1). (2.55)

Therefore, in order to find the average heat capacity in the temperature range from t 1 to t 2 according to the formula (2.59), it is necessary to first determine the average heat capacities and from the corresponding tables. After calculating the average heat capacity for a given process, the supplied heat is determined by the formula

q 1-2 = (t 2 - t 1). (2.56)

If the range of temperature change is small, then the dependence of the true heat capacity on temperature is close to linear, and the heat can be calculated as the product of the true heat capacity c(t cp) determined for the average gas temperature? t cp in a given temperature range, on the temperature difference:

q 1-2 = = . (2.57)

Such a calculation of heat is equivalent to calculating the area of ​​the trapezoid 1 "1" "22" (see Fig. 2.4) as the product of the median line of the trapezoid c (t cp) and its height DT.

The true heat capacity at an average temperature t cp in accordance with (4.56) has a value close to the average heat capacity in this temperature range.

For example, in accordance with Table C.4, the average molar isochoric heat capacity in the temperature range from 0 to 1000 ° C \u003d 23.283 kJ / (kmol.K), and the true molar isochoric heat capacity corresponding to an average temperature of 500 ° C for this temperature range is C mv \u003d 23.316 kJ / (kmol.K). The difference between these heat capacities does not exceed 0.2%.

Isochoric and isobaric heat capacity. Most often in practice, the heat capacities of isochoric and isobaric processes are used, occurring at a constant specific volume x = const and pressure p = const, respectively. These specific heat capacities are called isochoric c v and isobaric c p heat capacities, respectively. Using these heat capacities, any other types of heat capacities can be calculated.

Thus, an ideal gas is such an imaginary gas (gas model) whose state exactly corresponds to the Clapeyron equation of state, and the internal energy depends only on temperature.

As applied to an ideal gas, instead of partial derivatives (4.66) and (4.71), one should take total derivatives:

c x \u003d du / dT; (2.58)

p = dh / dT. (2.59)

It follows that c x and c p for an ideal gas, like u and h, depend only on temperature.

In the case of constant heat capacities, the internal energy and enthalpy of an ideal gas are determined by the expressions:

U = c x mT and u = c x T; (2.60)

H = c p mT and h = c p T. (2.61)

When calculating the combustion of gases, volumetric enthalpy, J / m 3, is widely used,

h" \u003d H / V 0 \u003d c p mT / V 0 \u003d c p c 0 T \u003d c "p T, (2.62)

where c "p \u003d cp c0 - volumetric isobaric heat capacity, J / (m 3 .K).

Mayer's equation. Let's establish a connection between the heat capacities of an ideal gas c x and c p . To do this, we use the PZT equation (4.68) for an ideal gas during an isobaric process

dq p \u003d c p dT \u003d du + pdx \u003d c x dT + pdx. (2.63)

Where do we find the difference in heat capacities

c p - c x \u003d pdx / dT \u003d p (x / T) p \u003d w p / dT (2.64)

(this relation for an ideal gas is a particular case of relation (2.75) for a real gas).

Differentiating the Clapeyron equation of state d(px) p = R dT under the condition of constant pressure, we obtain

dх / dT = R / p. (2.65)

Substituting this relation into equation (2.83), we obtain

c p - c x \u003d R. (2.66)

Multiplying all the quantities in this ratio by the molar mass M, we obtain a similar ratio for the molar heat capacities

Cm p - Cm x = Rm. (2.67)

Relations (2.65) and (2.66) are called Mayer's formulas (equations) for an ideal gas. This is due to the fact that Mayer used equation (2.65) to calculate the mechanical equivalent of heat.

The ratio of heat capacities c p / c x. In thermodynamics and its applications, not only the difference in heat capacities c p and c x, determined by the Mayer equation, but also their ratio c p / c x, which in the case of an ideal gas is equal to the ratio of heat to the change in SE in an isobaric process, is of great importance, i.e. it is ratio is a characteristic of an isobaric process:

k p \u003d k X \u003d q p / du \u003d c p dT / \u003d c p dT / c x dT \u003d c p / c x.

Therefore, if in the process of changing the state of an ideal gas, the ratio of heat to the change in SE is equal to the ratio c p /c x, then this process will be isobaric.

Since this ratio is often used and is included as an exponent in the equation of the adiabatic process, it is usually denoted by the letter k (without index) and called the adiabatic exponent

k \u003d q p / du \u003d c p / c x \u003d C m p / Cm x \u003d c "p / c" x. (2.68)

The values ​​of the true heat capacities and their ratio k of some gases in an ideal state (at p > 0 and T C = 0 ° C) are given in Table 3.1.

Table 3.1 - Some characteristics of ideal gases

	Chemical formula
	kJ/(kmolK)
water vapor
carbon monoxide
Oxygen
Carbon dioxide
Sulfur dioxide
Mercury vapor

On average, over all gases of the same atomicity, it is generally accepted that for monatomic gases k ? 1.67, for diatomic k? 1.40, for triatomic k? 1.29 (for water vapor, the exact value k = 1.33 is often taken).

Solving (2.65) and (2.67) together, we can express the heat capacities in terms of k and R:

Taking into account (2.69), equation (2.50) for the specific enthalpy takes the form

h = c p T = . (2.71)

For diatomic and polyatomic ideal gases, k depends on temperature: k = f(T). In accordance with equation (2.58)

k \u003d 1 + R / c x \u003d 1 + Rm / Cm x. (2.72)

Heat capacity of the gas mixture. To determine the heat capacity of a mixture of gases, it is necessary to know the composition of the mixture, which can be given by mass g i , molar x i or volume fractions r i, as well as the heat capacities of the mixture components, which are taken from the tables for the corresponding gases.

The specific heat capacity of a mixture consisting of N components for isoprocesses X = x, p = const is determined in terms of mass fractions by the formula

cXcm = . (2.73)

The molar heat capacity of a mixture is determined in terms of mole fractions

The volumetric heat capacity of the mixture is determined in terms of volume fractions by the formula

For ideal gases, molar and volume fractions are equal: x i = r i .

Calculation of heat through heat capacity. Here are formulas for calculating heat in various processes:

a) through the average specific heat and mass m

b) through the average molar heat capacity and the amount of substance m

c) through the average volumetric heat capacity and the volume V 0 reduced to normal conditions,

d) through the average molecular heat capacity and the number of molecules N

where DT \u003d T 2 - T 1 \u003d t 2 - t 1 - change in body temperature;

Average heat capacity in the temperature range from t 1 to t 2;

c(t cp) - true heat capacity, determined for the average body temperature t cp \u003d (t 1 + t 2) / 2.

According to Table C.4 of the heat capacities of air, we find the average heat capacities: = = 1.0496 kJ / (kgK); = 1.1082 kJ / (kgK). The average heat capacity in this temperature range is determined by the formula (4.59)

= (1.10821200 - 1.0496600) / 600 = 1.1668 kJ / (kgK),

where DT = 1200 - 600 = 600 K.

Specific heat through the average heat capacity in a given temperature range = 1.1668600 = 700.08 kJ / kg.

Now let's determine this heat according to the approximate formula (4.61) through the true heat capacity c(t cp), determined for the average heating temperature t cp = (t 1 + t 2) / 2 = (600 + 1200) / 2 = 900 o C.

The true heat capacity of air c p for 900 ° C according to table C.1 is 1.1707 kJ / (kgK).

Then the specific heat through the true heat capacity at the average heat supply temperature

q p \u003d c p (t cp) \u003d c p (900) DT \u003d 1.1707600 \u003d 702.42 kJ / kg.

The relative error in calculating heat according to an approximate formula through the true heat capacity at an average heating temperature e(q p) = 0.33%.

Therefore, in the presence of a table of true heat capacities, it is easiest to calculate the specific heat using formula (4.61) through the true heat capacity taken at the average heating temperature.

Based on experimental data, it has been established that the dependence of the true heat capacity of real gases on temperature is curvilinear, as shown in Fig. 6.6, and can be expressed as a power series With P = a +bt + dt 2 + ef 3 + .... (6.34)

where a, 6, d,... constant coefficients, the numerical values ​​of which depend on the type of gas and the nature of the process. In thermal calculations, the nonlinear dependence of heat capacity on temperature is often replaced by a linear one.

In this case, the true heat capacity is determined from

equations
(6.35)

Where t - temperature, °C;b= dc/ dt-angular straight line slope factor with n = a +bt.

Based on (6.20), we find the formula for the average heat capacity when it changes linearly with temperature according to (6.35)

(6.36)

If the process of temperature change proceeds in

interval ABOUT-t , then (6.36) takes the form
(6.37)

Heat capacity
is called the heat capacity of the average

temperature range
and heat capacity

- heat capacity average in the range 0- t.

The results of calculations of the true and average in the temperature range ABOUT-t mass or molar heat capacities at

constant volume and pressure, respectively, according to equations (6.34) and (6.37) are given in the reference literature. The main heat and cold engineering task is to determine the heat involved in the process. In accordance with the ratio q = c n dT and with a nonlinear dependence of the true heat capacity on temperature, the amount of heat is determined by the shaded elementary area in the diagram with coordinates With n T(Fig. 6.6). When the temperature changes from T 1 before T 2 in an arbitrary final process, the amount of input or output heat is determined, according to (6.38), as follows:

(6.38)

and is determined on the same diagram (Fig. 6.6) with an area of ​​​​12T 2 T 1 1. Substituting in (6.38) the value with n \u003d f (T) for a given gas according to relation (6.34) and integrating, we obtain a calculation formula for determining the heat in given interval of gas temperature change, which, however, follows from (6.16):

However, since in the reference literature there is only the average heat capacity in the temperature range 0- t, then the amount of heat in the process 12 can be determined not only by the previous formula, but as follows: Obviously, the ratio between the heat capacities is average in the temperature intervals T 1 - T 2 And 0- t:

The amount of heat supplied (removed) to m kg of the working fluid

The amount of heat supplied to V m 3 gas is determined by the formula

The amount of heat supplied (removed) to n moles of the working fluid is

6.10Molecular-kinetic theory of heat capacity

The molecular-kinetic theory of heat capacity is very approximate, since it does not consider the vibrational and potential components of internal energy. Therefore, according to this theory, the problem is to determine the distribution of thermal energy supplied to the substance between translational and rotational forms of internal kinetic energy. According to the Maxwell-Boltzmann distribution, if a certain amount of energy is imparted to a system of a very large number of microparticles, then it is distributed

between the translational and rotational motion of microparticles is proportional to their number of degrees of freedom. The number of degrees of freedom of a gas molecule (section 5.4) corresponds to the number of coordinates that determine its position in space.

A monatomic gas molecule has three degrees of freedom, since its position in space is determined by three coordinates, and for a monatomic gas these three degrees of freedom are the degrees of freedom of translational motion.

For a diatomic gas, the values ​​of the three coordinates of one atom do not yet determine the position of the molecule in space, since after determining the position of one atom, it must be taken into account that the second atom has the possibility of rotational motion. To determine the position in space of the second atom, it is necessary to know two of its coordinates (Fig. 6.7), and the third one will be determined from the equation known in analytical geometry

Where is the distance between atoms. Thus, with the known of the six coordinates, only five need to be known. Consequently, a diatomic gas molecule has five degrees of freedom, of which three are translational and two are rotational.

A triatomic gas molecule has six degrees of freedom - three translational and three rotational motions. This follows from the fact that to determine the position in space, it is necessary to know six coordinates of atoms, namely: three coordinates of the first atom, two coordinates of the second atom, and one coordinate of the third. Then the position of the atoms in space will be completely determined, since the distances between them
- are set.

If we take a gas of greater atomicity, that is, 4-atomic or more, then the number of degrees of freedom of such a gas will also be six, since the position of the fourth and each subsequent atom will be determined by its fixed distance from other atoms.

According to the molecular kinetic theory of matter, the average kinetic energy of the translational and rotational motions of each of the molecules is proportional to the temperature

and equal respectively And
is the number of degrees of freedom of rotational motion). Therefore, the kinetic energy of the translational and rotational motions of all molecules will be a linear function of temperature

J, (6.39)

J.

Equations (6.39) and (6.40) express the mentioned law of equipartition of energy over degrees of freedom, according to which the same average kinetic energy equal to 1/2 (kT) falls on each degree of freedom of translational and rotational motions of molecules.

The energy of vibrational motion of molecules is a complex increasing function of temperature, and only in some cases at high temperatures can it be approximately expressed by a formula similar to (6.40). The molecular-kinetic theory of heat capacity does not take into account the vibrational motion of molecules.

Repulsive and attractive forces act between two real gas molecules. For an ideal gas, there is no potential energy of interaction between molecules. In view of the foregoing, the internal energy of an ideal gas is equal to U=
. Because N= vnN A , That
The internal energy of one mole of an ideal gas, provided that the universal gas constant is determined by the product of two constants:
= kN A , is defined as follows:
,J/mol.

Differentiating with respect to T and knowing that du / dT = c r , we obtain the molar heat capacity of an ideal gas at a constant volume

Coefficient
called Poisson's ratio or adiabatic exponent.

For an ideal gas, the adiabatic index is a quantity that depends only on the atomic structure of the gas molecules, which is reflected in Table. 6.1. The symbolic value of the adiabatic exponent can be obtained from the Mayer equation With p - c v = R through the following transformations: kc v - c p = R, c v (k- l) - R, from where To= 1 + R/ c v . From the previous equality follows the expression of the isochoric heat capacity in terms of the adiabatic exponent cv = = R/(k- 1) and then the isobaric heat capacity: with r. = kR/(k- 1).

From Mayer's equation With R =
we obtain an expression for the molar heat capacity of an ideal gas at constant pressure
, J/(mol-K).

For approximate calculations at not very high temperatures, when the energy of the vibrational motion of atoms in molecules due to its smallness can be ignored, the obtained molar heat capacities can be used with v And With p as a function of the atomicity of gases. The values ​​of heat capacities are presented in table. 6.1.

Table 6.1

The values ​​of heat capacities according to molecular kineticgas theory

heat capacity Gas atomicity
	mole hail		mole hail
Monatomic gas Diatomic gas Triatomic or more atomic gas		12,5 20,8 29,1		20.8 29.1 37.4	1,67 1,40 1,28