True and average heat capacity. Heat capacity

The experimental values \u200b\u200bof heat dissipation at different temperatures are presented in the form of tables, graphs and empirical functions.

Distinguish true and middle heat capacity.

True heat capacity C is a heat capacity for a given temperature.

In engineering calculations, the average heat capacity at a given temperature range (T1; T2) is often used.

The average heat capacity is doubly: ,.

The lack of the latter designation is the odorlessness of the temperature range.

True and average heat capacity are associated with the relation:

The true heat capacity is the limit to which the average heat capacity seeks, in a given range of temperatures T1 ... T2, with Δt \u003d T2-T1

As experience shows, most gases have true heat capacity with increasing temperature. The physical explanation of this increase is as follows:

It is known that the gas temperature is not associated with the oscillatory motion of atoms and molecules, and depends on the kinetic energy E K of the transit movement of particles. But as the temperature increases, the heat supplied to the gas is becoming more and more redistributed in favor of the oscillatory movement, i.e. The temperature rise with the same heat supply as temperature grows slows down.

Typical dependence of heat capacity from temperature:

c \u003d C 0 + AT + BT 2 + DT 3 + ... (82)

where C 0, A, B, D is the empirical coefficients.

c is true heat capacity, i.e. The value of heat capacity for the specified temperature T.

For the heat capacity of the bitoproximating curve is polynomial in the form of a series of degrees t.

The approximating curve is carried out using special methods, for example, by the method of smallest squares. The essence of this method is that when using it, all points are approximately equidistant from the approximating curve.

For engineering calculations, as a rule, limited to the two first term in the right part, i.e. The dependence of heat capacity from temperature linear C \u003d C 0 + AT (83)

The average heat capacity is graphically defined as the middle line of the shaded trapezoid, as is known, the average line of the trapezium is defined as a substrate half asum.

Formulas are applied if empirical dependence is known.

In cases where the dependence of heat heat dissipation is not successful to satisfactorily approximate to the dependence C \u003d C 0 + AT, you can use the following formula:

This formula is applied in cases where the dependence C from T is significantly nonlinear.

From the molecular-kinetic theory of gases is known

U  \u003d 12.56t, u  - the internal energy of one kilomol of the perfect gas.

Previously obtained for perfect gas:

, ,

From the result obtained it follows that the heat capacity obtained using MTCs does not depend on the temperature.

Equation of Mayer: C  P -C  V \u003d R ,

c  P \u003d C  V + R  \u003d 12.56 + 8,31420.93.

As and the previous case, the molecular isobar heat capacity from the temperature does not depend on the MT gases.

The concept of ideal gas to the greatest degree corresponds to monatomic gases at low pressures, in practice it is necessary to deal with 2, 3rd ... atomic gases. For example, air, which consists of 79% of nitrogen (N 2), 21% oxygen (O 2) (in the engineering calculations, inert gases are not taken into account due to the small contents of their content).

It is possible for estimated calculations to use the following table:

In real gases, in contrast to the perfect, heat capacity may depend not only on temperature, but also on the volume and pressure of the system.

The heat capacity is the thermophysical characteristic that determines the body's ability to give or perceive heat to change the body temperature. The ratio of the amount of heat, subordinated (or allotted) in this process, is called the heat capacity of the body (tel system): C \u003d DQ / DT, where - the elementary amount of heat; - Elementary temperature change.

The heat capacity is numerically equal to the amount of heat that must be brought to the system so that under the given conditions, increase its temperature by 1 degree. The heat capacity will be J / K.

Depending on the quantitative unit of the body, to which the heat in thermodynamics is supplied, is distinguished by mass, bulk and molar heat capacity.

Mass heat capacity is a heat capacity, assigned to a unit of mass of the working fluid, C \u003d C / M

The unit of measurement of mass heat capacity is J / (kg × K). Mass heat capacity is also called specific heat capacity.

The volumetric heat capacity is the heat capacity, assigned to a unit of the volume of the working fluid, where the volume and density of the body under normal physical conditions. C '\u003d C / V \u003d \u200b\u200bC p. Volumetric heat capacity is measured in J / (m 3 × K).

Moldable heat capacity - heat capacity, attributed to the amount of working fluid (gas) in moles, C m \u003d C / N, where N is the amount of gas in the moles.

Mole heat capacity is measured in J / (mol × K).

Mass and molar heat capacity are associated with the following ratio:

The volumetric heat capacity of gases is expressed through molly as

Where M 3 / mol is a molar volume of gas under normal conditions.

Majer equation: C p - with V \u003d R.

Considering that the heat capacity is inconstant, and depends on the temperature and other thermal parameters, distinguish between the true and average heat capacity. In particular, if they want to emphasize the dependence of the heat capacity of the working fluid on temperature, then it is written as C (T), and the specific - as C (T). Usually, a true heat capacity is understood to understand the ratio of the elementary amount of heat, which is reported to the thermodynamic system in any process to an infinitely small increment of the temperature of this system caused by the trimmed heat. We will consider C (T) the true heat capacity of the thermodynamic system at a system temperature of T 1, and C (T) - the true specific heat capacity of the working fluid at its temperature is equal to T 2. Then the average specific capacity of the working fluid when it changes its temperature from T 1 to T 2 can be defined as

Typically, the tables are given the average values \u200b\u200bof the heat capacity C CP for different temperature ranges starting with T 1 \u003d 0 C. Therefore, in all cases where the thermodynamic process passes in the temperature range from T 1 to T 2, in which T 1 ≠ 0, number The specific heat of the Q of the process is determined using the table values \u200b\u200bof the average heat-capacity C CP as follows.

the value if it is determined in different parts of AB, AC, AD process AB, then This shows that in certain sections of the process, on which the temperature changes to 1 ° C, various amounts of heat are consumed. Therefore, the above formula does not determine the actual specific consumption of heat, but shows only how much heat average in the process AB is reported when the gas is heated by 1 o C.

Medium heat - The ratio of heat, communicated by the gas, to the change in its temperature, provided that the temperature difference is the ultimate magnitude. Under true heat capacity Gas understand the limit to which the average heat capacity seeks Δt. to zero. So, if in the process AA, the average heat capacity is the true heat capacity at the initial state A:

Hence, true heat capacity It is called the ratio of heat transmitted by gas in the process, to change its temperature, provided that the temperature difference is disappearing.

General formulas warmth. From the above formulas it follows that the heat transmitted gas in an arbitrary process can be determined by the formula:

or for arbitrary amount of gas

where - the average heat capacity in the process under consideration when it is changed. T 1. before T 2.. The warmth can also be determined by formulas:

where C is the true heat capacity.

Formulas for medium and true heat capacity. The heat capacity of real gases depends on pressure and temperature. Pressure dependence is often neglected. Dependence on temperature is significant and on the basis of experimental data is expressed by the equation of the species where a, B, D - Numeric coefficients depending on the nature of the gas and the nature of the process.

Specific heat:

The heat capacity referred to 1 kg of gas is called weight heat -. The heat capacity attributed to 1 m 3 of gas is called volume heat capacity - 3. The heat capacity attributed to 1 praying gas is called moldable heat capacity – .

Suppose for heating 1 kg of gas to 1 o C, it is necessary to joule heat. Because The mole contains a kilogram of gas, then for heating 1 praying 1 o C, it is necessary to more heat, i.e.

Now for heating 1 m 3 of gas to 1 o C, it is necessary to joule heat. Because Under normal conditions, 22.4 m 3 gas is contained, then for heating 1 praying 1 o, it is necessary to 22.4 times more, heat:

Comparing formulas (a) and (b), we will find the relationship between the weight and volumetric heat-strokes:

The dependence of heat capacity from the nature of the process. Consider two heat supply processes to gas:

a) heat is supplied to 1 kg of gas enclosed in a fixed piston cylinder (Fig. 5). Heat reported to gas will be equal , where - heat capacity when; and - initial and final gas temperature. When the temperature difference, we get that. Obviously, everything warm in this case will go to an increase in the internal energy of the gas.

Fig. 5. Fig. 6.

b) heat is supplied to 1 kg of gas enclosed in a cylinder with a movable piston (Fig. 6) and, in this case, will be equal , where - heat capacity when; and - initial and final gas temperature at. When we get that. In this case, the heated heat went to gas on an increase in the internal energy of gas (as in the first case), as well as to perform work when the piston moves. Therefore, to increase the temperature of 1 kg of gas to 1 ° C in the second case, more heat is needed than in the first, i.e. .

Considering other processes, it can be established that the heat capacity can take a variety of numeric values, because the amount of heat reported by gas depends on the nature of the process.

Connection between and , coefficient . When he heated 1 kg of gas to 1 ° C with when they are supplied. Part of it, equal, goes to an increase in internal energy, and part - to perform the work of expansion. Denote this work through. Because The heat spent on the heating of the gas and the performance must be in the amount equal to the suspended heat, then it can be written that

This is the amount of heat that must be reported to the system to increase its temperature by 1 ( TO) In the absence of useful work and the constancy of the corresponding parameters.

If as a system we take an individual substance, then general heat capacity system The heat capacity of 1 mol of substance () multiplied by the number of mole () is equal.

The heat capacity may be specific and molar.

Specific heat- This is the amount of heat required to heat the mass of the mass of the substance by 1 grad. (intensive value).

Molar heat capacity- this is the amount of heat required to heat one mole of substance by 1 grad..

Distinguish between true and average heat capacity.

The technique usually use the concept of medium heat capacity.

Average- This is a heat capacity for a certain temperature range.

If the system containing the amount of substance or mass is reported, the amount of heat, and the temperature of the system increased from before, then the average specific or molar heat capacity can be calculated:

True molar heat capacity - This is the ratio of an infinitely small amount of heat, reported 1 mol of substance at a certain temperature, to the increment of temperature, which is observed.

According to equation (19), heat capacity, as well as heat, is not a function of the state. With constant pressure or volume, according to equations (11) and (12), heat, and, therefore, the heat capacity acquire the properties of the state function, that is, become characteristic functions of the system. Thus, we obtain isoormal and isobaric heat capacity.

Isoormal heat - The amount of heat that must be reported to the system to increase the temperature to 1 if the process occurs at.

Osobaric heat capacity - The amount of heat that must be reported to the system to increase the temperature by 1 at.

The heat capacity depends not only on temperature, but also on the volume of the system, since between particles there are interaction forces that change when the distance between them changes, therefore private derivatives are used in equations (20) and (21).

The enthalpy of the ideal gas, as well as its internal energy, is the function only temperature:

and in accordance with the Mendeleev-Klapairone equation, then

Therefore, for the perfect gas in equations (20), (21), private derivatives can be replaced with complete differentials:

From the joint solution of equations (23) and (24), taking into account (22), we obtain the relationship between and for the perfect gas.

Sharing the variables in equations (23) and (24), it is possible to calculate the change in internal energy and enthalpy when heated 1 mol of perfect gas from temperature to

If at the specified temperature range, the heat capacity can be considered constant, then as a result of the integration, we obtain:

We establish the relationship between the average and true heat capacity. The change in entropy on one side is expressed by the equation (27), on the other -

Equating the right parts of the equations and expressing the average heat capacity, we have:

A similar expression can be obtained for medium isochoric heat capacity.

The heat capacity of most solid, liquid and gaseous substances increases with increasing temperature. The dependence of the heat capacity of solid, liquid and gaseous substances on temperature is expressed by an empirical equation of the form:

where but, b., c. and - empirical coefficients calculated on the basis of experimental data O, the coefficient refers to organic substances, and - to inorganic. The values \u200b\u200bof the coefficients for various substances are given in the directory and apply only for the specified temperature range.

The heat capacity of the ideal gas does not depend on temperature. According to the molecular-kinetic theory, the heat capacity coming on one degree of freedom is equal to (the degree of freedom is the number of independent modes for which the complex movement of the molecule can be decomposed). For a one-cattle molecule, a translational movement is characterized, which can be decomposed into three components in accordance with three mutually perpendicular directions on three axes. Therefore, the isoormal heat capacity of the same ideal gas is equal to

Then the isobaric heat capacity of the monoomic ideal gas according to (25) is determined by the equation

Die the perfect gas molecules in addition to the three degrees of freedom of translational movement have 2 degrees of freedom of rotational motion. Hence.

Considering that the heat capacity is inconstant, and depends on the temperature and other thermal parameters, distinguish between the true and average heat capacity. True heat capacity is expressed by equation (2.2) with certain parameters of the thermodynamic process, that is, in this state of the working fluid. In particular, if they want to emphasize the dependence of the heat capacity of the working fluid on temperature, they write it as, and the specific one - as. Usually, a true heat capacity is understood to understand the ratio of the elementary amount of heat, which is reported to the thermodynamic system in any process to an infinitely small increment of the temperature of this system caused by the trimmed heat. We will consider the heat capacity of the thermodynamic system at the temperature of the system equal, the actual specific heat capacity of the working fluid at its temperature equal. Then the average specific capacity of the working fluid when it changes its temperature to be able to determine how

Typically, the tables are given averages of heat capacity for different temperature ranges starting with. Therefore, in all cases, when the thermodynamic process passes in the temperature range, in which, the amount of specific heat from the process is determined using the table values \u200b\u200bof the average heat chamolesee:

The values \u200b\u200bof the average heat capacity and are found along tables.

2.3.teralness with constant volume and pressure

Of particular interest are the average and true heat capacity in the processes at a constant volume ( isoormal heatequal to the ratio of the specific amount of heat in isochoretum to changes in the temperature of the working fluid DT) and at constant pressure ( osobaric heat capacityequal to the ratio of the specific amount of heat in the isobaric process to changes in the temperature of the DT working fluid).

For ideal gases, the relationship between the isobaric and isochorean heatakes is established by the well-known major equation.

From the Mayer equation, it follows that the isobaric heat capacity is greater than isochorny to the value of the specific characteristic constant of the perfect gas. This is due to the fact that in isohorce () external work is not performed and the heat is consumed only to change the internal energy of the working fluid, while in the isobaric process () of the heat is consumed not only to change the internal energy of the working fluid, depending on its temperature, but also on the commitment of their external work.

For real gases, because when they are expanding, work is as if the work not only against external forces, but also internal work against the strength of the interaction between gas molecules, to which is additionally spent heat consuming.

In the heat engineering, the ratio of heat-capacity, which is called the Poisson coefficient (adiabuding indicator). In tab. 2.1 are valid values \u200b\u200bobtained experimentally at a temperature of 15 ° C.

The heat capacity is inhabited by temperature, therefore, the indicator is adiabybently dependent on temperature.

It is known that with an increase in temperature, the heat capacity increases. Therefore, with increasing temperatures, approaching one. However, there is always more units. Typically, the dependence of the adiabatic temperature is expressed by the formula

and since