Goal of the work
Experimentally determine the values of the average heat capacity of air in the temperature range from t 1 to t 2, establish the dependence of the heat capacity of air on temperature.
1. Determine the power spent on gas heating from t 1
before t 2 .
2. Fix the values of air flow in a given time interval.
Lab Preparation Guidelines
1. Work through the section of the course “Heat capacity” according to the recommended literature.
2. Familiarize yourself with this methodological guide.
3. Prepare protocols for laboratory work, including the necessary theoretical material related to this work (calculation formulas, diagrams, graphs).
Theoretical introduction
Heat capacity- the most important thermophysical quantity, which is directly or indirectly included in all heat engineering calculations.
Heat capacity characterizes the thermophysical properties of a substance and depends on the molecular weight of the gas μ , temperature t, pressure R, the number of degrees of freedom of the molecule i, from the process in which heat is supplied or removed p = const, v =const. The heat capacity depends most significantly on the molecular weight of the gas μ . So, for example, the heat capacity for some gases and solids is
So the less μ , the less substance is contained in one kilomol and the more heat must be supplied to change the temperature of the gas by 1 K. That is why hydrogen is a more efficient coolant than, for example, air.
Numerically, heat capacity is defined as the amount of heat that must be brought to 1 kg(or 1 m 3), a substance to change its temperature by 1 K.
Since the amount of heat supplied dq depends on the nature of the process, then the heat capacity also depends on the nature of the process. The same system in different thermodynamic processes has different heat capacities - cp, cv, c n. Of greatest practical importance are cp And cv.
According to the molecular-kinematic theory of gases (MKT), for a given process, the heat capacity depends only on the molecular weight. For example, heat capacity cp And cv can be defined as
For air ( k = 1,4; R = 0,287 kJ/(kg· TO))
kJ/kg
For a given ideal gas, the heat capacity depends only on temperature, i.e.
The heat capacity of the body in this process called the ratio of heat dq obtained by the body with an infinitesimal change in its state to a change in body temperature by dt
True and average heat capacity
Under the true heat capacity of the working fluid is understood:
The true heat capacity expresses the value of the heat capacity of the working fluid at a point for given parameters.
The amount of transferred heat. expressed through the true heat capacity, can be calculated by the equation
Distinguish:
Linear dependence of heat capacity on temperature
Where A- heat capacity at t= 0 °С;
b = tgα - slope factor.
Nonlinear dependence of heat capacity on temperature.
For example, for oxygen, the equation is written as
kJ/(kg K)
Under medium heat capacity with t understand the ratio of the amount of heat in process 1-2 to the corresponding change in temperature
kJ/(kg K)
The average heat capacity is calculated as:
Where t = t 1 + t 2 .
Calculation of heat according to the equation
difficult, since the tables give the value of heat capacity. Therefore, the heat capacity in the range from t 1 to t 2 must be determined by the formula
.
If the temperature t 1 and t 2 is determined experimentally, then for m kg gas, the amount of heat transferred should be calculated according to the equation
Medium with t And With the true heat capacities are related by the equation:
For most gases, the higher the temperature t, the higher the heat capacity with v , with p. Physically, this means that the hotter the gas, the more difficult it is to heat it further.
Heat capacity is a thermophysical characteristic that determines the ability of bodies to give or receive heat in order to change the body temperature. The ratio of the amount of heat supplied (or removed) in a given process to a change in temperature is called the heat capacity of the body (system of bodies): C = dQ / dT, where is the elementary amount of heat; - an elementary change in temperature.
The heat capacity is numerically equal to the amount of heat that must be supplied to the system in order to increase its temperature by 1 degree under given conditions. The unit of heat capacity is J/K.
Depending on the quantitative unit of the body to which heat is supplied in thermodynamics, mass, volume and molar heat capacities are distinguished.
Mass heat capacity is the heat capacity per unit mass of the working fluid, c \u003d C / m
The unit of mass heat capacity is J/(kg×K). Mass heat capacity is also called specific heat capacity.
Volumetric heat capacity is the heat capacity per unit volume of the working fluid, where and are the volume and density of the body under normal physical conditions. C'=c/V=c p . Volumetric heat capacity is measured in J / (m 3 × K).
Molar heat capacity - heat capacity, related to the amount of the working fluid (gas) in moles, C m = C / n, where n is the amount of gas in moles.
Molar heat capacity is measured in J/(mol×K).
Mass and molar heat capacities are related by the following relation:
The volumetric heat capacity of gases is expressed in terms of molar as
Where m 3 / mol is the molar volume of gas under normal conditions.
Mayer equation: C p - C v \u003d R.
Considering that the heat capacity is not constant, but depends on temperature and other thermal parameters, a distinction is made between true and average heat capacity. In particular, if you want to emphasize the dependence of the heat capacity of the working fluid on temperature, then write it as C(t), and specific - as c(t). Usually, true heat capacity is understood as the ratio of the elementary amount of heat that is reported to a thermodynamic system in any process to an infinitesimal increase in the temperature of this system caused by the imparted heat. We will consider C(t) the true heat capacity of the thermodynamic system at a system temperature equal to t 1 , and c(t) - the true specific heat capacity of the working fluid at its temperature equal to t 2 . Then the average specific heat of the working fluid when its temperature changes from t 1 to t 2 can be determined as
Usually, the tables give the average values of the heat capacity c cf for various temperature intervals, starting from t 1 \u003d 0 0 C. Therefore, in all cases when the thermodynamic process takes place in the temperature range from t 1 to t 2, in which t 1 ≠ 0, the amount The specific heat q of the process is determined using the tabular values of the average heat capacities c cf as follows.
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 we take an individual substance as a system, then total heat capacity of the system equals the heat capacity of 1 mole of a substance () times the number of moles ().
Heat capacity can be specific and molar.
Specific heat is the amount of heat required to raise the temperature of a unit mass of a substance by 1 hail(intense value).
Molar heat capacity is the amount of heat required to raise the temperature of one mole of a substance by 1 hail.
Distinguish between true and average heat capacity.
In engineering, the concept of average heat capacity is usually used.
Medium is the heat capacity for a certain temperature range.
If a system containing an amount of a substance or a mass was told the amount of heat , and the temperature of the system increased from to , then you can calculate the average specific or molar heat capacity:
True molar heat capacity- this is the ratio of an infinitesimal amount of heat imparted by 1 mole of a substance at a certain temperature to the temperature increase that is observed in this case.
According to equation (19), heat capacity, like heat, is not a state function. At constant pressure or volume, according to equations (11) and (12), heat, and, consequently, heat capacity acquire the properties of a state function, that is, they become characteristic functions of the system. Thus, we obtain isochoric and isobaric heat capacities.
Isochoric heat capacity- the amount of heat that must be reported to the system in order to increase the temperature by 1 if the process occurs at .
Isobaric heat capacity- the amount of heat that must be reported to the system in order to increase the temperature by 1 at .
The heat capacity depends not only on temperature, but also on the volume of the system, since there are interaction forces between particles that change with a change in the distance between them, therefore partial derivatives are used in equations (20) and (21).
The enthalpy of an ideal gas, like its internal energy, is a function of temperature only:
and in accordance with the Mendeleev-Clapeyron equation, then
Therefore, for an ideal gas in equations (20), (21), partial derivatives can be replaced by total differentials:
From the joint solution of equations (23) and (24), taking into account (22), we obtain the equation of the relationship between and for an ideal gas.
By dividing the variables in equations (23) and (24), we can calculate the change in internal energy and enthalpy when 1 mole of an ideal gas is heated from temperature to
If the heat capacity can be considered constant in the indicated temperature range, then as a result of integration we obtain:
Let us establish the relationship between the average and true heat capacity. The change in entropy, on the one hand, is expressed by equation (27), on the other hand,
Equating the right parts of the equations and expressing the average heat capacity, we have:
A similar expression can be obtained for the average 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 A, b, c and - empirical coefficients calculated on the basis of experimental data on , and the coefficient refers to organic substances, and - to inorganic. The values of the coefficients for various substances are given in the handbook and are applicable only for the specified temperature range.
The heat capacity of an ideal gas does not depend on temperature. According to the molecular kinetic theory, the heat capacity per one degree of freedom is equal to (the degree of freedom is the number of independent types of motion into which the complex motion of a molecule can be decomposed). A monatomic molecule is characterized by translational motion, which can be decomposed into three components in accordance with three mutually perpendicular directions along three axes. Therefore, the isochoric heat capacity of a monatomic ideal gas is
Then the isobaric heat capacity of a monatomic ideal gas according to (25) is determined by the equation
Diatomic molecules of an ideal gas, in addition to three degrees of freedom of translational motion, also have 2 degrees of freedom of rotational motion. Hence.
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.
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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.
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.
The internal energy of the system can change as a result of heat transfer. That is, if heat is supplied to the system in the amount of dQ, and work is not performed dW = 0, then according to the I law of thermodynamics
dU = dQ – dW = dQ
Heat - a way to change the internal energy of the system without changing the external parameters (dV = 0 ® dW = 0), this microscopic way to convert energy.
When a certain amount of heat dQ is absorbed by the system, its internal energy increases by dU (according to formula (6.32.)). An increase in internal energy leads to an increase in the intensity of the movement of the particles that make up the system. According to the findings of statistical physics, the average speed of molecules is related to temperature
Those. absorption by the system of a certain amount of heat dQ leads to an increase in the temperature of the system by an amount dT proportional to dQ.
dT = const . dQ (6.33)
Relation (6.33) can be rewritten in another form:
dQ=C. dT or , (6.34)
where C is a constant called heat capacity systems.
So, heat capacity - this is the amount of heat required to heat the thermodynamic system by one degree on the Kelvin scale.
The heat capacity of the system depends on:
a) the composition and temperature of the system;
b) system size;
c) the conditions under which the transfer of heat occurs.
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Scheme 6.6. Types of heat capacity
Those. C (heat capacity), like Q, is a function of the process, not a state, and refers to extensive parameters.
According to the amount of the heated substance, they distinguish:
1) specific heat C sp, referred to 1 kg or 1 g of a substance;
2) molar (molar) heat capacity C m, referred to 1 mol of a substance.
Dimension (C beat) = J / g. TO
(C m) \u003d J / mol. TO
There is a relationship between specific and molar heat capacities
C m \u003d C beats. M, (6.35)
where M is the molar mass.
When describing physical and chemical processes, the molar heat capacity C m is usually used (we will not write the index in the future).
There are also middle And true heat capacity.
Average heat capacity is the ratio of a certain amount of heat to the temperature difference
(6.36)
True heat capacity C called the ratio of an infinitesimal amount of heat dQ, which must be brought to one mole of a substance, to an infinitesimal temperature increment - dT.
Let us establish the relationship between the true and average heat capacities.
Firstly,
Second, we express Q from formula (6.36) 
(6.37). On the other hand, from formula (6.34) ® dQ = CdT (6.38). We integrate (6.38) in the interval T 1 - T 2 and obtain
Equate the right parts of expressions (6.37) and (6.39)
From here 
(6.40)
This equation relates the average heat capacity to the true C.
The average heat capacity is calculated in the temperature range from T 1 to T 2 . Often the interval is chosen from OK to T, i.e. the lower limit T 1 = OK, and the upper has a variable value, i.e. from a certain interval we pass to an indefinite one. Then equation (6.40) takes the form:
The calculation can be carried out graphically if the values of the true heat capacity at several temperatures are known. The dependence C = f(T) is represented by curve AB in Fig. 1. 1.
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Rice. 6.7. Graphical definition of the average heat capacity
The integral in expression (6.40) is the area of the figure T 1 ABT 2.
Thus, by measuring the area, we determine
(6.42)
Consider the value of the heat capacity of the system under certain conditions:
According to the I law of thermodynamics dQ V = dU. For simple systems, internal energy is a function of volume and temperature U = U (V,T)
Heat capacity under these conditions
(6.43)
dQ p = dH. For simple systems H = H(p,T);
Heat capacity
(6.44)
C p and C V - heat capacities at constant p and V.
If we consider 1 mole of a substance, i.e. C p and C V - molar heat capacities
dQ V = C V dT, dQ p = C p dT (6.45)
For "n" mole of substance dQ V = nC V dT, dQ p = nC p dT
Based on expression (6.45), we find
(6.46)
Knowing the dependence of the heat capacity of a substance on temperature, according to formula (6.46), one can calculate the change in the enthalpy of the system in the interval T 1 ¸T 2. T 1 = OK or 298.15 K is chosen as the base temperature. In this case, the difference in enthalpies H (T) - H (298) is called the high-temperature component of enthalpy.
Let's find the connection between С р and С V . From expressions (6.43) and (6.44) we can write:
From the I law of thermodynamics, taking into account only mechanical work for a simple system, for which U = U(V,T)
dQ = dU + pdV = 
those. 
(6.49)
Substitute dQ from expression (6.46) into (6.48) and (6.49) and get:
For a simple system, the volume can be considered as a function of pressure and temperature, i.e.
V = V(p,T) ® dV = 
under the condition p = const dp = 0, 
those. 
From here 
,
Thus 
(6.51)
For 1 mole of ideal gas pV = RT,
C p – C V =
For 1 mole of real gas and applying the van der Waals equation leads to the following expression:
C p – C V = 
For real gases C p - C V > R. This difference increases with increasing pressure, since with increasing pressure increases , associated with the interaction of real gas molecules with each other.
For a solid at ordinary temperature C p – C V< R и составляет примерно 1 Дж/(моль. К). с понижением температуры разность С p – C V уменьшается и при Т ® ОК С p – C V ® 0.
The heat capacity has the property of additivity, i.e. heat capacity of a mixture of two substances
(6.52)
In general
,
where x i - the proportion of substances "I" in the mixture.
Heat capacity is one of the most important thermodynamic characteristics of individual substances.
Currently, there are accurate methods for measuring heat capacity in a wide temperature range. The theory of heat capacity for a simple solid at low pressures has been developed quite satisfactorily. According to the molecular kinetic theory of heat capacity, for one mole of gas, there is R/2 for each degree of freedom. Those. since the molar heat capacity of an ideal gas at constant volume
C V \u003d C n + C in + C to + C e, (6.53)
where C n is the heat capacity of the gas associated with the translational motion of molecules,
From in - with rotational,
C to - with oscillatory,
and С e - with electronic transitions, then for a monatomic ideal gas С V = 3/2R,
for diatomic and linear triatomic molecules
C V \u003d 5 / 2R + C to
for nonlinear polyatomic molecules
C V \u003d 3R + C to
The heat capacity C k, associated with the oscillatory motion of atoms in a molecule, obeys the laws of quantum mechanics and does not correspond to the law of uniform distribution of energy over degrees of freedom.
C e in formula (6.53) is not taken into account, C e is the heat capacity associated with electronic transitions in the molecule. The transition of electrons to a higher level under the action of heat transfer is possible only at temperatures above 2000 K.
The heat capacity of solids with an atomic crystal lattice can be calculated using the Debye equation:
C V \u003d C D (x), 
,
where q is the characteristic temperature;
n m is the maximum characteristic vibration frequency of atoms in a molecule.
As the temperature rises, C V of solids with an atomic crystal lattice tends to the limit value C V ® 3R. At very low temperatures
C V ~ T 3 (T< q/12).
The heat capacities C p according to the experimental values of C V (or vice versa) for substances with an atomic crystal lattice can be calculated using the equation:
C p \u003d C V (1 + 0.0214C V )
For a complex solid or liquid substance, a good theory does not yet exist. If experimental data on heat capacity are not available, then it can be estimated using empirical rules
1) Dulong and Petit's rule: atomic heat capacity at constant volume for any simple solid is approximately 25 J/(mol K)
The rule holds at high temperatures (close to the melting point of the solid) for elements whose atomic mass is greater than that of potassium. As Boltzmann showed, it can be qualitatively substantiated by the kinetic theory:
C V » 25 J/(mol. K)(3R)
2) The Neumann-Kopp rule (additivity rule) is based on the assumption of the invariability of the heat capacity of elements during the formation of chemical bonds
With sv-va \u003d 25n
where n is the number of atoms in the molecule.
Heat capacities closer to the experimental values are obtained according to the Neumann-Kopp rule, if we take the values of atomic heat capacities presented in Table 1 for light elements. 6.1.
Table 6.1.
Atomic heat capacities for light elements
For other elements, C p 0 » 25.94 J/(mol. K).
3) The additivity rule underlies the Kelly formula, which is valid for high-boiling pure inorganic liquids (BeO, BeCl 2 , MgBr 2, etc.):
where n is the number of atoms in the molecule that make up the molecule of the inorganic substance.
For molten elements with d- and f-electrons, C at reaches 42¸50 J / (mol. K).
4) Approximate calculation method for organic liquids, using atomic-group components of heat capacities
The latter were obtained by analyzing the experimental data of a large number of compounds, some of which are summarized in Table. 6.2.
Table 6.2.
Some values of the atomic group components of the heat capacities
| atom or group | C p, J / (mol. K) | atom or group | C p, J / (mol. K) |
| –CH 3 | 41,32 | -ABOUT- | 35,02 |
| -CH 2 - | 26,44 | -S- | 44,35 |
| CH– | 22,68 | –Cl | 35,98 |
| –CN | 58,16 | –Br | 15,48 |
| –OH 2 | 46,02 | C 6 H 5 - | 127,61 |
| C=O(esters) | 60,75 | –NH 2 (amines) | 63,6 |
| C=O(ketones) | 61,5 | –NO 2 | 64,02 |
Dependence of heat capacity on temperature
The heat capacity of solid, liquid and gaseous substances increases with temperature. Only the heat capacities of monatomic gases are practically independent of T (for example, He, Ar and other noble gases). The most complex C(T) dependence is observed for a solid. Dependence С(Т) is studied experimentally, because theory is not well developed.
Usually, the dependence of atomic and molar heat capacities on temperature is expressed in the form of interpolation equations.
C p \u003d a + b. T + s. T 2 (for organic substances) (6.53)
C p \u003d a + b. T + c / . T -2 (for inorganic substances)
The coefficients a, b, c, c / - constant values characteristic of a given substance are calculated on the basis of experimental data and are valid in a certain temperature range.
