Objective
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 is needed 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.
The experimental values of heat capacities at various temperatures are presented in the form of tables, graphs and empirical functions.
Distinguish between true and average heat capacity.
The true heat capacity C is the heat capacity for a given temperature.
In engineering calculations, the average value of heat capacity in a given temperature range (t1;t2) is often used.
The average heat capacity is denoted in two ways: ,.
The disadvantage of the latter notation is that the temperature range is not specified.
The true and average heat capacities are related by the relation:
The true heat capacity is the limit to which the average heat capacity tends, in a given temperature range t1…t2, at ∆t=t2-t1
As experience shows, for most gases, the true heat capacities increase with increasing temperature. The physical explanation for this increase is as follows:
It is known that the gas temperature is not related to the oscillatory motion of atoms and molecules, but depends on the kinetic energy E k of the translational motion of particles. But as the temperature rises, the heat supplied to the gas is more and more redistributed in favor of oscillatory motion, i.e. the temperature rise with the same heat input slows down as the temperature rises.
Typical dependence of heat capacity on temperature:
c=c 0 + at + bt 2 + dt 3 + … (82) 
where c 0 , a, b, d are empirical coefficients.
c - True heat capacity, i.e. heat capacity value for a given temperature T.
For the heat capacity of the bitoptimizing curve, this is a polynomial in the form of a series in powers of t.
The fitting curve is carried out using special methods, for example, the least squares method. 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, they are limited to the first two terms on the right side, i.e. the dependence of heat capacity on temperature is assumed to be linear c=c 0 + at (83)
The average heat capacity is graphically defined as the middle line of the shaded trapezoid, as you know, the middle line of the trapezoid is defined as half the sum of the bases.
Formulas are applied if the empirical dependence is known.
In cases where the dependence of heat capacity on temperature cannot be satisfactorily approximated to the dependence c \u003d c 0 + at, the following formula can be used:
This formula is used in cases where the dependence of c on t is essentially non-linear.
From the molecular kinetic theory of gases, it is known
U \u003d 12.56T, U - internal energy of one kilomole of ideal gas.
Previously obtained for an ideal gas:
,
,
It follows from the result obtained that the heat capacity obtained using the MCT does not depend on temperature.
Mayer equation: c p -c v =R ,
c p \u003d c v + R \u003d 12.56 + 8.314 20.93.
As in the previous case, according to the MKT of gases, the molecular isobaric heat capacity does not depend on temperature.
The concept of an ideal gas is most consistent with monatomic gases at low pressures; in practice, one has to deal with 2, 3 ... atomic gases. For example, air, which by volume consists of 79% nitrogen (N 2), 21% oxygen (O 2) (in engineering calculations, inert gases are not taken into account due to the smallness of their content).
You can use the following table for estimates:
|
monatomic | ||
|
diatomic | ||
|
triatomic |
In real gases, in contrast to the ideal, the heat capacities can depend not only on temperature, but also on the volume and pressure of the system.
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 O-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 O-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 the 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
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=
.
As N=
vnN A
,
then 
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 |
||
Heat capacity is the ratio of the amount of heat imparted to the system to the temperature increase observed in this case (in the absence of a chemical reaction, the transition of a substance from one state of aggregation to another and at A "= 0.)
Heat capacity is usually calculated per 1 g of mass, then it is called specific (J / g * K), or per 1 mol (J / mol * K), then it is called molar.
Distinguish average and true heat capacity.
Middle heat capacity is the heat capacity in the temperature range, i.e. the ratio of the heat imparted to the body to the increment in its temperature by ΔT
True The heat capacity of a body is the ratio of an infinitesimal amount of heat received by the body to the corresponding increase in its temperature.
It is easy to establish a connection between the average and true heat capacity:
substituting the values of Q into the expression for the average heat capacity, we have:
The true heat capacity depends on the nature of the substance, the temperature, and the conditions under which heat is transferred to the system.
So, if the system is enclosed in a constant volume, i.e. for isochoric process we have:
If the system expands or contracts while the pressure remains constant, i.e. for isobaric process we have:
But ΔQ V = dU, and ΔQ P = dH, therefore
C V = (∂U/∂T) v , and C P = (∂H/∂T) p
(if one or more variables are held constant while others change, then the derivatives are said to be partial with respect to the changing variable).
Both ratios are valid for any substances and any states of aggregation. To show the relationship between C V and C P, it is necessary to differentiate the expression for the enthalpy H \u003d U + pV /
For an ideal gas pV=nRT
for one mole or
The difference R is the work of the isobaric expansion of 1 mole of an ideal gas as the temperature rises by one unit.
For liquids and solids, due to a small change in volume when heated, С P = С V
Dependence of the thermal effect of a chemical reaction on temperature, Kirchhoff's equations.
Using Hess's law, one can calculate the thermal effect of a reaction at the temperature (usually 298K) at which the standard heats of formation or combustion of all participants in the reaction are measured.
But more often it is necessary to know the thermal effect of a reaction at different temperatures.
Consider the reaction:
ν A A+ν B B= ν C С+ν D D
Let us denote by H the enthalpy of the participant in the reaction per 1 mole. The total change in the enthalpy ΔΗ (T) of the reaction is expressed by the equation:
ΔΗ \u003d (ν C H C + ν D H D) - (ν A H A + ν B H B); va, vb, vc, vd - stoichiometric coefficients. x.r.
If the reaction proceeds at constant pressure, then the change in enthalpy will be equal to the heat effect of the reaction. And if we differentiate this equation with respect to temperature, we get:
Equations for isobaric and isochoric process
and 
called Kirchhoff equations(in differential form). They allow qualitatively evaluate the dependence of the thermal effect on temperature.
The influence of temperature on the thermal effect is determined by the sign of the value ΔС p (or ΔС V)
At ∆С p > 0 value , that is, with increasing temperature thermal effect increases
at ∆С p< 0 that is, as the temperature increases, the thermal effect decreases.
at ∆С p = 0- thermal effect of the reaction does not depend on temperature
That is, as follows from this, ΔС p determines the sign in front of ΔН.
Heat capacity is the ability to absorb some amount of heat during heating or give it away when cooled. The heat capacity of a body is the ratio of an infinitesimal amount of heat that a body receives to the corresponding increase in its temperature indicators. The value is measured in J/K. In practice, a slightly different value is used - specific heat capacity.
Definition
What does specific heat capacity mean? This is a quantity related to a single amount of a substance. Accordingly, the amount of a substance can be measured in cubic meters, kilograms, or even in moles. What does it depend on? In physics, the heat capacity depends directly on which quantitative unit it refers to, which means that they distinguish between molar, mass and volumetric heat capacity. In the construction industry, you will not meet with molar measurements, but with others - all the time.
What affects specific heat capacity?
You know what heat capacity is, but what values \u200b\u200baffect the indicator is not yet clear. The value of specific heat capacity is directly affected by several components: the temperature of the substance, pressure and other thermodynamic characteristics.
As the temperature of the product rises, its specific heat capacity increases, however, certain substances differ in a completely non-linear curve in this dependence. For example, with an increase in temperature indicators from zero to thirty-seven degrees, the specific heat capacity of water begins to decrease, and if the limit is between thirty-seven and one hundred degrees, then the indicator, on the contrary, will increase.
It is worth noting that the parameter also depends on how the thermodynamic characteristics of the product (pressure, volume, and so on) are allowed to change. For example, the specific heat at a stable pressure and at a stable volume will be different.
How to calculate the parameter?
Are you interested in what is the heat capacity? The calculation formula is as follows: C \u003d Q / (m ΔT). What are these values? Q is the amount of heat that the product receives when heated (or released by the product during cooling). m is the mass of the product, and ΔT is the difference between the final and initial temperatures of the product. Below is a table of the heat capacity of some materials.
What can be said about the calculation of heat capacity?
Calculating the heat capacity is not an easy task, especially if only thermodynamic methods are used, it is impossible to do it more precisely. Therefore, physicists use the methods of statistical physics or knowledge of the microstructure of products. How to calculate for gas? The heat capacity of a gas is calculated from the calculation of the average energy of thermal motion of individual molecules in a substance. The movements of molecules can be of a translational and rotational type, and inside a molecule there can be a whole atom or vibration of atoms. Classical statistics says that for each degree of freedom of rotational and translational movements, there is a molar value, which is equal to R / 2, and for each vibrational degree of freedom, the value is equal to R. This rule is also called the equipartition law.
In this case, a particle of a monatomic gas differs by only three translational degrees of freedom, and therefore its heat capacity should be equal to 3R/2, which is in excellent agreement with experiment. Each diatomic gas molecule has three translational, two rotational and one vibrational degrees of freedom, which means that the equipartition law will be 7R/2, and experience has shown that the heat capacity of a mole of a diatomic gas at ordinary temperature is 5R/2. Why was there such a discrepancy in theory? Everything is due to the fact that when establishing the heat capacity, it will be necessary to take into account various quantum effects, in other words, to use quantum statistics. As you can see, heat capacity is a rather complicated concept.
Quantum mechanics says that any system of particles that oscillate or rotate, including a gas molecule, can have certain discrete energy values. If the energy of thermal motion in the installed system is insufficient to excite oscillations of the required frequency, then these oscillations do not contribute to the heat capacity of the system.
In solids, the thermal motion of atoms is a weak oscillation around certain equilibrium positions, this applies to the nodes of the crystal lattice. An atom has three vibrational degrees of freedom and, according to the law, the molar heat capacity of a solid body is equal to 3nR, where n is the number of atoms present in the molecule. In practice, this value is the limit to which the heat capacity of the body tends at high temperatures. The value is achieved with normal temperature changes in many elements, this applies to metals, as well as simple compounds. The heat capacity of lead and other substances is also determined.
What can be said about low temperatures?
We already know what heat capacity is, but if we talk about low temperatures, how will the value be calculated then? If we are talking about low temperature indicators, then the heat capacity of a solid body then turns out to be proportional T 3 or the so-called Debye's law of heat capacity. The main criterion for distinguishing high temperatures from low ones is the usual comparison of them with a parameter characteristic of a particular substance - this can be the characteristic or Debye temperature q D . The presented value is set by the vibration spectrum of atoms in the product and depends significantly on the crystal structure.
In metals, conduction electrons make a certain contribution to the heat capacity. This part of the heat capacity is calculated using the Fermi-Dirac statistics, which takes electrons into account. The electronic heat capacity of a metal, which is proportional to the usual heat capacity, is a relatively small value, and it contributes to the heat capacity of the metal only at temperatures close to absolute zero. Then the lattice heat capacity becomes very small and can be neglected.
Mass heat capacity
Mass specific heat capacity is the amount of heat that is required to be brought to a unit mass of a substance in order to heat the product per unit temperature. This value is denoted by the letter C and it is measured in joules divided by a kilogram per kelvin - J / (kg K). This is all that concerns the heat capacity of the mass.
What is volumetric heat capacity?
Volumetric heat capacity is a certain amount of heat that needs to be brought to a unit volume of production in order to heat it per unit temperature. This indicator is measured in joules divided by a cubic meter per kelvin or J / (m³ K). In many building reference books, it is the mass specific heat capacity in work that is considered.
Practical application of heat capacity in the construction industry
Many heat-intensive materials are actively used in the construction of heat-resistant walls. This is extremely important for houses that are characterized by periodic heating. For example, oven. Heat-intensive products and walls built from them perfectly accumulate heat, store it during heating periods of time and gradually release heat after the system is turned off, thus allowing you to maintain an acceptable temperature throughout the day.
So, the more heat is stored in the structure, the more comfortable and stable the temperature in the rooms will be.
It should be noted that ordinary brick and concrete used in housing construction have a significantly lower heat capacity than expanded polystyrene. If we take ecowool, then it is three times more heat-consuming than concrete. It should be noted that in the formula for calculating the heat capacity, it is not in vain that there is mass. Due to the large huge mass of concrete or brick, in comparison with ecowool, it allows accumulating huge amounts of heat in the stone walls of structures and smoothing out all daily temperature fluctuations. Only a small mass of insulation in all frame houses, despite the good heat capacity, is the weakest area for all frame technologies. To solve this problem, impressive heat accumulators are installed in all houses. What it is? These are structural parts that are characterized by a large mass with a fairly good heat capacity index.
Examples of heat accumulators in life
What could it be? For example, some internal brick walls, a large stove or fireplace, concrete screeds.
Furniture in any house or apartment is an excellent heat accumulator, because plywood, chipboard and wood can actually store heat only per kilogram of weight three times more than the notorious brick.
Are there any drawbacks to thermal storage? Of course, the main disadvantage of this approach is that the heat accumulator needs to be designed at the stage of creating a frame house layout. All due to the fact that it is very heavy, and this will need to be taken into account when creating the foundation, and then imagine how this object will be integrated into the interior. It is worth saying that it is necessary to take into account not only the mass, it will be necessary to evaluate both characteristics in the work: mass and heat capacity. For example, if you use gold with an incredible weight of twenty tons per cubic meter as a heat storage, then the product will function as it should only twenty-three percent better than a concrete cube, which weighs two and a half tons.
Which substance is most suitable for a heat storage?
The best product for a heat accumulator is not concrete and brick at all! Copper, bronze and iron do a good job of this, but they are very heavy. Oddly enough, but the best heat accumulator is water! The liquid has an impressive heat capacity, the largest among the substances available to us. Only helium gases (5190 J / (kg K) and hydrogen (14300 J / (kg K)) have more heat capacity, but they are problematic to apply in practice. If you wish and need, see the heat capacity table of the substances you need.
