The ratio of the amount of heat received by a body with an infinitesimal change in its state to the associated change in body temperature is called heat capacity bodies in this process:
Typically, heat capacity is referred to a unit amount of a substance and, depending on the chosen unit, is distinguished:
specific mass heat capacityc , referred to 1 kg of gas,
J/(kg K);
specific volumetric heat capacityc´, referred to the amount of gas contained in 1 m 3 volume under normal physical conditions, J/(m 3 K);
specific molar heat capacity, referred to one kilomole, J/(kmol·K).
The relationship between specific heat capacities is established by obvious relationships: ;
Here is the gas density under normal conditions.
The change in body temperature with the same amount of heat imparted depends on the nature of the process occurring during this process, therefore heat capacity is a function of the process. This means that the same working fluid, depending on the process, requires a different amount of heat to heat it by 1 K. Numerically, the value of c varies from +∞ to -∞.
In thermodynamic calculations, the following are of great importance:
heat capacity at constant pressure
equal to the ratio of the amount of heat imparted to the body in a process at constant pressure to the change in body temperature dT
heat capacity at constant volume
equal to the ratio of the amount of heat , supplied to the body in the process at a constant volume, to a change in body temperature .
According to the first law of thermodynamics for closed systems, in which equilibrium processes occur 
, And
For an isochoric process ( v=const) this equation takes the form 
, and, taking into account (1.5), we obtain that
,
that is, the heat capacity of a body at constant volume is equal to the partial derivative of its internal energy with respect to temperature and characterizes the rate of growth of internal energy in an isochoric process with increasing temperature.
For an ideal gas
For the isobaric process () from equation (2.16) and (2.14) we obtain
This equation shows the relationship between heat capacities with p And cv. For an ideal gas it is greatly simplified. Indeed, the internal energy of an ideal gas is determined only by its temperature and does not depend on volume, therefore, and in addition, it follows from the equation of state 
, where
This relationship is called the Mayer equation and is one of the main ones in the technical thermodynamics of ideal gases.
In progress v=const the heat imparted to the gas goes only to change its internal energy, while in the process R= const heat is spent both to increase internal energy and to do work against external forces. That's why with p more cv on the amount of this work.
For real gases, since when they expand (at p=const) work is done not only against external forces, but also against the forces of attraction acting between molecules, which causes additional heat consumption.
Typically, heat capacities are determined experimentally, but for many substances they can be calculated using statistical physics methods.
The numerical value of the heat capacity of an ideal gas can be found by the classical theory of heat capacity, based on the theorem on the uniform distribution of energy over the degrees of freedom of molecules. According to this theorem, the internal energy of an ideal gas is directly proportional to the number of degrees of freedom of molecules and energy kT/2, per one degree of freedom. For 1 mole of gas
,
Where No- Avogadro's number; i- the number of degrees of freedom (the number of independent coordinates that must be specified in order to completely determine the position of the molecule in space).
A monatomic gas molecule has three degrees of freedom, corresponding to three components in the direction of the coordinate axes, into which translational motion can be decomposed. A diatomic gas molecule has five degrees of freedom, since in addition to translational motion, it can rotate about two axes perpendicular to the line connecting the atoms (the energy of rotation around the axis connecting the atoms is zero if the atoms are considered points). The molecule of a triatomic and generally polyatomic gas has six degrees of freedom: three translational and three rotational.
Since for an ideal gas 
, then the molar heat capacities of mono-, di- and polyatomic gases are equal, respectively:
;
; 
.
The results of the classical theory of heat capacity are in fairly good agreement with experimental data in the room temperature region (Table 2.1), but the main conclusion about temperature independence is not confirmed by experiment. The discrepancies, especially significant in the region of low and fairly high temperatures, are associated with the quantum behavior of molecules and are explained within the framework of the quantum theory of heat capacity.
Heat capacity of some gases at t = 0°C in an ideal gas state
Goal of the work: Study of thermal processes in an ideal gas, familiarization with the Clément-Desormes method and experimental determination of the ratio of the molar heat capacities of air at constant pressure and constant volume.
Description of the installation and method of studying the process
Appearance working panel and circuit diagram experimental setup FPT1-6n is shown in Fig. 8: 1 – “NETWORK” switch for powering the installation; 2 – “Compressor” switch for pumping air into the working vessel (capacity with volume V = 3500 cm3), located in the cavity of the housing; 3 – valve K1, necessary to prevent the release of pressure from the working vessel after the compressor stops; 4 – pneumatic toggle switch “Atmosphere”, allowing a short time connect the working vessel to the atmosphere; 5 – pressure meter using a pressure sensor in the working vessel;
Rice. 8. Appearance of the working panel
6 – two-channel temperature meter, allowing you to measure the temperature inside the environment and the temperature inside the working vessel.
The state of a certain mass of gas is determined by three thermodynamic parameters: pressure R, volume V and temperature T. The equation that establishes the relationship between these parameters is called the equation of state. For ideal gases, such an equation is the Clapeyron-Mendeleev equation:
Where m– mass of gas; μ – molar mass; R= 8.31 J/mol∙K – universal gas constant.
Any change in the state of a thermodynamic system associated with a decrease or increase in at least one of the parameters p, V, T is called a thermodynamic process.
Isoprocesses– these are processes occurring under one constant parameter:
isobaric – at p = const;
isochoric – with V = const;
isothermal – at T = const.
The adiabatic process occurs without heat exchange with environment, therefore, to implement it, the system is thermally insulated or the process is carried out so quickly that heat exchange does not have time to occur. During an adiabatic process, all three parameters change R, V, T.
During adiabatic compression of an ideal gas, its temperature increases, and during expansion it decreases. In Fig. 9 in the coordinate system R And V shows the isotherm ( рV = const) and adiabatic ( рV γ = const). It can be seen from the figure that the adiabat is steeper than the isotherm. This is explained by the fact that during adiabatic compression, an increase in gas pressure occurs not only due to a decrease in its volume, as with isothermal compression, but also due to an increase in temperature.
Rice. 9. рV = const; рV γ = const
Heat capacity a substance (body) is called a value equal to the amount of heat required to heat it by one Kelvin. It depends on body weight, its chemical composition and the type of heat process. The heat capacity of one mole of a substance is called molar heat capacity C μ.
According to the first law of thermodynamics, the amount of heat dQ, communicated to the system, is spent on increasing internal energy dU systems and performance of work by the system dA against external forces
dQ = dU + dA. (2)
Using the first law of thermodynamics (2) and the Clapeyron-Mendeleev equation (1), we can derive an equation describing the adiabatic process - the Poisson equation
рV γ = const,
or in other parameters:
TV γ -1 = const,
T γ p 1-γ = const.
In these equations - the adiabatic exponent
γ = С р / С v,
where C v and C p are the molar heat capacities at constant volume and pressure, respectively.
For an ideal gas, the heat capacities C p and C v can be calculated theoretically. When heating a gas at a constant volume (isochoric process), the work done by the gas dA = рdV is equal to zero, so the molar heat capacity
, (3)
Where i– number of degrees of freedom – the number of independent coordinates with which you can uniquely specify the position of the molecule; index V means isochoric process.
With isobaric heating ( p = const) the amount of heat supplied to the gas is spent on increasing the internal energy and performing the work of expansion of the gas:
.
The heat capacity of a mole of gas is equal to
Equation (5) is called Mayer's equation. Consequently, the difference in molar heat capacities C p – C v = R is numerically equal to the work of expansion of one mole of an ideal gas when it is heated by one Kelvin at constant pressure. This is the physical meaning of the universal gas constant R.
For ideal gases the ratio γ = C p / C v = (i + 2) / i depends only on the number of degrees of freedom of gas molecules, which, in turn, is determined by the structure of the molecule, i.e. the number of atoms that make up a molecule. A monatomic molecule has 3 degrees of freedom (inert gases). If a molecule consists of two atoms, then the number of degrees of freedom consists of the number of degrees of freedom of translational motion (i post = 3) of the center of mass and rotational (i time = 2) motion of the system around two axes perpendicular to the axis of the molecule, i.e. equals 5. For tri- and polyatomic molecules i = 6 (three translational and three rotational degrees of freedom).
In this work, the coefficient γ for air is determined experimentally.
If a certain amount of air is pumped into a vessel using a pump, the pressure and temperature of the air inside the vessel will increase. Due to the heat exchange of air with the environment, after some time the temperature of the air in the vessel will be equal to the temperature T0 external environment.
The pressure established in the vessel is equal to р 1 = р 0 + р′, Where p 0- Atmosphere pressure, R'– additional pressure. Thus, the air inside the vessel is characterized by the parameters ( р 0 + р′), V 0, T 0, and the equation of state has the form
. (6)
If you open the “ATMOSPHERE” toggle switch for a short time (~3 s), the air in the vessel will expand. This expansion process can be considered as adding additional volume to the vessel V′. The pressure in the vessel will become equal to atmospheric P 0, the temperature will drop to T 1, and the volume will be equal V 0 + V′. Therefore, at the end of the process the equation of state will have the form
. (7)
Dividing expression (7) by expression (6), we obtain
. (8)
Expansion occurs without heat exchange with the external environment, i.e. the process is adiabatic, therefore, for the initial and final states of the system the relation is valid
. (9)
An ideal gas is a mathematical model of a gas in which the potential energy of the molecules is assumed to be negligible compared to their kinetic energy. There are no forces of attraction or repulsion between molecules, collisions of particles with each other and with the walls of the vessel are absolutely elastic, and the interaction time between molecules is negligible compared to the average time between collisions.
2. What are the degrees of freedom of molecules? How is the number of degrees of freedom related to Poisson's ratio γ?
The number of degrees of freedom of a body is the number of independent coordinates that must be specified in order to completely determine the position of the body in space. For example, a material point moving arbitrarily in space has three degrees of freedom (coordinates x, y, z).
Molecules of a monatomic gas can be considered as material points on the grounds that the mass of such a particle (atom) is concentrated in a nucleus whose dimensions are very small (10 -13 cm). Therefore, a monatomic gas molecule can have only three degrees of freedom of translational motion.
Molecules consisting of two, three or more atoms cannot be likened to material points. A diatomic gas molecule, to a first approximation, consists of two tightly bound atoms located at some distance from each other
3. What is the heat capacity of an ideal gas during an adiabatic process?
Heat capacity is a value equal to the amount of heat that must be imparted to a substance to increase its temperature by one kelvin.
4. In what units are pressure, volume, temperature, and molar heat capacities measured in the SI system?
Pressure – kPa, volume – dm 3, temperature – in Kelvin, molar heat capacities – J/(molK)
5. What are the molar heat capacities Cp and Cv?
A gas has a heat capacity at constant volume Cv and heat capacity at constant pressure Cr.
At a constant volume, the work of external forces is zero, and the entire amount of heat imparted to the gas from the outside goes entirely to increasing its internal energy U. Hence, the molar heat capacity of a gas at a constant volume C v is numerically equal to the change in the internal energy of one mole of gas ∆U when its temperature increases by 1 K:
∆U=i/2*R(T+1)-i/2RT=i/2R
Thus, the molar heat capacity of a gas at constant volume
WITH v=i/2R
specific heat capacity at constant volume
WITH v=i/2*R/µ
When a gas is heated at constant pressure, the gas expands; the amount of heat imparted to it from the outside goes not only to increase its internal energy U, but also to perform work A against external forces. Consequently, the heat capacity of a gas at constant pressure is greater than the heat capacity at constant volume by the amount of work A performed by one mole of gas during expansion resulting from an increase in its temperature by 1 K at constant pressure P:
C p = WITH v+A
It can be shown that for a mole of gas the work is A=R, then
C p = WITH v+R=(i+2)/2*R
Using the relationship between specific and molar heat capacities, we find for the specific heat capacity:
C p = (i+2)/2*R
Direct measurement of specific and molar heat capacities is difficult, since the heat capacity of the gas will be a tiny fraction of the heat capacity of the container in which the gas is located, and therefore the measurement will be extremely inaccurate.
It is easier to measure the ratio of greatness C p / WITH v
γ=C p / WITH v=(i+2)/i.
This ratio depends only on the number of degrees of freedom of the molecules that make up the gas.
Specific heat capacity of a substance- a value equal to the amount of heat required to heat 1 kg of a substance by 1 K:
The unit of specific heat capacity is joule per kilogram kelvin (J/(kg K)).
Molar heat capacity- a value equal to the amount of heat required to heat 1 mole of a substance by 1 K:
Where ν =m/M is the amount of substance.
The unit of molar heat capacity is joule per mole kelvin (J/(mol K)).
The specific heat capacity c is related to the molar heat capacity C m, the relation
where M is the molar mass of the substance.
Heat capacities are identified at constant volume and constant pressure if, during the process of heating a substance, its volume or pressure is maintained constant. Let us write down the expression of the first law of thermodynamics for one mole of gas, taking into account (1) and δA=pdV
If the gas is heated at a constant volume, then dV = 0 and the work of external forces is also zero. Then the heat imparted to the gas from the outside only goes to increase its internal energy:
(4) i.e., the molar heat capacity of a gas at a constant volume C V is equal to the change in the internal energy of one mole of gas with an increase in its temperature by 1 K. Since U m =( i/2)RT ,
If the gas is heated at constant pressure, then expression (3) can be represented in the form
Considering that (U m / dT) does not depend on the type of process (the internal energy of an ideal gas does not depend on either p or V, but is determined only by temperature T) and is always equal to C V, and differentiating the Clapeyron-Mendeleev equation pV m = RT by T (p=const), we get
Expression (6) is called Mayer's equation; it says that C p is always greater than C V exactly by the molar gas constant. This is explained by the fact that in order to heat a gas at a constant pressure, an additional amount of heat is required to perform the work of expansion of the gas, since the constancy of the pressure is ensured by an increase in the volume of the gas. Using (5), formula (6) can be written as
When studying thermodynamic processes, it is important to know the characteristic ratio of C p to C V for each gas:
(8)
called adiabatic index. From the molecular kinetic theory of ideal gases, the numerical values of the adiabatic exponent are known; they depend on the number of atoms in the gas molecule:
Monatomic gas γ = 1,67;
Diatomic gas γ = 1,4;
Tri- and polyatomic gas γ = 1,33.
(The adiabatic exponent is also denoted by k)
11. Warmth. The first law of thermodynamics.
The internal energy of a thermodynamic system can change in two ways: through work done on the system and through heat exchange with the environment. The energy that a body receives or loses in the process of heat exchange with the environment is called amount of heat or simply warmth.
The unit of measurement in (SI) is joule. The calorie is also used as a unit of heat measurement.
The first law of thermodynamics is one of the basic principles of thermodynamics, which is essentially the law of conservation of energy as applied to thermodynamic processes.
The first law of thermodynamics was formulated in the middle of the 19th century as a result of the work of J. R. Mayer, Joule and G. Helmholtz. The first law of thermodynamics is often formulated as the impossibility of the existence of a perpetual motion machine of the 1st kind, which would do work without drawing energy from any source.
Formulation
The amount of heat received by the system goes to change its internal energy and perform work against external forces.
The first law of thermodynamics can be formulated as follows:
“The change in the total energy of the system in a quasi-static process is equal to the amount of heat Q imparted to the system, in sum with the change in energy associated with the amount of substance N at the chemical potential, and the work A” performed on the system by external forces and fields, minus the work A performed the system itself against external forces":
For an elementary amount of heat, elementary work, and a small increment (total differential) of internal energy, the first law of thermodynamics has the form:
Dividing the work into two parts, one of which describes the work done on the system, and the second - the work done by the system itself, emphasizes that these works can be done by forces of different nature due to different sources of forces.
It is important to note that and are complete differentials and and are not. The heat increment is often expressed in terms of temperature and entropy increment: .
INTRODUCTION
According to the first law of thermodynamics, the amount of energy imparted to the system in the process of heat exchange dQ goes to change its internal energy dU and to the system performing work dA against external forces:
The amount of heat required to heat one (kilo)mole of gas by one degree is determined by the molar heat capacity - WITH.
The magnitude of the heat capacity depends on the heating conditions. There are two types of heat capacities: C p - molar heat capacity at constant pressure and C v - molar heat capacity at constant volume, related by the equation:
С p =С v +R, (2)
where R is the universal gas constant, numerically equal to the work done when heating one mole of an ideal gas by one kelvin at constant pressure.
A process that occurs without heat exchange with the environment (dQ = 0) is called adiabatic. It is described by the Poisson equation:
The work of an adiabatic process, as follows from the First Law of Thermodynamics (3), is accomplished only due to changes in internal energy:
The total work of an adiabatic process can be calculated using the formula:
(5)
Instruments and accessories: liquid pressure gauge, closed glass bottle with three-way valve, pump.
THEORY OF THE METHOD AND DESCRIPTION OF THE INSTALLATION.
Method for determining C p / C v , used in the work is based on the process of adiabatic expansion of air.
The installation (Fig. 22) consists of a thick-walled cylinder 2, connected to the injection pump 3 and open U-shaped water pressure gauge 1. Three way valve 4 allows you to connect the cylinder to a pump or atmosphere.
Let us denote the mass of gas in a cylinder at atmospheric pressure - m 1.
If you connect a cylinder to a pump and pump up air, the pressure in the cylinder will increase and become equal p 1 =p 0 +h 1 , Where h 1- excess above atmospheric pressure p 0, measured by pressure gauge, (p 0, And h 1 must be expressed in the same units).
Note. Since the air in the cylinder heats up during injection, measure the excess pressure h 1 should be done when the air temperature in the cylinder becomes equal to room temperature (after 1-2 minutes).
Gas mass m 1 will now occupy a volume V 1 less than the volume of the cylinder.
Its condition is characterized by the following parameters: p 1, V 1, T 1 (Fig. 23). If you briefly connect the balloon to the atmosphere using a tap, the air will expand rapidly (i.e., adiabatically). Part of air mass m will come out of the container. The remaining air mass m 1, which occupied part of the volume of the cylinder before opening the valve, will again occupy the entire volume Vk = V2. The pressure in the cylinder will become equal to atmospheric (p 2 =p 0). The air temperature as a result of its adiabatic expansion will be below room temperature. Thus, at the moment the tap is closed, the air is in state II (p 2, V 2, T 2).
For gas mass m 1, according to Poisson's law (3), we obtain:
Since the temperature in states I and III is the same, then according to the Boyle-Mariotte law:
Comparing equalities (6) and (7), we obtain:
Let's take logarithm of this expression
and solve it relatively
Considering that p 1 =p 0 +h 1; p 2 = p 0 ; p 3 =p 0 +h 2 we get:
Since the pressures differ slightly from each other, approximately in the last expression logarithms can be replaced by numbers:
or
To calculate the work of adiabatic expansion, we use formula (5). Since according to Poisson's law
then formula (5) will take the form:
A= 
Where V≈V k, specified on the installation.
COMPLETING OF THE WORK
1. Using a tap, connect the cylinder to the pump and pump air until the difference in liquid levels in the pressure gauge becomes 20-30 cm.
2. Close the tap and wait until the liquid levels in the pressure gauge are established. Count the difference in liquid levels in the pressure gauge elbows h 1(count along the lower edge of the meniscus).
3. Open the tap and at the moment when the liquid levels in both elbows of the pressure gauge are equal, quickly close it.
4. After waiting 1-2 minutes until the air in the cylinder warms up to room temperature, measure the difference in liquid levels in both elbows of the pressure gauge h 2
5. Use a barometer to measure atmospheric pressure r 0 .
6. Enter the data into the table.
7. Repeat the experiment (steps 1-4) at least five times.
| №№ | h 1, mm water. Art. | h 2, mm water. Art. | h 1 -h 2, mm water. Art. | |||
COMPUTING
1. Calculate the value for each measurement using formula (8).