DEFINITION
Describes adiabatic process, flowing in . Adiabatic is a process in which there is no heat exchange between the system under consideration and environment: .
Poisson's equation looks like:
Here, the volume occupied by the gas is its, and the value is called the adiabatic exponent.
Adiabatic exponent in Poisson's equation
In practical calculations it is convenient to remember that for ideal gas the adiabatic index is equal to , for a diatomic one – , and for a triatomic one – .
What to do with real gases, When important role do the forces of interaction between molecules begin to play? In this case, the adiabatic index for each gas under study can be obtained experimentally. One such method was proposed in 1819 by Clément and Desormes. We fill the cylinder with cold gas until the pressure in it reaches . Then we open the tap, the gas begins to expand adiabatically, and the pressure in the cylinder drops to atmospheric pressure. After the gas is isochorically heated to ambient temperature, the pressure in the cylinder will increase to . Then the adiabatic exponent can be calculated using the formula:
The adiabatic index is always greater than 1, therefore, during adiabatic compression of a gas - both ideal and real - to a smaller volume, the gas temperature always increases, and during expansion the gas cools. This property of the adiabatic process, called pneumatic flint, is used in diesel engines, where the combustible mixture is compressed in the cylinder and ignited by high temperature. Let us recall the first law of thermodynamics: , where - , and A is the work performed on it. Since the work done by the gas only goes to change it internal energy- and therefore temperature. From the Poisson equation we can obtain a formula for calculating the work of a gas in an adiabatic process:
Here n is the amount of gas in moles, R is the universal gas constant, T is absolute temperature gas
Poisson's equation for an adiabatic process is used not only in engine calculations internal combustion, but also in the design of refrigeration machines.
It is worth remembering that the Poisson equation accurately describes only an equilibrium adiabatic process consisting of continuously alternating equilibrium states. If in reality we open the valve in the cylinder so that the gas expands adiabatically, an unsteady transient process will arise with gas vortices, which will die out due to macroscopic friction.
Examples of problem solving
EXAMPLE 1
| Exercise | A monatomic ideal gas was adiabatically compressed so that its volume doubled. How will the gas pressure change? |
| Solution | The adiabatic exponent for a monatomic gas is equal to . However, it can also be calculated using the formula:
where R is the universal gas constant, and i is the degree of freedom of the gas molecule. For a monatomic gas, the degree of freedom is 3: this means that the center of the molecule can make forward movements three coordinate axes. Therefore, the adiabatic index:
Let us represent the states of the gas at the beginning and end of the adiabatic process through the Poisson equation:
|
| Answer | The pressure will decrease by 3.175 times. |
EXAMPLE 2
| Exercise | 100 moles of a diatomic ideal gas were adiabatically compressed at a temperature of 300 K. At the same time, the gas pressure increased 3 times. How has the work of gas changed? |
| Solution | The degree of freedom of a diatomic molecule, since the molecule can move translationally along three coordinate axes and rotate around two axes. |
Purpose of the work: get acquainted with the adiabatic process, determine the adiabatic index for air.
Equipment: cylinder with valve, compressor, pressure gauge.
THEORETICAL INTRODUCTION
Adiabatic process is a process that takes place in thermodynamic system without heat exchange with the environment. Thermodynamic system is a system containing huge amount particles. For example, a gas whose number of molecules is comparable to the Avagadro number 6.02∙10 23 1/mol. Although the movement of each particle obeys Newton's laws, there are so many of them that it is impossible to solve the system of dynamic equations to determine the parameters of the system. Therefore, the state of the system is characterized by thermodynamic parameters, such as pressure P, volume V, temperature T.
According to first law of thermodynamics, which is the law of conservation of energy in thermodynamic processes, heat Q, supplied to the system, is spent on doing work A and the change in internal energy Δ U
Q=A+ D U. (1)
Heat is the amount of energy of chaotic motion transferred to the thermodynamic system. The supply of heat leads to an increase in temperature: , where n– amount of gas, WITH− molar heat capacity, depending on the type of process. Internal energy of an ideal gas is the kinetic energy of the molecules. It is proportional to temperature: , where Cv– molar heat capacity during isochoric heating. Job elementary change in volume by pressure forces is equal to the product of pressure and change in volume: dA= PdV.
For an adiabatic process occurring without heat exchange ( Q= 0), work is done due to changes in internal energy, A = − D U. During adiabatic expansion, the work done by the gas is positive, so the internal energy and temperature decrease. When compressed, the opposite is true. All rapidly occurring processes can be fairly accurately considered adiabatic.
Let us derive the equation of the adiabatic process of an ideal gas. To do this, we apply the equation of the first law of thermodynamics for an elementary adiabatic process dA= − dU, which takes the form РdV =−n С v dT. Let us add to this differential equation one more, obtained by differentiating the Mendeleev–Clapeyron equation ( PV=νRT): PdV +VdP =nR dT. By excluding one of the parameters in two equations, for example, temperature, we obtain a relationship for the other two parameters 
. Integrating and potentiating, we obtain the adiabatic equation in terms of pressure and volume:
P V g = const.
Likewise:
T V g -1 = const, P g -1 T -- g = const. (2)
Here 
– adiabatic exponent, equal to ratio heat capacities of gas during isobaric and isochoric heating.
Let us obtain a formula for the adiabatic exponent in molecular kinetic theory. Molar heat capacity by definition is the amount of heat required to heat one mole of a substance by one Kelvin. During isochoric heating, heat is spent only to increase internal energy 
. Substituting heat, we get .
During isobaric heating of a gas under constant pressure conditions, an additional part of the heat is spent on the work of volume change 
. Therefore, the amount of heat ( dQ = dU + dA) obtained by isobaric heating by one Kelvin will be equal to 
. Substituting into the heat capacity formula, we get 
.
Then adiabatic exponent can be determined theoretically by the formula
Here i – number of degrees of freedom gas molecules. This is the number of coordinates sufficient to determine the position of the molecule in space or the number of constituent energy components of the molecule. For example, for monoatomic molecule kinetic energy can be represented as the sum of three energy components corresponding to motion along three coordinate axes, i= 3. For a rigid diatomic molecule, two more energy components should be added rotational movement, since there is no rotational energy about the third axis passing through the atoms. So for diatomic molecules i= 5. For air as a diatomic gas, the theoretical value of the adiabatic index will be equal to g = 1.4.
The adiabatic exponent can be determined experimentally by the Clément–Desormes method. Air is pumped into the balloon, compressing it to a certain pressure. R 1, a little more atmospheric. When compressed, the air heats up slightly. After establishing thermal equilibrium balloon on short time open. In this expansion process 1–2 the pressure drops to atmospheric R 2 =P atm, and the mass of gas under study, which previously occupied part of the volume of the cylinder V 1, expands, occupying the entire cylinder V 2 (Fig. 1). The process of air expansion (1−2) occurs quickly; it can be considered adiabatic, occurring according to equation (2)
. (4)
In the adiabatic expansion process, the air cools. After closing the valve, the cooled air in the cylinder through the walls of the cylinder is heated to laboratory temperature T 3 = T 1. This is an isochoric process 2–3
. (5)
Solving equations (4) and (5) together, excluding temperatures, we obtain the equation 
, from which the adiabatic index should be determined γ
. The pressure sensor does not measure the absolute pressure, which is written in the process equations, but the excess pressure above atmospheric pressure. That is R 1 = Δ R 1 +R 2, and R 3 =Δ R 3 +R 2. Moving on to excess pressures, we get 
. Excess pressures are small compared to atmospheric pressure R 2. Let us expand the terms of the equation into a series according to the relation 
. After reduction by R 2 we obtain the calculation formula for the adiabatic exponent
. (6)
The laboratory installation (Fig. 2) consists of a glass cylinder, which communicates with the atmosphere through a valve Atmosphere. Air is pumped into the cylinder by a compressor with the tap open. TO. After pumping, to avoid air leakage, close the tap.
GETTING THE WORK DONE
1. Connect the installation to a 220 V network.
Open the cylinder tap. Turn on the compressor, pump air to excess pressure in the range of 4–11 kPa. Close the cylinder tap. Wait 1.5–2 minutes, record the pressure value Δ R 1 to the table.
2. Turn the valve Atmosphere until it clicks, the valve opens and closes. There will be an adiabatic release of air with a decrease in temperature. Monitor the increase in pressure in the cylinder as it heats up. Measure highest pressure Δ R 3 after thermal equilibrium has been established. Write the result in the table.
Repeat the experiment at least five times, changing the initial pressure in the range of 4–11 kPa.
| Δ R 1, kPa | |||||
| Δ R 3, kPa | |||||
| γ |
Turn off the installation.
3. Make calculations. Determine the adiabatic index in each experiment using formula (6). Write it down in the table. Determine the average value of the adiabatic index<γ >
4. Estimate the random measurement error using the formula for direct measurements
. (7)
5. Write the result in the form g = <g> ± dg. R= 0.9. Compare result with theoretical value adiabatic index of a diatomic gas g theory = 1,4.
Draw conclusions.
TEST QUESTIONS
1. Give the definition of an adiabatic process. Write down the first law of thermodynamics for an adiabatic process. Explain the change in gas temperature during adiabatic processes of compression and expansion.
2. Derive the equation of the adiabatic process for the parameters pressure – volume.
3. Derive the equation of the adiabatic process for the parameters pressure – temperature.
4. Define the number of degrees of freedom of molecules. How does the internal energy of an ideal gas depend on the type of molecules?
5. How are processes carried out with air in the Clément – Desormes cycle, how do pressures and temperatures change in the processes?
6. Derive a calculation formula for the experimental determination of the adiabatic index.
Related information.
Calculation of the pressure in the front of the air shock wave during the destruction of the container is carried out according to formulas (3.12), (3.45), in the latter of which the value aMQ v n is replaced by E, the value of the coefficient b 1 = 0.3.
A serious danger is posed by the scattering of fragments resulting from the destruction of the container. The movement of a fragment with a known initial speed can be described by a system of equations of the form
\s\up15(x" = -\f((0C1S1 \b (x" -\f((0C2S2 \b (x"2 + y"2 (3.45)
where m is the mass of the fragment, kg; C 1 , C 2 - coefficients drag and the lifting force of the fragment, respectively; S 1 , S 2 - area of the frontal and lateral surface of the fragment, m 2 ; r 0 - air density, kg/m 3 ; a - angle of departure of the fragment; x, y - coordinate axes.
The solution to this system of equations is shown in Fig. 3.7.
In approximate calculations to estimate the scattering range of fragments, it is allowed to use the relation
where L m is the maximum scattering range of fragments, m; V 0 - initial speed flight of fragments, m/s; g = 9.81 m/s 2 - free fall acceleration.
Relationship (3.46) was obtained for the case of fragments flying in airless space. At large quantities V 0 it overestimates the value of L m . The range L m determined in this way should be limited from above by the value L *
L m £ L * = 238 3.47,
where E is the energy of the explosion under consideration, J; Q v tr is the heat of the TNT explosion (Table 2), J/kg. The L * values were obtained during explosions of TNT charges in a metal shell (bombs, shells).
If a container explodes with compressed flammable gas explosion energy E, J, is found according to the relation
E= 
+ MQ v p 3.48,
where M = awM 0 - mass of gas participating in the explosion, kg; Q v p - heat of explosion of flammable gas, J/kg; a, w - coefficients determined according to (3.32), (3.45);
The mass of gas in the container before the explosion is M 0 = Vr 0, where the values of P 0, P g, V have the same meaning as in formula (3.46), and the value of r 0 is the gas density at atmospheric pressure.
As noted in Section 3.4, the adiabatic index of hot water explosion products g » 1.25. More exact values The adiabatic indices of some gases used to calculate the consequences of an explosion are given in Table 3.8.
Table 3.8
In the case under consideration, the relation E » E uv + E osc + E t also holds, where E is the explosion energy, E uv = b 1 E - the energy spent on the formation of an air shock wave, E osc = b 2 E - kinetic energy of fragments , E t = b 0 E - energy going to thermal radiation. According to the data here, the coefficients b 1 = 0.2, b 2 = 0.5, b 3 = 0.3.
Calculation of the pressure in the front of the air shock wave and the range of dispersal of fragments at known values explosion energy E and coefficients b 1 , b 2 , b 3 are given by analogy with the considered case of explosion of a container with an inert gas.
It is necessary to note the difference between the events that occur during depressurization of vessels containing gas under pressure and vessels containing liquefied gases. If in the first case the main damaging factor is shell fragments, then in the second, fragments may not form, since when the seal of cylinders with liquefied gases is broken, their internal pressure almost simultaneously with depressurization becomes equal to the external one and then the processes of outflow of liquefied gas from the destroyed balloon into the environment and its evaporation. Moreover, in the event of an explosion, the main damaging factors are shock wave and thermal radiation.
Federal Agency for Education
Saratov State Technical University
DETERMINATION OF THE ADIABATH INDICATOR
FOR AIR
Guidelines for performing laboratory work
by courses "Thermal engineering", " Technical thermodynamics
And heating engineering for students
specialties 280201
daytime and correspondence forms training
Approved
editorial and publishing council
Saratovwhom state
technical university
Saratov 2006
Purpose of the work: familiarization with the methodology and experimental determination of the adiabatic index for air, study of the basic laws for adiabatic, isochoric and isothermal processes of change in the state of working fluids.
BASIC CONCEPTS
Adiabatic processes are the processes of changing the state of the working fluid (gas or steam) that occur without the supply or removal of heat from it.
Necessary and sufficient condition adiabatic process is the analytical expression dq =0, meaning that there is absolutely no heat transfer in the process, i.e. q =0. At dq =0 for reversible processes Tds =0, i.e. ds =0; this means that for reversible adiabatic processes s = const . In other words, a reversible adiabatic process is at the same time isoentropic.
The equation that relates the change in the main thermodynamic parameters in an adiabatic process, i.e., the adiabatic equation has the form:
font-size:14.0pt">where k - adiabatic (isentropic) index:
Font-size:14.0pt">The adiabatic equation can be obtained in another form, using the relationship between the main thermodynamic parameters:
font-size:14.0pt">The dependency is obtained similarly:
font-size:14.0pt">Work in an adiabatic process can be determined from the equation of the first law of thermodynamics:
font-size:14.0pt">When
font-size:14.0pt">or
font-size:14.0pt">Replacing
font-size:14.0pt">we get:
font-size:14.0pt">Replacing and in this equation, we get, J/kg:
font-size:14.0pt">Using the relationship between thermodynamic parameters, we can obtain another expression for the operation of the adiabatic process. Taking out brackets in equation (4), we will have:
font-size:14.0pt">but
font-size:14.0pt">then
font-size:14.0pt">Graphic display of the adiabatic process in p - v - and T - s -coordinates are shown in Fig. 1.
In p - v - coordinates the adiabatic curve is exponential function, from where , where a is a constant value.
In p - v - in coordinates, the adiabat is always steeper than the isotherm, since EN-US style="font-size:16.0pt"">cp>cv . Process 1-2 corresponds to expansion, process 1-2¢ - compression. Area of the site under the adiabatic curve in p,v - coordinates is numerically equal to the work of the adiabatic process (“ L "in Fig. 1).
In T - s -coordinates, the adiabatic curve is vertical line With . The area under the process curve is degenerate, which corresponds to zero heat of the adiabatic process.
 |
Fig.1. Adiabatic process of changing the state of a gas
in p -v - and T -s - diagrams
Real processes occurring with working fluids in heat engines are close to the adiabatic process. For example, expansion of gases and vapors in turbines and cylinders of heat engines, compression of gases and vapors in compressors of heat engines and refrigeration machines.
Approximately the size k can be estimated from the atomicity of the gas (or the main gases in the mixture), neglecting the temperature dependence:
for monatomic gases: font-size:14.0pt">for diatomic gases: font-size:14.0pt">for triatomic and polyatomic gases: 
.
At known composition gas, the adiabatic index can be calculated exactly from the tabulated values of heat capacities depending on temperature.
The adiabatic exponent can also be determined from the differential relations of thermodynamics. Unlike the ideal gas theory, differential equations of thermodynamics make it possible to obtain general patterns changes in parameters for real gases. Differential equations of thermodynamics are obtained by partial differentiation of the combined equation of the first and second laws of thermodynamics:
font-size:14.0pt">by several state parameters at once.
Apparatus differential equations thermodynamics allows, in particular, to establish a number of important relationships for the heat capacities of real gases.
One of them is a relationship of the form:
font-size:14.0pt">Relation (7) establishes a connection between heat capacities cp, cv and basic parameter changes p and v in an adiabatic process font-size:14.0pt">and isothermal process
.Considering that the adiabatic exponent , equation (7) can be rewritten as:
font-size:14.0pt">The last expression can be used to experimentally determine the adiabatic index.
EXPERIMENTAL PROCEDURE
To determine the true adiabatic index of sufficiently rarefied real gases using equation (8), accurate measurements of the thermodynamic parameters p are required, v, T and their partial derivatives. But if we substitute small finite increments into equation (8), then when 
the average value of the adiabatic index will be equal to:
https://pandia.ru/text/79/436/images/image034_1.gif" width="12" height="23 src=">font-size:14.0pt">When p2=rbar, that is, equal to barometric pressure,
Font-size:14.0pt">where p u 1, p u 3 – excess pressure in states 1, 3.
It is obvious that with a decrease in excess pressureр u 1 value km will approach the true value for atmospheric air.
The laboratory installation (Fig. 2) has a constant volume vessel 1, taps 2, 3. Air is pumped into the vessel by compressor 4. The air pressure in the vessel is measured U -shaped pressure gauge 5. The vessel is not isothermal, therefore the air in it assumes an equilibrium temperature state with the environment as a result of heat exchange. The air temperature in the vessel is controlled using a mercury thermometer 6 with a division value of 0.01° C.
|
Fig.2. Diagram of a laboratory setup for determining the indicator
adiabats of air: 1 – vessel; 2, 3 – taps; 4 – compressor;
5 - U-shaped pressure gauge; 6 – thermometer
Figure 3 shows the thermodynamic processes occurring in the air during the experiment: process 1-2 – adiabatic expansion of air when it is partially released from the vessel; 2-3 – isochoric heating of air to ambient temperature; 1-3 - effective (resulting) process of isothermal expansion of air.
| |
|
|
|
| |
| |
|
When executing of this work there are no dangerous and harmful factors and cannot arise. However, the pressure in the vessel with a manually driven compressor should be raised gradually by rotating the compressor flywheel. This will prevent water from being knocked out of the pressure gauge.
PROCEDURE FOR PERFORMANCE OF THE WORK
Familiarize yourself with the installation diagram and inspect it to determine its readiness for operation.
Determine from the barometer and record in the measurement report the atmospheric pressure pbar, temperature
t and relative humidity in the laboratory. Open tap 2 (Fig. 2) and with tap 3 closed, rotating the flywheel of compressor 4, pump air into vessel 1. As noted above, p u 1 should be as small as possible. Therefore, having created a slight excess pressure in the vessel, stop the air supply and close valve 2.The pressure is maintained for some time necessary to establish thermal equilibrium with the environment, as evidenced by the constant readings of pressure gauge 5. Write down the p value
u 1. Then open and when you reach atmospheric pressure immediately close tap 3. The air remaining in the vessel as a result of adiabatic expansion and cooling upon expiration will begin to heat up due to the isochoric supply of heat from the environment. This process is observed by a noticeable increase in pressure in the vessel to p u 3. Repeat the experiment 5 times.The results obtained are entered into the measurement protocol in the form of Table 1.
Table 1
t ,° C | pu 1, Pa | pu 3, Pa |
|
||
PROCESSING OF EXPERIMENTAL RESULTS
Exercise:
1. Determine the values of the adiabatic index in each experiment according to (8) and the probable (average) value of the air adiabatic index:
font-size:14.0pt">where n – number of experiments,
and compare the obtained value with the table (Table 2):
Font-size:14.0pt">2. Carry out a study of the processes of adiabatic expansion, subsequent isochoric heating of air and an effective isothermal process, which is the result of the first two real processes.
Table 2
Physical properties of dry air under normal conditions
Temperature t, °C |
heat capacity, kJ/(kmol× K) | Mass heat capacity, kJ/(kg× K) | Volumetric heat capacity, kJ/(m3× K) | Adiabatic exponent k |
|||
m from pm | m with vm | from pm | with vm | from ¢pm | with ¢ vm |
||
To do this, it is necessary to average the thermodynamic parameters p, T in over the number of experiments characteristic points 1, 2, 3 (Fig. 3) and from them calculate the caloric characteristics: heat, work, change in internal energy, change in enthalpy and entropy in each of the indicated thermodynamic processes. Compare the caloric characteristics of a real isothermal process (characteristics calculated from calculated relationships) and an effective isothermal process (characteristics that are the sum of the corresponding characteristics of adiabatic and isochoric processes).
Draw conclusions.
Directions:
The equation of the isochoric process has the form:
font-size:14.0pt">CALCULATION OF DETERMINATION ERROR
VALUES OF THE ADIABATIC INDICATOR
1. Absolute and relative errors experimental determination adiabatic index
k according to (9), (10) and tabular data are determined by the formulas:font-size:14.0pt">where k table – tabular value of the adiabatic exponent.
2. Absolute error in determining the adiabatic index based on the results of measuring excess pressure p
u 1 and p u 3 (9) is calculated by the formula:font-size:14.0pt">where D r u = D r u 1 = D r u 3 - absolute error overpressure measurements by U -shaped pressure gauge, which can be taken equal to 1 mm water. Art.
Relative error, %, of determining the adiabatic index based on measurement results:
font-size:14.0pt">SELF-TEST QUESTIONS
1. Indicate the difference in the concepts of adiabatic and isentropic processes.
2. What thermodynamic quantity is called the adiabatic exponent? Explain physical meaning adiabatic index.
3. Tell us about the device experimental setup and experimental methodology.
4. Why for an adiabatic process, except for the condition q =0, an additional condition is imposed dq =0?
5. Write the adiabatic equations.
6. Derive an expression for the operation of an adiabatic process.
7. Write and explain the expression for the change in internal energy in all thermodynamic processes.
8. Write and explain the expression for the enthalpy change in general view.
9. Write an expression for the change in entropy in general form. Obtain simplified expressions for particular thermodynamic processes.
10. How is the isochoric process characterized, and what are its equations, work, and heat?
11. What is it characterized by? isothermal process, and what are its equation, work, heat?
12. What is called a particular thermodynamic process of changing the state of a gas? List them.
13. What is the essence of the theory of differential equations of thermodynamics? Write the combined equation of the first and second laws of thermodynamics.
14. Draw the adiabatic curve in p - v - and T - s -coordinates. Why in p - v - in coordinates does the adiabat always steepen than the isotherm?
15. What do the areas under the curves of thermodynamic processes show in p - v - and T - s -coordinates?
16. Draw the isochore curve in
17. Draw the isotherm curve in p - v - and T - s -coordinates.
LITERATURE
1. Kirillin thermodynamics. , . 3rd ed., revised. and additional M. Nauka, 19с.
2. Nashchokin thermodynamics and heat transfer: a textbook for universities. . 3rd ed., corrected. and additional M. graduate School, 19s.
3. Gortyshov and the technique of thermophysical experiment. , ; edited by . M: Energoatomizdat, 1985. P.35-51.
4. Thermal engineering: a textbook for universities. edited by . 2nd ed., revised. M. Energoatomizdat, 19с.
DETERMINATION OF THE ADIABATH INDICATOR FOR AIR
Guidelines for performing laboratory work
by courses "Thermal engineering", " Technical thermodynamics
And heating engineering ", "Hydraulics and heat engineering"
Compiled by: SEDELKIN Valentin Mikhailovich
KULESHOV Oleg Yurievich
KAZANTSEVA Irina Leonidovna
Reviewer
License ID No. 000 dated 11/14/01
Signed for printing Format 60´ 84 1/16
Boom. type. Conditional oven l. Academic ed. l.
Circulation Order Free
Saratov State Technical University
Saratov, Politekhnicheskaya st., 77
Printed at RIC SSTU. Saratov, Politekhnicheskaya st., 77
Adiabatic exponent(sometimes called Poisson's ratio) - ratio of heat capacity at constant pressure() to heat capacity at constant volume (). Sometimes it is also called isentropic expansion factor. Designated Greek letter(gamma) or (kappa). The letter symbol is primarily used in chemical engineering disciplines. In heat engineering, the Latin letter is used.
Equation:
, is the heat capacity of the gas, is the specific heat capacity (the ratio of heat capacity to unit mass) of the gas, the indices and indicate the condition of constant pressure or constant volume, respectively.To understand this relationship, consider the following experiment:
A closed cylinder with a fixed piston contains air. The pressure inside is equal to the pressure outside. This cylinder is heated to a certain, required temperature. While the piston cannot move, the volume of air in the cylinder remains unchanged, while the temperature and pressure increase. When the required temperature is reached, heating stops. At this moment, the piston is “freed” and, thanks to this, begins to move outward without heat exchange with the environment (the air expands adiabatically). While performing work, the air inside the cylinder is cooled below the previously reached temperature. To return the air to a state where its temperature again reaches the required value mentioned above (with the piston still “freed”), the air must be heated. For this heating, it is necessary to supply approximately 40% from the outside (for diatomic gas - air) more heat than was supplied during the previous heating (with the piston fixed). In this example, the amount of heat supplied to a cylinder with a fixed piston is proportional to , whereas total quantity heat input is proportional to . Thus, the adiabatic exponent in this example is 1.4.
Another way to understand the difference between and is that it applies when work is done on a system that is forced to change its volume (that is, by the movement of a piston that compresses the contents of the cylinder), or if work is done on a system by changing its temperature ( that is, by heating the gas in the cylinder, which forces the piston to move). applies only if - and this expression denotes the work done by the gas - is equal to zero. Let's consider the difference between heat input with a fixed piston and heat input with a free piston. In the second case, the gas pressure in the cylinder remains constant, and the gas will both expand, doing work on the atmosphere, and increase its internal energy (with increasing temperature); the heat that is supplied from the outside only partially goes towards changing the internal energy of the gas, while the rest of the heat goes towards doing work by the gas.
| Adiabatic exponents for various gases | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Pace. | Gas | γ | Pace. | Gas | γ | Pace. | Gas | γ | ||
| −181 °C | H 2 | 1.597 | 200 °C | Dry air | 1.398 | 20°C | NO | 1.400 | ||
| −76 °C | 1.453 | 400 °C | 1.393 | 20°C | N2O | 1.310 | ||||
| 20°C | 1.410 | 1000 °C | 1.365 | −181 °C | N 2 | 1.470 | ||||
| 100 °C | 1.404 | 2000 °C | 1.088 | 15 °C | 1.404 | |||||
| 400 °C | 1.387 | 0°C | CO2 | 1.310 | 20°C | Cl2 | 1.340 | |||
| 1000 °C | 1.358 | 20°C | 1.300 | −115 °C | CH 4 | 1.410 | ||||
| 2000 °C | 1.318 | 100 °C | 1.281 | −74 °C | 1.350 | |||||
| 20°C | He | 1.660 | 400 °C | 1.235 | 20°C | 1.320 | ||||
| 20°C | H2O | 1.330 | 1000 °C | 1.195 | 15 °C | NH 3 | 1.310 | |||
| 100 °C | 1.324 | 20°C | CO | 1.400 | 19 °C | Ne | 1.640 | |||
| 200 °C | 1.310 | −181 °C | O2 | 1.450 | 19 °C | Xe | 1.660 | |||
| −180 °C | Ar | 1.760 | −76 °C | 1.415 | 19 °C | Kr | 1.680 | |||
| 20°C | 1.670 | 20°C | 1.400 | 15 °C | SO 2 | 1.290 | ||||
| 0°C | Dry air | 1.403 | 100 °C | 1.399 | 360°C | Hg | 1.670 | |||
| 20°C | 1.400 | 200 °C | 1.397 | 15 °C | C2H6 | 1.220 | ||||
| 100 °C | 1.401 | 400 °C | 1.394 | 16 °C | C 3 H 8 | 1.130 | ||||
Relations for an ideal gas
For an ideal gas, the heat capacity does not depend on temperature. Accordingly, enthalpy can be expressed as and internal energy can be represented as . Thus, we can also say that the adiabatic exponent is the ratio of enthalpy to internal energy:
On the other hand, heat capacities can also be expressed through the adiabatic exponent () and the universal gas constant ():
It can be quite difficult to find information about table values, while table values are given more often. In this case you can use the following formula to define:
where is the amount of substance in moles.
Relationships using degrees of freedom
The adiabatic exponent () for an ideal gas can be expressed in terms of the number of degrees of freedom () of gas molecules:
orThermodynamic expressions
Values obtained using approximate relationships (in particular), in many cases, are not accurate enough for practical engineering calculations, such as calculations of flow rates through pipelines and valves. It is preferable to use experimental values than those obtained using approximate formulas. Strict relation values can be calculated by determining from properties expressed as:
Values are easy to measure, while values for must be determined from formulas like this. See here ( English) to get more detailed information on the relationships between heat capacities.
Adiabatic process
where is the pressure and is the volume of the gas.
Experimental determination of the adiabatic index value
Since the processes occurring in small volumes of gas during passage sound wave, are close to adiabatic, the adiabatic index can be determined by measuring the speed of sound in the gas. In this case, the adiabatic index and the speed of sound in the gas will be related by the following expression:
where is the adiabatic exponent; - Boltzmann constant; - universal gas constant; - absolute temperature in kelvins; - molecular weight; - molar mass.
Another way to experimentally determine the value of the adiabatic exponent is the Clément-Desormes method, which is often used in educational purposes when executing laboratory work. The method is based on studying the parameters of a certain mass of gas passing from one state to another by two successive processes: adiabatic and isochoric.
The laboratory setup includes a glass bottle connected to a pressure gauge, a tap and a rubber bulb. The bulb is used to pump air into the balloon. A special clamp prevents air leakage from the cylinder. The pressure gauge measures the difference in pressure inside and outside the cylinder. The valve can release air from the cylinder into the atmosphere.
Let the cylinder initially be at atmospheric pressure and room temperature. The process of performing work can be divided into two stages, each of which includes an adiabatic and isochoric process.
1st stage:
With the tap closed, pump into the cylinder small quantity air and clamp the hose with a clamp. At the same time, the pressure and temperature in the cylinder will increase. This is an adiabatic process. Over time, the pressure in the cylinder will begin to decrease due to the fact that the gas in the cylinder will begin to cool due to heat exchange through the walls of the cylinder. In this case, the pressure will decrease as the volume is built. This is an isochoric process. After waiting until the air temperature inside the cylinder is equal to the ambient air temperature, we will record the pressure gauge readings.
2nd stage:
Now open tap 3 for 1-2 seconds. The air in the balloon will expand adiabatically to atmospheric pressure. At the same time, the temperature in the cylinder will decrease. Then we close the tap. Over time, the pressure in the cylinder will begin to increase due to the fact that the gas in the cylinder will begin to heat up due to heat exchange through the walls of the cylinder. In this case, the pressure will increase again at a constant volume. This is an isochoric process. After waiting until the air temperature inside the cylinder compares with the ambient air temperature, we record the pressure gauge reading. For each branch of 2 stages, you can write the corresponding adiabatic and isochore equations. The result is a system of equations that include the adiabatic exponent. Their approximate solution leads to the following calculation formula for the desired value.
