Adiabatic change of state
An adiabatic or adiabatic change of state ( Greek α a , German 'not' and διαβαίνειν diabaínein 'go through') is a thermodynamic process in which a system is transferred from one state to another without exchanging heat with its surroundings. In this sense, adiabatic and “heat-tight” are used synonymously. The property of such a system not to exchange heat with the environment is called adiabasis . In contrast, in diabatic and diathermic processes, heat is exchanged with the environment (see, for example, isothermal change in state ) .
Adiabatic changes in state, in which the system is almost in equilibrium at any point in time from the beginning to the end of the change, are called quasi-static ; their course can be represented by a curve in the state space. If the quasi-static change of state is controlled solely by changing external parameters of the system by means of idealized external devices, then these curves are called adiabatic . External parameters are quantities that describe the external, idealized secondary conditions of the thermodynamic system; such as the volume of the system or the components of the magnetic field strength of an external magnetic field.
Thought experiments with adiabatic changes of state are fundamental for the determination of the postulates of thermodynamics . They provide the connection between the work done on a system and the internal energy of the system. The Carnot cycle , which is often used in the literature of thermodynamics, includes the adiabatic compression and expansion of the working gas. In the axiomatic structure of thermodynamics, adiabatic changes of state are of central importance. The conditions for adiabatic changes of state are never quite achieved in practice. However, this idealization provides usable to good descriptions for many real processes: for example, for processes that run rapidly, where there is insufficient time for temperature equalization, or for changes to systems in particularly heat-insulating containers.
The concept of the adiabatic change of state developed together with the gas and heat theory in the 19th century.
The calculation of the speed of sound in air inspired Laplace and others to investigate the adiabatic state changes in gases. In 1802 he led a too small calculated speed of sound due to the fact that there will be no temperature compensation in the rapid expansion and compression of air and the Boyle-Mariotte , who does not meet here. In 1802 Biot and in 1808 Poisson published calculations on temperature changes during adiabatic compression in sound waves. During this time, the first detailed measurements of the specific heat capacities of gases were carried out. Ch.B.Clement and N. Desormes published in 1819 for first air measured values of the ratio of heat capacity at constant pressure to that at constant volume . In 1823 Poisson calculated the speed of sound using this value and a theory by Laplace.
In 1823 Poisson argued for an understanding of heat as a function of state using adiabatic volume changes. In 1824 Nicolas Léonard Sadi Carnot used in his work Réflexions sur la puissance motrice du feu adiabatic changes of state to move the working material of his ideal heat engine between the two heat reservoirs.
The term adiabatic for a change of state without heat transfer can be found in literature from the second half of the 19th century, for example Rankine speaks of adiabatic curves in the work On the theory of explosive gas engines 1866 .
In 1909, the mathematician C. Carathéodory published a paper on an axiomatic foundation of thermodynamics. In this work, adiabatic changes of state in simple thermodynamic systems are of central importance. In a more recent work on the second law of thermodynamics and entropy from 1999, Lieb and Yngvason use adiabatic state changes to define the relation of adiabatic reachability in the thermodynamic state space.
The definition of the adiabatic change of state given in the introduction includes many types of thermodynamic processes, including those that are not quasi-static. When reading various textbooks, a different impression can arise, since adiabatic changes of state are often only considered in connection with quasi-static processes in simple systems.
According to the definition, in the event of an adiabatic change in state, energy may only be added to or removed from the thermodynamic system by means of mechanical, electrical or magnetic work ; the system must be insulated against heat flows of any form ; In the ideal case there should be no heat conduction , no convective heat transfer and no heat radiation between the system and the outside world.
In reality, complete thermal insulation cannot be achieved, but real processes can take place adiabatically to a good approximation if
- they take place in a well-insulated container, e.g. in a Dewar or an adiabatic calorimeter ,
- the change of state takes place so quickly that little heat can flow in or out in the short time (e.g. in an internal combustion engine, in an air pump or in the propagation of sound ) or
- the volume of the system is very large, so that heat flows at its edge play practically no role (e.g. in the case of thermally rising air parcels).
Compression and expansion of gases
The compression of the air in an air pump is approximately an adiabatic change of state. The work done on the pump increases the internal energy of the air, and so does the temperature of the air. If the compression is repeated quickly, for example when inflating a bicycle tire, the temperature increase on the pump can be clearly felt. The additional internal energy is also known colloquially as heat of compression or heat of compression .
A pneumatic lighter quickly compresses air to less than one twentieth of its original volume. The air becomes so hot that the stored scale glows and a fire can then be kindled.
Conversely, if internal energy is converted into volume work when an air volume expands, the temperature of the expanding gas falls. The cooling of air masses during thermal lift or when ascending the edges of mountains is known. Adiabatic expansion and thus cooling of the air also occurs on the upper side of the wings of commercial aircraft, see dynamic lift . The cooling becomes visible when the saturation concentration , which decreases with the temperature , falls below the existing air humidity and clouds or fog form.
The Gay-Lussac experiment is also an adiabatic change of state. However, when the gas expands, no internal energy is converted into external work. The process is not quasi-static, only at the beginning and the end is the system in thermodynamic equilibrium. Ideal gases do not change their temperature.
It is an adiabatic change of state when frictional work is carried out on a thermally isolated system and the system is in thermodynamic equilibrium at the beginning and at the end of the work process.
Carrying out the experiment in James Prescott Joule's classical experiment to determine the heat equivalent is one such process. Joule's system consisted of a copper tank with water and an integrated agitator. Using the stirrer and a device with weights, precisely measurable mechanical work was converted into internal energy of the system (especially water). Joule measured the temperature before and after the friction work. In his test report from 1850 he also goes into his precautions for thermal insulation of the system.
Instead of a defined amount of mechanical work, a measured amount of electrical work can also be performed on a system, for example to determine the heat capacity of a substance. The agitator is replaced by an electric heater; the thermodynamic system consists of a container, the substance and the heater. For thermal insulation, the system is best placed in a Dewar or an adiabatic calorimeter. If the system is in thermodynamic equilibrium before and after the introduction of electrical energy, there is an adiabatic change in state.
An accumulator as a thermodynamic system can deliver energy to the outside in the form of electrical work or it can be supplied with energy from the outside by means of electrical work, depending on the direction of the current in the electrical supply lines. The supply or withdrawal of electrical energy leads to a change in the amount of substance on the electrodes. If the accumulator is thermally insulated by an adiabatic calorimeter, it is a question of adiabatic state changes. Since resistive losses always occur during the process, the system heats up to a greater or lesser extent. The change in state is not reversible because of these ohmic losses. Such adiabatic changes in state are brought about and measured during safety tests on batteries, for example.
In thermodynamics, adiabatic changes of state are often considered, in which the initial state consists of two systems, each of which is in thermodynamic equilibrium. The two systems are viewed as a single composite system. The change of state takes place in which the systems are connected to one another without any effort and then interact with one another; for example through thermal contact or by removing a partition between the systems - e.g. B. Open a isolation valve. The processes involved are irreversible and can be very violent. The change is complete when the entire system no longer changes after the coupling, i.e. has found a new thermodynamic equilibrium.
In order for the change of state to be adiabatic, the assembled system must be thermally isolated from the environment; Dewar vessels or adiabatic calorimeters are suitable for this. Here are some examples of such state changes:
- Determination of the heat of solution : One system consists of a known amount of water and the other a known amount of table salt. Both systems are at room temperature. Then the salt is added to the water - coupling the systems. It dissolves in the water, then the temperature of the salt solution is measured.
- Heat of neutralization determination : one system is an acid and the other a lye. The acid and alkali are then carefully stirred together and the temperature of the mixture monitored until it no longer changes. The heat of neutralization can be determined from the change in temperature and the amount of substance.
- Chemical reaction : At the beginning there is a system with two moles of hydrogen and a system with one mole of oxygen. If a separating valve between the two systems is opened, an oxyhydrogen gas mixture is created, which ultimately reacts to form water. In the final state, the system consists of water.
- Phases in equilibrium : one system is water in the liquid phase, the second water vapor in a container. After the two systems are connected, an equilibrium is established between the liquid and the gaseous phase.
Heat balance between two systems
Let A and B be two simple systems, separate at the start of the process and each in thermal equilibrium. A has a higher temperature than B.
- Irreversible heat balance : If the systems are pushed together so that they touch, or if they are thermally connected to one another via a copper wire , heat flows irreversibly from system A to system B until both systems have the same final temperature.
- Reversible heat balance : If the thermal energy is transported from the warmer to the colder system by an ideal heat engine - the Carnot process - the temperatures of the systems converge. In addition to the heat transfer, work is done by the overall system, i.e. energy is withdrawn from it. If this process is repeated until the temperatures of the two systems are the same, the final temperature is lower than in the case of irreversible heat exchange, since the energy of the overall system has decreased, but the entropy has remained the same. Conversely, the temperature difference between systems A and B can also be increased again if the machine between the systems is operated as a heat pump with the addition of external work.
An adiabatic change of state is a change of a thermodynamic system that is isolated apart from work processes from a state of equilibrium to a state of equilibrium ; Mechanical or electrical work can be performed on the system from the outside , or it can perform such work. If energy is supplied to the system through work, then there is ; if the system does work, then it is ; if no work is done, it is . For the structure of thermodynamics it is important that this definition of the adiabatic change of state does without the term heat .
A state of a thermodynamic system is said to be adiabatically attainable from the state if there is an adiabatic change of state, which has an initial and an end state.
The following two postulates apply to a thermodynamic system:
- For every two states of equilibrium and the following applies: can be reached from adiabatically or from .
- If there are two adiabatic changes of state with the same initial state and the same final state , then the work of one change of state is equal to the work of the other.
For two states and it may be that both of a work is adiabatic accessible, and of a work is adiabatic reached, the following applies .
Definition of internal energy
Based on these postulates it is possible to introduce the internal energy as a state function for a thermodynamic system. A value is set arbitrarily for any state of equilibrium . The internal energy for any state then results as follows:
- is from by an adiabatic change in state with a working distance, then applies .
- otherwise needs of by adiabatic change of state with a work to be accessible and it is .
The amount of heat as a physical quantity
The internal energy is only fixed up to one constant. With it, for a general change of state of a thermally non-insulated system from an initial state to an end state and a work, the heat supplied to the system is carried out
Are defined. For an adiabatic change of state follows .
Adiabatic changes of state are not only needed to define internal energy and heat as physical quantities - as just shown - they can also serve to introduce temperature and entropy within the framework of an axiomatic structure of thermodynamics. Here are two approaches worth mentioning:
The mathematician Constantin Carathéodory makes the following statement in his investigation of the fundamentals of thermodynamics:
"In any environment of an arbitrarily prescribed initial state there are states that cannot be approximated arbitrarily by adiabatic state changes."
as an axiom to the beginning of a mathematical model for thermodynamic systems. This axiom is equivalent to the second law of thermodynamics .
In his work, Carathéodory introduces the concept of the simple thermodynamic system : it is a system in which each state of equilibrium is uniquely defined by specifying a value for the internal energy and a tuple of external parameters for the external constraints.
In a more recent work on the second law of thermodynamics, Elliot H. Lieb and Jakob Yngvason develop a mathematical model for thermodynamic systems without the implicit assumptions made by Caratheodory about analytical properties of the state functions. This model is based on the quasi-order in the state space given by the adiabatic accessibility ; see also the adjacent schematic drawing. A presentation in German with clear application examples can be found in a textbook by André Thess.
With additional axioms for the scaling and coupling of thermodynamic systems, Lieb and Yngvason define the state function of entropy for states of equilibrium and, above all, the temperature . The monotony applies to entropy : In a simple thermodynamic system, a state of equilibrium can be reached adiabatically from a state if and only if .
An adiabatic change of state is called quasi-static if the system is almost in thermodynamic equilibrium at all times during the change. In this case, the equilibrium points assumed during the change describe a coherent path in the state space . This path is called adiabatic when the quasi-static change of state is controlled solely by changing external parameters of the system by means of idealized external devices. External parameters are quantities that describe the external, idealized secondary conditions of the thermodynamic system; such as the volume of the system or the components of the magnetic field strength of an external magnetic field. The course of state changes in processes with frictional work, on the other hand, is not referred to as adiabatic.
If the external constraints of a system can be described by external parameters , then there is a function for each external parameter , so that in the case of an adiabatic change of state along an adiabatic, the work done on the system is equal to the path integral over a 1-form.
Simple thermodynamic systems are those systems in which the external parameters together with the internal energy uniquely determine a state of equilibrium; the sizes then form a coordinate system in . Examples of not simple thermodynamic systems are the overall system of two simple systems isolated from one another or, because of the hysteresis, a system with ferromagnetic material.
In simple thermodynamic systems, the quasi-static adiabatic state changes are always reversible . In simple thermodynamic systems the adiabats are thus also curves of constant entropy ; in these systems the adiabats are identical to the isentropes . Because of the great practical importance of these simple systems, adiabats and isentropes are often used synonymously in the literature. This can be confusing, however, since adiabatic and isentropic systems can be different in thermodynamic systems that are not simple . In addition, the introduction of entropy in thermodynamics presupposes the concept of the adiabatic change of state.
Adiabats of the ideal gas
The equilibrium states of the simple thermodynamic system consisting of a fixed amount of substance of a gas in a container with the changeable volume form a two-dimensional state space . If the temperature of the gas and the volume are chosen as coordinates for the points in , the result for the work done on the system by changing the volume is :
The last equality is only valid for an ideal gas , where the pressure is given by the state function of the ideal gas , with the amount of substance and the gas constant . Furthermore, in the case of an ideal gas, the change in internal energy is independent of the volume and proportional to the change in temperature.
is the constant molar heat capacity at constant volume.
The following applies to adiabatic processes
- and thus for the ideal gas
This equation is fulfilled if and only if
applies; That means: if the system is at the beginning of an adiabatic change of state at the point and at the end is the volume , then the final temperature is calculated as:
In the last equation, the exponent was by the frequently used here adiabatic with words, is for air . If the points in are described by the coordinates or , the equation reads
They follow from the first relationship using the equation of state for ideal gases. These equations are called adiabatic equations or Poisson's equations . Each is a conditional equation for the adiabats of an ideal gas in the -, - or -diagram.
Numerical examples for air
With local pressure changes in the course of sound propagation in the air, with the rise of large air masses in the atmosphere, with heat engines (see also compression ratio ) or with an air pump, there are expansions or compressions of air masses, which can often be described as adiabatic changes of state to a good approximation .
The final values or for such a change in state can be calculated from the initial values , the compression ratio and the adiabatic exponent . For some example values of values calculated according to the above adiabatic equation can be found in the following table.
|Cases of adiabatic compression|
|Sound from a speaking
person at a distance of 1 m
|Volume reduction by 10%|
|Volume reduction by 50%|
|Bicycle pump from 1 to 5 bar absolute, i.e. 4 bar overpressure|
|Otto engine (compression 1:10)|
|Diesel engine (compression 1:20)|
|(High pressure compressor fills a diving cylinder from 1 bar ambient pressure to 199 bar overpressure)|
|Compression 1: 100 (there is even more extreme compression, e.g. on re-entry )|
|Cases of adiabatic expansion|
|Ascent of dry air by approx. 100 m
in the lower earth atmosphere
|Ascent of dry air by approx. 1000 m
in the lower earth atmosphere
|Outflow of compressed air, pressure reduction from 10 bar to 1 bar|
|(Discharge of compressed air, pressure reduction from 200 bar to 1 bar)|
The values calculated here apply to idealized air, i.e. under the following assumptions:
- ideal gas, so
- low pressure (<100 bar) (cases with higher pressure are in brackets) and
- sufficiently high temperature (> 200 K)
- a temperature-independent molar heat capacity , in fact this increases slowly with the temperature for nitrogen and oxygen.
- diatomic molecules such as nitrogen N 2 , oxygen O 2 (1- and 3-atom molecules have different adiabatic exponents ); The adiabatic exponent is constant only with noble gases; with polyatomic molecules it slowly falls with increasing temperature.
- dry air, i.e. free from water vapor (if air contains water vapor, condensation can occur, both when the pressure increases and when the temperature drops below the dew point )
- The walls of the container move at speeds that are less than the mean thermal speed of the molecules. The speed of sound can be regarded as a practical limit speed , as it is of the order of magnitude of thermal speeds.
An adiabatic change of state can be viewed and described macroscopically on a system made up of many particles. The changes in the microscopic range can be traced in the following example, where an adiabatic cylinder is assumed in which an ideal, highly diluted gas at ambient temperature is compressed to half its volume by a piston in the course of a change in state. This results in an increase in pressure and an increase in temperature, which can be explained as follows:
- The pressure arises from the fact that the gas particles (atoms in noble gases , molecules in other ideal gases ) move very quickly and regularly collide with the wall of the cylinder or the piston and then bounce off again.
- If the volume is halved, the distance for the gas particles between the cylinder walls and the piston is halved. Since the same number of particles hit the piston twice as often when the path is halved, the macroscopically measurable pressure is doubled. This pressure increase can be calculated from the general gas equation and corresponds to the pressure increase with an isothermal compression , i.e. H. this pressure increase is retained even if the gas is allowed to cool to ambient temperature after adiabatic compression.
- As the piston compresses the volume, it moves. The gas particles that then hit the piston get a higher speed when they rebound. This proportion of the supplied energy increases the temperature of the gas in the piston. The speed of movement of the piston is irrelevant for the temperature increase in a heat-tight cylinder (assuming adiabatic compression). In the case of slow movement, many particle collisions with a small increase in speed take place, or in the case of fast movement there are few collisions with a large increase in speed.
Because the gas particles collide with each other and exchange momentum , the pressure increase is not only evident on the piston, but on the entire inner wall of the cylinder.
- Günther Ludwig : Introduction to the basics of theoretical physics . tape 4 . Vieweg & Sohn, Braunschweig 1979, ISBN 3-528-09184-3 , XIV Thermodynamik (The book shows the fundamental meaning of adiabatic changes of state for thermodynamics. When reading it should be noted: Ludwig uses the term work process instead of adiabatic change of state - the terms denote not exactly the same - and for quasi-static adiabatic change of state along an adiabatic the term adiabatic process .).
- André Thess: The entropy principle - thermodynamics for the dissatisfied . Oldenbourg Wissenschaftsverlag, Munich 2014, ISBN 978-3-486-76045-3 , Adiabatic Accessibility (explains in Chapter 2 the meaning and variety of adiabatic changes in state ).
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- The work done is written here as a differential form, real work results as an integral of this one-form over the process path. Since the differential form for the work performed cannot generally be written as the total differential of a state function, the symbol is used here instead of the symbol .
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