Adiabatic process


An adiabatic process occurs without transferring heat or mass between a thermodynamic system and its surroundings. Unlike an isothermal process, an adiabatic process transfers energy to the surroundings only as work. It also conceptually undergirds the theory used to expound the first law of thermodynamics and is therefore a key thermodynamic concept.
Some chemical and physical processes occur too rapidly for energy to enter or leave the system as heat, allowing a convenient "adiabatic approximation". For example, the adiabatic flame temperature uses this approximation to calculate the upper limit of flame temperature by assuming combustion loses no heat to its surroundings.
In meteorology and oceanography, adiabatic cooling produces condensation of moisture or salinity, oversaturating the parcel. Therefore, the excess must be removed. There, the process becomes a pseudo-adiabatic process whereby the liquid water or salt that condenses is assumed to be removed upon formation by idealized instantaneous precipitation. The pseudoadiabatic process is only defined for expansion because a compressed parcel becomes warmer and remains undersaturated.

Description

A process without transfer of heat or matter to or from a system, so that, is called adiabatic, and such a system is said to be adiabatically isolated. The assumption that a process is adiabatic is a frequently made simplifying assumption. For example, the compression of a gas within a cylinder of an engine is assumed to occur so rapidly that on the time scale of the compression process, little of the system's energy can be transferred out as heat to the surroundings. Even though the cylinders are not insulated and are quite conductive, that process is idealized to be adiabatic. The same can be said to be true for the expansion process of such a system.
The assumption of adiabatic isolation is useful and often combined with other such idealizations to calculate a good first approximation of a system's behaviour. For example, according to Laplace, when sound travels in a gas, there is no time for heat conduction in the medium, and so the propagation of sound is adiabatic. For such an adiabatic process, the modulus of elasticity can be expressed as, where is the ratio of specific heats at constant pressure and at constant volume and is the pressure of the gas.

Various applications of the adiabatic assumption

For a closed system, one may write the first law of thermodynamics as :, where denotes the change of the system's internal energy, the quantity of energy added to it as heat, and the work done by the system on its surroundings.
Naturally occurring adiabatic processes are irreversible.
The transfer of energy as work into an adiabatically isolated system can be imagined as being of two idealized extreme kinds. In one such kind, no entropy is produced within the system, and the work is only pressure-volume work. In nature, this ideal kind occurs only approximately because it demands an infinitely slow process and no sources of dissipation.
The other extreme kind of work is isochoric work, for which energy is added as work solely through friction or viscous dissipation within the system. A stirrer that transfers energy to a viscous fluid of an adiabatically isolated system with rigid walls, without phase change, will cause a rise in temperature of the fluid, but that work is not recoverable. Isochoric work is irreversible. The second law of thermodynamics observes that a natural process, of transfer of energy as work, always consists at least of isochoric work and often both of these extreme kinds of work. Every natural process, adiabatic or not, is irreversible, with, as friction or viscosity are always present to some extent.

Adiabatic heating and cooling

The adiabatic compression of a gas causes a rise in temperature of the gas. Adiabatic expansion against pressure, or a spring, causes a drop in temperature. In contrast, free expansion is an isothermal process for an ideal gas.
Adiabatic heating occurs when the pressure of a gas is increased by work done on it by its surroundings, e.g., a piston compressing a gas contained within a cylinder and raising the temperature where in many practical situations heat conduction through walls can be slow compared with the compression time. This finds practical application in diesel engines which rely on the lack of heat dissipation during the compression stroke to elevate the fuel vapor temperature sufficiently to ignite it.
Adiabatic heating occurs in the Earth's atmosphere when an air mass descends, for example, in a katabatic wind, Foehn wind, or chinook wind flowing downhill over a mountain range. When a parcel of air descends, the pressure on the parcel increases. Because of this increase in pressure, the parcel's volume decreases and its temperature increases as work is done on the parcel of air, thus increasing its internal energy, which manifests itself by a rise in the temperature of that mass of air. The parcel of air can only slowly dissipate the energy by conduction or radiation, and to a first approximation it can be considered adiabatically isolated and the process an adiabatic process.
Adiabatic cooling occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand, thus causing it to do work on its surroundings. When the pressure applied on a parcel of air is reduced, the air in the parcel is allowed to expand; as the volume increases, the temperature falls as its internal energy decreases. Adiabatic cooling occurs in the Earth's atmosphere with orographic lifting and lee waves, and this can form pileus or lenticular clouds.
Adiabatic cooling does not have to involve a fluid. One technique used to reach very low temperatures is via adiabatic demagnetisation, where the change in magnetic field on a magnetic material is used to provide adiabatic cooling. Also, the contents of an expanding universe can be described as an adiabatically cooling fluid.
Rising magma also undergoes adiabatic cooling before eruption, particularly significant in the case of magmas that rise quickly from great depths such as kimberlites.
In the Earth's convecting mantle beneath the lithosphere, the mantle temperature is approximately an adiabat. The slight decrease in temperature with shallowing depth is due to the decrease in pressure the shallower the material is in the Earth.
Such temperature changes can be quantified using the ideal gas law, or the hydrostatic equation for atmospheric processes.
In practice, no process is truly adiabatic. Many processes rely on a large difference in time scales of the process of interest and the rate of heat dissipation across a system boundary, and thus are approximated by using an adiabatic assumption. There is always some heat loss, as no perfect insulators exist.

Ideal gas (reversible process)

The mathematical equation for an ideal gas undergoing a reversible adiabatic process can be represented by the polytropic process equation
where is pressure, is volume, and for this case, where
being the specific heat for constant pressure, being the specific heat for constant volume, is the adiabatic index, and is the number of degrees of freedom.
For a monatomic ideal gas,, and for a diatomic gas . Note that the above formula is only applicable to classical ideal gases and not Bose–Einstein or Fermi gases.
For reversible adiabatic processes, it is also true that
where T is an absolute temperature. This can also be written as

Example of adiabatic compression

The compression stroke in a gasoline engine can be used as an example of adiabatic compression. The model assumptions are: the uncompressed volume of the cylinder is one litre ; the gas within is the air consisting of molecular nitrogen and oxygen only ; the compression ratio of the engine is 10:1 ; and the uncompressed gas is at approximately room temperature and pressure.
so our adiabatic constant for this example is about 6.31 Pa m4.2.
The gas is now compressed to a 0.1 L volume. The adiabatic constant remains the same, but with the resulting pressure unknown
so solving for P2:
or 25.1 bar. Note that this pressure increase is more than a simple 10:1 compression ratio would indicate; this is because the gas is not only compressed, but the work done to compress the gas also increases its internal energy, which manifests itself by a rise in the gas temperature and an additional rise in pressure above what would result from a simplistic calculation of 10 times the original pressure.
We can solve for the temperature of the compressed gas in the engine cylinder as well, using the ideal gas law, PV = nRT. Our initial conditions being 100 kPa of pressure, 1 L volume, and 300 K of temperature, our experimental constant is:
We know the compressed gas has = 0.1 L and =, so we can solve for temperature:
That is a final temperature of 753 K, or 479 °C, or 896 °F, well above the ignition point of many fuels. This is why a high-compression engine requires fuels specially formulated to not self-ignite, or that a supercharger with an intercooler to provide a pressure boost but with a lower temperature rise would be advantageous. A diesel engine operates under even more extreme conditions, with compression ratios of 16:1 or more being typical, in order to provide a very high gas temperature, which ensures immediate ignition of the injected fuel.

Adiabatic free expansion of a gas

For an adiabatic free expansion of an ideal gas, the gas is contained in an insulated container and then allowed to expand in a vacuum. Because there is no external pressure for the gas to expand against, the work done by or on the system is zero. Since this process does not involve any heat transfer or work, the first law of thermodynamics then implies that the net internal energy change of the system is zero. For an ideal gas, the temperature remains constant because the internal energy only depends on temperature in that case. Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible.

Derivation of ''P''–''V'' relation for adiabatic heating and cooling

The definition of an adiabatic process is that heat transfer to the system is zero,. Then, according to the first law of thermodynamics,
where is the change in the internal energy of the system and is work done by the system. Any work done must be done at the expense of internal energy, since no heat is being supplied from the surroundings. Pressure–volume work done by the system is defined as
However, does not remain constant during an adiabatic process but instead changes along with.
It is desired to know how the values of and relate to each other as the adiabatic process proceeds. For an ideal gas the internal energy is given by
where is the number of degrees of freedom divided by two, is the universal gas constant and is the number of moles in the system.
Differentiating equation yields
Equation is often expressed as because.
Now substitute equations and into equation to obtain
factorize :
and divide both sides by :
After integrating the left and right sides from to and from to and changing the sides respectively,
Exponentiate both sides, substitute with, the heat capacity ratio
and eliminate the negative sign to obtain
Therefore,
and

Derivation of ''P''–''T'' relation for adiabatic heating and cooling

Substituting the ideal gas law into the above, we obtain
which simplifies to

Derivation of discrete formula and work expression

The change in internal energy of a system, measured from state 1 to state 2, is equal to
At the same time, the work done by the pressure–volume changes as a result from this process, is equal to
Since we require the process to be adiabatic, the following equation needs to be true
By the previous derivation,
Rearranging gives
Substituting this into gives
Integrating we obtain the expression for work,
Substituting in second term,
Rearranging,
Using the ideal gas law and assuming a constant molar quantity,
By the continuous formula,
or
Substituting into the previous expression for,
Substituting this expression and in gives
Simplifying,

Graphing adiabats

An adiabat is a curve of constant entropy in a diagram. Some properties of adiabats on a P–V diagram are indicated. These properties may be read from the classical behaviour of ideal gases, except in the region where PV becomes small, where quantum effects become important.
  1. Every adiabat asymptotically approaches both the V axis and the P axis.
  2. Each adiabat intersects each isotherm exactly once.
  3. An adiabat looks similar to an isotherm, except that during an expansion, an adiabat loses more pressure than an isotherm, so it has a steeper inclination.
  4. If isotherms are concave towards the north-east direction, then adiabats are concave towards the east north-east.
  5. If adiabats and isotherms are graphed at regular intervals of entropy and temperature, respectively, then as the eye moves towards the axes, it sees the density of isotherms stay constant, but it sees the density of adiabats grow. The exception is very near absolute zero, where the density of adiabats drops sharply and they become rare.
The right diagram is a P–V diagram with a superposition of adiabats and isotherms:
The isotherms are the red curves and the adiabats are the black curves.
The adiabats are isentropic.
Volume is the horizontal axis and pressure is the vertical axis.

Etymology

The term adiabatic is an anglicization of the Greek term ἀδιάβατος "impassable".
It is used in the thermodynamic sense by Rankine, and adopted by Maxwell in 1871.
The etymological origin corresponds here to an impossibility of transfer of energy as heat and of transfer of matter across the wall.
The Greek word ἀδιάβατος is formed from privative ἀ- and διαβατός, "passable", in turn deriving from διά, and βαῖνειν.

Conceptual significance in thermodynamic theory

The adiabatic process has been important for thermodynamics since its early days. It was important in the work of Joule because it provided a way of nearly directly relating quantities of heat and work.
Energy can enter or leave a thermodynamic system enclosed by walls that prevent mass transfer only as heat or work. Therefore, a quantity of work in such a system can be related almost directly to an equivalent quantity of heat in a cycle of two limbs. The first limb is an isochoric adiabatic work process increasing the system's internal energy; the second, an isochoric and workless heat transfer returning the system to its original state. Accordingly, Rankine measured quantity of heat in units of work, rather than as a calorimetric quantity. In 1854, Rankine used a quantity that he called "the thermodynamic function" that later was called entropy, and at that time he wrote also of the "curve of no transmission of heat", which he later called an adiabatic curve. Besides its two isothermal limbs, Carnot's cycle has two adiabatic limbs.
For the foundations of thermodynamics, the conceptual importance of this was emphasized by Bryan, by Carathéodory, and by Born. The reason is that calorimetry presupposes a type of temperature as already defined before the statement of the first law of thermodynamics, such as one based on empirical scales. Such a presupposition involves making the distinction between empirical temperature and absolute temperature. Rather, the definition of absolute thermodynamic temperature is best left till the second law is available as a conceptual basis.
In the eighteenth century, the law of conservation of energy was not yet fully formulated or established, and the nature of heat was debated. One approach to these problems was to regard heat, measured by calorimetry, as a primary substance that is conserved in quantity. By the middle of the nineteenth century, it was recognized as a form of energy, and the law of conservation of energy was thereby also recognized. The view that eventually established itself, and is currently regarded as right, is that the law of conservation of energy is a primary axiom, and that heat is to be analyzed as consequential. In this light, heat cannot be a component of the total energy of a single body because it is not a state variable but, rather, a variable that describes a transfer between two bodies. The adiabatic process is important because it is a logical ingredient of this current view.

Divergent usages of the word ''adiabatic''

This present article is written from the viewpoint of macroscopic thermodynamics, and the word adiabatic is used in this article in the traditional way of thermodynamics, introduced by Rankine. It is pointed out in the present article that, for example, if a compression of a gas is rapid, then there is little time for heat transfer to occur, even when the gas is not adiabatically isolated by a definite wall. In this sense, a rapid compression of a gas is sometimes approximately or loosely said to be adiabatic, though often far from isentropic, even when the gas is not adiabatically isolated by a definite wall.
Quantum mechanics and quantum statistical mechanics, however, use the word adiabatic in a very different sense, one that can at times seem almost opposite to the classical thermodynamic sense. In quantum theory, the word adiabatic can mean something perhaps near isentropic, or perhaps near quasi-static, but the usage of the word is very different between the two disciplines.
On the one hand, in quantum theory, if a perturbative element of compressive work is done almost infinitely slowly, it is said to have been done adiabatically. The idea is that the shapes of the eigenfunctions change slowly and continuously, so that no quantum jump is triggered, and the change is virtually reversible. While the occupation numbers are unchanged, nevertheless there is change in the energy levels of one-to-one corresponding, pre- and post-compression, eigenstates. Thus a perturbative element of work has been done without heat transfer and without introduction of random change within the system. For example, Max Born writes "Actually, it is usually the 'adiabatic' case with which we have to do: i.e. the limiting case where the external force acts very slowly. In this case, to a very high approximation
that is, there is no probability for a transition, and the system is in the initial state after cessation of the perturbation. Such a slow perturbation is therefore reversible, as it is classically."
On the other hand, in quantum theory, if a perturbative element of compressive work is done rapidly, it randomly changes the occupation numbers of the eigenstates, as well as changing their shapes. In that theory, such a rapid change is said not to be adiabatic, and the contrary word diabatic is applied to it. One might guess that perhaps Clausius, if he were confronted with this, in the now-obsolete language he used in his day, would have said that "internal work" was done and that 'heat was generated though not transferred'.
In classical thermodynamics, such a rapid change would still be called adiabatic because the system is adiabatically isolated, and there is no transfer of energy as heat. The strong irreversibility of the change, due to viscosity or other entropy production, does not impinge on this classical usage.
Thus for a mass of gas, in macroscopic thermodynamics, words are so used that a compression is sometimes loosely or approximately said to be adiabatic if it is rapid enough to avoid heat transfer, even if the system is not adiabatically isolated. But in quantum statistical theory, a compression is not called adiabatic if it is rapid, even if the system is adiabatically isolated in the classical thermodynamic sense of the term. The words are used differently in the two disciplines, as stated just above.