An adiabatic process is a process that occurs without the transfer of heat or matter between
a system and its surroundings. A key concept in thermodynamics, adiabatic transfer provides
a rigorous conceptual basis for the theory used to expound the first law of thermodynamics.
It is also key in a practical sense, that many rapid chemical and physical processes
are described using the adiabatic approximation; such processes are usually followed or preceded
by events that do involve heat transfer. Adiabatic processes are primarily and exactly
defined for a system contained by walls that are completely thermally insulating and impermeable
to matter; such walls are said to be adiabatic. An adiabatic transfer is a transfer of energy
as work across an adiabatic wall or sector of a boundary.
Approximately, a transfer may be regarded as adiabatic if it happens in an extremely
short time, so that there is no opportunity for significant heat exchange.
The adiabatic flame temperature is a virtual quantity. It is the temperature that would
be achieved by a flame in the absence of heat loss to the surroundings.
Description For an adiabatic transfer of energy as work,
the notation Q = 0 indicates that no energy is transferred as heat across the adiabatic
boundary or wall. An ideal or fictive quasi-static adiabatic
transfer of energy as work that occurs without friction or viscous dissipation within the
system is said to be isentropic, with ΔS = 0.
For a natural process of transfer of energy as heat, driven by a finite temperature difference,
entropy is both transferred with the heat and "generated" within the system. Such a
process is neither adiabatic nor isentropic, having Q > 0, and ΔS > 0 according to the
second law of thermodynamics. A natural adiabatic process is irreversible
and is not isentropic. Adiabatic transfer of energy as work can be analyzed into two
extreme component kinds. One extreme kind is without friction or viscous dissipation
within the system, and this is usually pressure-volume work, denoted customarily by P dV. This is
an ideal that does not occur unaccompanied in nature. It may be regarded as "reversible".
The other extreme kind is isochoric work, for which dV = 0, solely through friction
or viscous dissipation within the system. Isochoric work is irreversible. The second
law of thermodynamics observes that a natural process of transfer of energy as work, exactly
considered, always consists at least of isochoric work and often of both of these extreme kinds
of work. Every natural process, adiabatic or not, exactly considered, is irreversible,
with ΔS > 0, however slight may be the friction or viscosity.
Adiabatic heating and cooling Adiabatic changes in temperature occur due
to changes in pressure of a gas while not adding or subtracting any heat. In contrast,
free expansion is an isothermal process for an ideal gas.
Adiabatic heat occurs when the pressure of a gas is increased from work done on it by
its surroundings, e.g., a piston compressing a gas contained within an adiabatic cylinder.
This finds practical application in Diesel engines which rely on the lack of quick heat
dissipation during their compression stroke to elevate the fuel vapor temperature sufficiently
to ignite it. Adiabatic heating also occurs in the Earth's
atmosphere when an air mass descends, for example, in a katabatic wind or Foehn or chinook
wind flowing downhill over a mountain range. When a parcel of air descends, the pressure
on the parcel increases. Due to this increase in pressure, the parcel's volume decreases
and its temperature increases, thus increasing the internal energy.
Adiabatic cooling occurs when the pressure of a substance is decreased as it does work
on its surroundings. Adiabatic cooling occurs in the Earth's atmosphere with orographic
lifting and lee waves, and this can form pileus or lenticular clouds if the air is cooled
below the dew point. When the pressure applied on a parcel of air decreases, the air in the
parcel is allowed to expand; as the volume increases, the temperature falls and internal
energy decreases. Adiabatic cooling does not have to involve
a fluid. One technique used to reach very low temperatures is 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. 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
The mathematical equation for an ideal gas undergoing a reversible adiabatic process
is
where P is pressure, V is volume, and
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 Let's now look at a common example of adiabatic
compression- the compression stroke in a gasoline engine. We will make a few simplifying assumptions:
that the uncompressed volume of the cylinder is 1000 cm3, that the gas within is nearly
pure nitrogen, and that the compression ratio of the engine is 10:1. The uncompressed gas
is at approximately room temperature and pressure.
so our adiabatic constant for this experiment is about 1.58 billion.
The gas is now compressed to a 100cc volume. The new volume is 100 ccs, but the constant
for this experiment is still 1.58 billion:
so solving for P:
or about 362 PSI or 24.5 atm. 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 has also heated the gas and the hotter gas will have
a greater pressure even if the volume had not changed.
We can solve for the temperature of the compressed gas in the engine cylinder as well, using
the ideal gas law. Our initial conditions are 100,000 pa of pressure, 1000 cc volume,
and 300 K of temperature, so our experimental constant is:
We know the compressed gas has V = 100 cc and P = 2.50E6 pascals, so we can solve for
temperature by simple algebra:
That's a final temperature of 751 K, or 477 °C, or 892 °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 and inter cooler to provide a lower temperature
at the same pressure would be advantageous. A diesel engine operates under even more extreme
conditions, with compression ratios of 20:1 or more being typical, in order to provide
a very high gas temperature which ensures immediate ignition of 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 continuous formula 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 dU is the change in the internal energy of the system and δW is work done by the
system. Any work done must be done at the expense of internal energy U, since no heat
δQ is being supplied from the surroundings. Pressure-volume work δW done by the system
is defined as
However, P does not remain constant during an adiabatic process but instead changes along
with V. It is desired to know how the values of dP
and dV 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, R is the universal gas constant
and n is the number of moles in the system. Differentiating Equation and use of the ideal
gas law, , yields
Equation is often expressed as because . Now substitute equations and into equation
to obtain
factorize ::
and divide both sides by PV:
After integrating the left and right sides from to V and from to P and changing the sides
respectively,
Exponentiate both sides, and substitute with , the heat capacity ratio
and eliminate the negative sign to obtain
Therefore,
and
Derivation of discrete formula 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,
Substituting ,
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
on the P-V diagram. Properties of adiabats on a P-V diagram are:
Every adiabat asymptotically approaches both the V axis and the P axis.
Each adiabat intersects each isotherm exactly once.
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.
If isotherms are concave towards the "north-east" direction, then adiabats are concave towards
the "east north-east". If adiabats and isotherms are graphed severally
at regular changes 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 following 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, literally means 'not to be passed through'. It is formed from the
ancient Greek privative "α" + διαβατός, "able to be passed through", in turn deriving
from διὰ-, and βαῖνειν, thus ἀδιάβατος . According to Maxwell, and to Partington,
the term was introduced by Rankine. The etymological origin corresponds here to
an impossibility of transfer of energy as heat and of transfer of matter across the
wall. Usage 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. In this sense, a rapid compression of a gas is approximately adiabatic
even when the gas is not adiabatically isolated. Quantum statistical theory, however, uses
the word adiabatic in an almost opposite sense. On one hand in that theory, if a perturbative
element of compressive work is done almost infinitely slowly on an adiabatically isolated
mass of gas, 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. The occupation numbers are unchanged, though the energy levels
of one-to-one corresponding, pre-and post-change, eigenstates are changed. 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 that theory, if the perturbative
element of compressive work is done rapidly, the compressing piston imparts energy by collision
to the molecules of the gas. This randomly changes the occupation numbers of the eigenstates,
as well as changing their shapes. Further collisions of gas molecules involve viscosity,
with further random changes in occupation numbers. In that theory, such a rapid change
is said not to be 'adiabatic', and the contrary word 'diabatic' is even applied to it. One
might guess that Clausius, if he were confronted with this, would perhaps have said that internal
work was done and that, in the language he used, '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, with viscosity, does not alter this classical usage.
But the words are used differently in quantum theory, as stated just above.
Thus for a mass of gas, in macroscopic thermodynamics, words are so used that a compression is sometimes
approximately said to be adiabatic if it is sufficiently rapid, even if the gas is not
adiabatically isolated. But in quantum statistical theory, a compression is said not to be adiabatic
if it is sufficiently rapid, even if the gas is adiabatically isolated.
See also Cyclic process
First law of thermodynamics Heat burst
Isobaric process Isenthalpic process
Isentropic process Isochoric process
Isothermal process Polytropic process
Entropy Quasistatic process
Total air temperature Magnetic refrigeration
References
Silbey, Robert J.; et al.. Physical chemistry. Hoboken: Wiley. p. 55. ISBN 978-0-471-21504-2.
Broholm, Collin. "Adiabatic free expansion." Physics & Astronomy @ Johns Hopkins University.
N.p., 26 Nov. 1997. Web. 14 Apr. *Nave, Carl Rod. "Adiabatic Processes." HyperPhysics.
N.p., n.d. Web. 14 Apr. 2011. [1]. Thorngren, Dr. Jane R.. "Adiabatic Processes."
Daphne – A Palomar College Web Server. N.p., 21 July
1995. Web. 14 Apr. 2011. [2]. External links
Article in HyperPhysics Encyclopaedia