# Adiabatic process

:"This article covers adiabatic processes in

thermodynamics . For adiabatic processes inquantum mechanics , seeadiabatic process (quantum mechanics) . For atmospheric adiabatic processes, seelapse rate ."In

thermodynamics , an**adiabatic process**or an**isocaloric process**is athermodynamic process in which noheat is transferred to or from the workingfluid . The term "adiabatic" literally means impassable (from Greek ἀ-διὰ-βαῖνειν, not-through-to pass), corresponding here to an absence ofheat transfer . Conversely, a process that involves heat transfer (addition or loss of heat to the surroundings) is generally called**diabatic**. Linguistically, "diabatic" is the opposite of "adiabatic" by the absence of the initial "a" (which conveys negation in Greek).For example, an

**adiabatic boundary**is a boundary that is impermeable to heat transfer and the system is said to be adiabatically (or thermally) insulated; an insulated wall approximates an adiabatic boundary. Another example is theadiabatic flame temperature , which is the temperature that would be achieved by a flame in the absence of heat loss to the surroundings. An adiabatic process that is reversible is also called anisentropic process . Additionally, an adiabatic process that is irreversible and extracts no work is in anisenthalpic process, such as viscous drag, progressing towards a nonnegative change in entropy.One opposite extreme—allowing heat transfer with the surroundings, causing the temperature to remain constant—is known as an

isothermal process . Since temperature is thermodynamically conjugate toentropy , the isothermal process is conjugate to the adiabatic process for reversible transformations.A transformation of a thermodynamic system can be considered adiabatic when it is quick enough that no significant heat is transferred between the system and the outside. At the opposite extreme, a transformation of a thermodynamic system can be considered isothermal if it is slow enough so that the system's temperature remains constant by heat exchange with the outside.

**Adiabatic heating and cooling**Adiabatic changes in temperature occur due to changes in

pressure of agas while not adding or subtracting anyheat .**Adiabatic heating**occurs when the pressure of a gas is increased from work done on it by its surroundings, ie apiston .Diesel engines rely on adiabatic heating during their compression stroke to elevate the temperature sufficiently to ignite the fuel. Similarlyjet engines rely upon adiabatic heating to create the correct compression of the air to enable fuel to be injected and ignition to then occur.Adiabatic heating also occurs in the

Earth's atmosphere when anair mass descends, for example, in akatabatic wind orFoehn wind flowing downhill.**Adiabatic cooling**occurs when the pressure of a substance is decreased as it does work on its surroundings. Adiabatic cooling does not have to involve a fluid. One technique used to reach very low temperatures (thousandths and even millionths of a degree above absolute zero) is adiabatic demagnetisation, where the change inmagnetic field on a magnetic material is used to provide adiabatic cooling. Adiabatic cooling also occurs in theEarth's atmosphere withorographic lifting andlee waves , and this can form pileus orlenticular cloud s if the air is cooled below thedew point .Rising magma also undergoes adiabatic cooling before eruption.

Such temperature changes can be quantified using the

ideal gas law , or thehydrostatic equation for atmospheric processes.It should be noted that no process is truly adiabatic. Many processes are close to adiabatic and can be easily approximated by using an adiabatic assumption, but there is always some heat loss. "There is no such thing as a perfect insulator".

**Ideal gas (reversible case only)**The mathematical equation for an ideal fluid undergoing a reversible (i.e., no entropy generation) adiabatic process is: $P\; V^\{gamma\}\; =\; operatorname\{constant\}\; qquad$where "P" is pressure, "V" is volume, and: $gamma\; =\; \{C\_\{P\}\; over\; C\_\{V\; =\; frac\{alpha\; +\; 1\}\{alpha\},$$C\_\{P\}$ being the

specific heat for constant pressure and$C\_\{V\}$ being the specific heat for constant volume.$alpha$ comes from the number of degrees of freedom divided by 2 (3/2 for monatomic gas, 5/2 for diatomic gas).For a monatomic ideal gas, $gamma\; =\; 5/3$, and for a diatomic gas (such asnitrogen andoxygen , the main components of air) $gamma\; =\; 7/5$. 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

: $P^\{gamma-1\}T^\{-gamma\}=\; operatorname\{constant\}$

: $VT^alpha\; =\; operatorname\{constant\}$

where "T" is an absolute temperature.

This can also be written as

: $TV^\{gamma\; -\; 1\}\; =\; operatorname\{constant\}$

**Derivation of continuous formula**The definition of an adiabatic process is that heat transfer to the system is zero, $delta\; Q=0$. Then, according to the

first law of thermodynamics ,:$ext\{(1)\}\; qquad\; d\; U\; +\; delta\; W\; =\; delta\; Q\; =\; 0,$

where "dU" is the change in the internal energy of the system and "δW" is work done"by" the system. Any work ("δW") 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

:$ext\{(2)\}\; qquad\; delta\; W\; =\; P\; ,\; dV.$

However, "P" does not remain constant during an adiabatic process butinstead 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

:$ext\{(3)\}\; qquad\; U\; =\; alpha\; n\; R\; T,$

where "R" is the

universal gas constant and "n" is thenumber of moles in the system (a constant).Differentiating Equation (3) and use of the

ideal gas law , $P\; V\; =\; n\; R\; T$, yields:$ext\{(4)\}\; qquad\; d\; U\; =\; alpha\; n\; R\; ,\; dT\; =\; alpha\; ,\; d\; (P\; V)\; =\; alpha\; (P\; ,\; dV\; +\; V\; ,\; dP).$

Equation (4) is often expressed as $d\; U\; =\; n\; C\_\{V\}\; ,\; d\; T$because $C\_\{V\}\; =\; alpha\; R$.

Now substitute equations (2) and (4) into equation (1) to obtain

: $-P\; ,\; dV\; =\; alpha\; P\; ,\; dV\; +\; alpha\; V\; ,\; dP,$

simplify:

: $-\; (alpha\; +\; 1)\; P\; ,\; dV\; =\; alpha\; V\; ,\; dP,$

and divide both sides by "PV":

: $-(alpha\; +\; 1)\; \{d\; V\; over\; V\}\; =\; alpha\; \{d\; P\; over\; P\}.$

After integrating the left and right sides from $V\_0$ to V and from $P\_0$ to P and changing the sides respectively,

: $ln\; left(\; \{P\; over\; P\_0\}\; ight)\; =\; \{-\{alpha\; +\; 1\; over\; alpha\; ln\; left(\; \{V\; over\; V\_0\}\; ight).$

Exponentiate both sides,

: $left(\; \{P\; over\; P\_0\}\; ight)\; =left(\; \{V\; over\; V\_0\}\; ight)^\{-\{alpha\; +\; 1\; over\; alpha,$

and eliminate the negative sign to obtain

: $left(\; \{P\; over\; P\_0\}\; ight)=left(\; \{V\_0\; over\; V\}\; ight)^\{alpha\; +\; 1\; over\; alpha\}.$

Therefore,

: $left(\; \{P\; over\; P\_0\}\; ight)\; left(\; \{V\; over\; V\_0\}\; ight)^\{alpha+1\; over\; alpha\}\; =\; 1$

and

: $P\; V^\{alpha+1\; over\; alpha\}\; =\; P\_0\; V\_0^\{alpha+1\; over\; alpha\}\; =\; P\; V^gamma\; =\; operatorname\{constant\}.$

**Derivation of discrete formula**The change in internal energy of a system, measured from state 1 to state 2, is equal to

:$ext\{(1)\}\; qquad\; delta\; U\; =\; alpha\; R\; n\_2T\_2\; -\; alpha\; R\; n\_1T\_1\; =\; alpha\; R\; (n\_2T\_2\; -\; n\_1T\_1)$

At the same time, the work done by the pressure-volume changes as a result from this process, is equal to

:$ext\{(2)\}\; qquad\; delta\; W\; =\; P\_2V\_2\; -\; P\_1V\_1$

Since we require the process to be adiabatic, the following equation needs to be true

:$ext\{(3)\}\; qquad\; delta\; U\; +\; delta\; W\; =\; 0$

Substituting (1) and (2) in (3) leads to

:$alpha\; R\; (n\_2T\_2\; -\; n\_1T\_1)\; +\; (P\_2V\_2\; -\; P\_1V\_1)\; =\; 0\; qquad\; qquad\; qquad$

or

:$frac\; \{(P\_2V\_2\; -\; P\_1V\_1)\}\; \{-(n\_2T\_2\; -\; n\_1T\_1)\}\; =\; alpha\; R\; qquad\; qquad\; qquad$

If it's further assumed that there are no changes in molar quantity (as often in practical cases), the formula is simplified tothis one::$frac\; \{(P\_2V\_2\; -\; P\_1V\_1)\}\; \{-(T\_2\; -\; T\_1)\}\; =\; alpha\; n\; R\; qquad\; qquad\; qquad$

**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 (just like isotherms).

#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 (more vertical).

#If isotherms are concave towards the "north-east" direction (45 °), then adiabats are concave towards the "east north-east" (31 °).

#If adiabats and isotherms are graphed severally at regular changes of entropy and temperature, respectively (like altitude on a contour map), then as the eye moves towards the axes (towards the south-west), 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 (seeNernst's theorem ).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.

**ee also***

Cyclic process

*First law of thermodynamics

*Isobaric process

*Isochoric process

*Isothermal process

*Polytropic process

*Thermodynamic entropy

*Quasistatic equilibrium

*Total air temperature

*Adiabatic engine

*Magnetic_refrigeration **References****External links**[

*http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html#c1: Article in HyperPhysics Encyclopaedia*]

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

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