Isentropic exponent

Special cases of the polytropic change of state :
n = 0: isobaric ,
n = 1: isothermal ,
n = κ: isentropic ,
n = ∞: isochoric

Isentropic exponent for gases at normal pressure

Temp	gas	κ		Temp	gas	κ		Temp	gas	κ
-200 ° C	H 2	1.65		0 ° C	Air dry	1.40		-180 ° C	N 2	1.43
-73 ° C		1.44		400 ° C		1.37		20 ° C		1.40
20 ° C		1.41		1000 ° C		1.32		500 ° C		1.36
1000 ° C		1.36		2000 ° C		1.30		1000 ° C		1.32
2000 ° C		1.31		−55 ° C	CO 2	1.35		2000 ° C		1.30
-250 to 1500 ° C	Hey	1.67		20 ° C		1.29		20 ° C	CH 4	1.31
				400 ° C		1.24		350 ° C		1.18
100 ° C	H 2 O	1.33		1000 ° C		1.18		20 ° C	H 2 S	1.33
200 ° C		1.32		2000 ° C		1.16		500 ° C		1.25
500 ° C		1.28		20 ° C	CO	1.40		20 ° C	NH 3	1.32
1000 ° C		1.23		1000 ° C		1.32		450 ° C		1.20
2000 ° C		1.19		2000 ° C		1.29		-100 to 500 ° C	Ne, Ar Xe, Kr	1.67
20 ° C	NO	1.39		-180 ° C	O 2	1.44
2000 ° C		1.29		20 ° C		1.40		20 ° C	SO 2	1.28
20 ° C	N 2 O	1.28		400 ° C		1.34		250 ° C		1.22
250 ° C		1.22		1000 ° C		1.31		15 ° C	C 2 H 6	1.20
20 ° C	NO 2	1.29		2000 ° C		1.28		15 ° C	C 3 H 8	1.13

Isentropic exponent for supercritical gases at 200 bar pressure

Temp	gas	κ		Temp	gas	κ		Temp	gas	κ
126.2 K	N 2	2.07		154.6 K	O 2	2.25		304.1K	CO 2	2.36
300 K		1.67		300 K		1.77		500 K		1.50
600 K		1.43		600 K		1.41		700 K		1.28
2000 K		1.30		1000 K		1.33		1100 K		1.20
638.9 K	H 2 O *	10.7		5.2 K	Hey	1.13		126.2 K	Ar	2.07
700 K		1.95		300 K		1.65		300 K		2.23
900 K		1.41		700 K		1.66		500 K		1.81
1200 K		1.28		1500 K		1.66		700 K		1.72
33.15K	H 2	1.51		132.9 K	CO	2.54		190.6 K	CH 4	2.00
300 K		1.42		300 K		1.69		300 K		1.91
600 K		1.39		400 K		1.53		400 K		1.47
1000 K		1.38		500 K		1.47		600 K		1.24
* gaseous above 638.9 K (not supercritical)

The isentropic exponent (also called adiabatic exponent or heat capacity ratio ) denoted by the symbol  ( Kappa ) or  ( Gamma ), is the dimensionless ratio of the heat capacities of gases at constant pressure ( C _p ) and at constant volume ( C _V ): ${\displaystyle \kappa }$ ${\displaystyle \gamma }$

${\displaystyle \kappa ={\frac {C_{p}}{C_{V}}}}$

The quotient is a strongly temperature-dependent material property of real gases . Alternatively, it can also be calculated as the quotient of the mass-related or molar heat capacities at constant pressure or constant volume. ${\displaystyle \kappa }$

Its name was given as an exponent in the isentropic equation or adiabatic equation for ideal gases : ${\displaystyle \kappa }$

${\displaystyle \ pV^{\kappa }=\mathrm {const.} }$

Isentropic changes of state are adiabatic and reversible and thus leave the entropy constant. You step z. B. approximately occurs in large-scale air currents, which is why κ is often referred to in meteorology as the adiabatic exponent , adiabatic coefficient or adiabatic index . In technology, an adiabatic change of state (e.g. in a steam turbine ) is usually not reversible, since friction, throttling and shock processes produce entropy (see " Adiabatic machine " and " Second law of thermodynamics "). These changes of state can approximately be described by a polytropic with a polytropic exponent n that differs from κ . The isentropic is the special case of a polytropic with (see picture). ${\displaystyle n=\kappa }$

The isentropic exponent also determines the speed of sound , since the rapid pressure and density fluctuations associated with sound are approximately isentropic. The isentropic exponent can be measured with the help of the Rüchardt experiment .

Calculated heat capacity ratio

The value of the isentropic exponent depends on the degree of freedom of the gas particles and the degree of freedom of a gas molecule depends on the geometry and the bond strength of the atoms. Gas molecules with more atoms have a higher degree of freedom. The degree of freedom is made up of the translational, rotational, and oscillation or vibration degrees of freedom. Translation is excited at all temperatures. Rotation already takes place with lower, vibration of linear molecules from medium, vibration of rigid molecules only at higher temperatures. Therefore, the heat capacity of polyatomic gases increases with increasing temperature. In other words: with decreasing temperature, more and more degrees of freedom “freeze” and the isentropic exponent increases.

In all gases, the isobaric heat capacity runs parallel to the isochoric heat capacity over a wide temperature range. This is why the gas constant , i.e. the difference between isobaric and isochoric molar heat , remains the same over a large temperature range .

The isentropic exponent can also be roughly described as follows:

${\displaystyle \kappa ={\frac {f+2}{f}}}$

The degree of freedom can be roughly described as follows:

${\displaystyle f=3\cdot N-r}$

The degree of freedom f of a body indicates how many possibilities of movement this body has within a coordinate system. The individual mass point has 3 degrees of freedom, it can move in space along the x, y and z axes. It has no freedom of rotation, because a point cannot rotate. A system of N points has 3N degrees of freedom. If there are rigid bonds between the points r, the number of degrees of freedom is reduced to 3N - r. Rigid bodies have angled bonds.

Dry air consists mainly of diatomic molecules and has an isentropic exponent of 1.4. This corresponds to the theoretical value for 3 translational and 2 rotational degrees of freedom in the kinetic gas theory , since with diatomic molecules a rotation about the connecting axis is not possible. At very low temperatures, hydrogen (H ₂ ) has the same value as the monatomic noble gases, because then the rotation itself is stopped. Rotation of polyatomic molecules and oscillation of linear or slightly angled molecules are already excited below normal temperature , oscillation of rigid molecules only above normal temperature. At much higher temperatures, dissociation and ionization lead to even more degrees of freedom. In the atmosphere, the expansion and cooling of the humid air can lead to condensation of the water. The exponent becomes lower due to the released heat of condensation.

Isentropic exponent ; Degree of freedom ; Atomic bonds ; of gases under normal conditions ${\displaystyle \kappa }$ ${\displaystyle f}$
${\displaystyle r}$

Gas molecule	${\displaystyle \kappa ={\frac {f+2}{f}}}$	${\displaystyle f=3\cdot N-r}$	Examples
1-atom	${\displaystyle {\frac {5}{3}}=1{,}{\overline {6}}}$	${\displaystyle 3=3\cdot 1-0}$	Helium, argon
2-atom	${\displaystyle {\frac {7}{5}}=1{,}4}$	${\displaystyle 5=3\cdot 2-1}$	N 2 , O 2 , H 2 , CO, NO
3-atom, rigid (angled)	${\displaystyle {\frac {8}{6}}=1{,}{\overline {3}}}$	${\displaystyle 6=3\cdot 3-3}$	H 2 O vapor at 100 ° C, H 2 S
3-atom, not rigid (linear)	${\displaystyle {\frac {9}{7}}\approx 1{,}29}$	${\displaystyle 7=3\cdot 3-2}$	CO 2 , SO 2 , N 2 O, NO 2
${\displaystyle R=C_{\mathrm {p(mol)} }-C_{\mathrm {V(mol)} }}$; ( Gas constant = 8.314 J / molK)

literature

• Alfred Böge: Manual of mechanical engineering. Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-1025-0 .

Web links

• Video: Determination of Cp / Cv according to Rüchardt . Institute for Scientific Film (IWF) 2004, made available by the Technical Information Library (TIB), doi : 10.3203 / IWF / C-14879 .

swell

1. ↑ ^{a b} NIST Standard Reference Database Number 69
2. ↑ Engineering Toolbox: High temperature cp values
3. ↑ https://link.springer.com/content/pdf/bbm%3A978-3-658-03169-5%2F1.pdf
4. ↑ dtv atlas of physics; Mechanics, acoustics, thermodynamics, optics. Volume 1, Munich 1987ff, ISBN 3-423-03226-X , pp. 49 and 109.