Adiabatikus kitevő

Adiabatikus kitevő különböző gázokra.^[1]^[2]
Hőm.	Gáz	κ	Hőm.	Gáz	κ	Hőm.	Gáz	κ
–181 °C	H₂	1,597	200 °C	Száraz levegő	1,398	20 °C	NO	1,40
–76 °C		1,453	400 °C		1,393	20 °C	N₂O	1,31
20 °C		1,41	1000 °C		1,365	–181 °C	N₂	1,47
100 °C		1,404	2000 °C		1,088	15 °C	N₂	1,404
400 °C		1,387	0 °C	CO₂	1,310	20 °C	Cl₂	1,34
1000 °C		1,358	20 °C		1,30	–115 °C	CH₄	1,41
2000 °C		1,318	100 °C		1,281	–74 °C		1,35
20 °C	He	1,66	400 °C		1,235	20 °C		1,32
20 °C	H₂O	1,33	1000 °C		1,195	15 °C	NH₃	1,310
100 °C		1,324	20 °C	CO	1,40	19 °C	Ne	1,64
200 °C		1,310	–181 °C	O₂	1,45	19 °C	Xe	1,66
–180 °C	Ar	1,76	–76 °C		1,415	19 °C	Kr	1,68
20 °C	Ar	1,67	20 °C		1,40	15 °C	SO₂	1,29
0 °C	Száraz levegő	1,403	100 °C		1,399	360 °C	Hg	1,67
20 °C		1,40	200 °C		1,397	15 °C	C₂H₆	1,22
100 °C		1,401	400 °C		1,394	16 °C	C₃H₈	1,13

Az adiabatikus kitevő vagy fajhőviszony az állandó nyomáson és állandó térfogaton mért fajhő hányadosa. Az adiabatikus kitevőt általában a görög κ-val (kappa) jelölik:

\kappa ={\frac {c_{p}}{c_{v}}},

ahol $c\,$ a gáz fajhője (vagy hőkapacitása), a $p\,$ és $v\,$ index pedig az állandó nyomást, illetve az állandó térfogatot jelöli.

Ideális gázok esete

Ideális gázoknál a fajhő nem változik a hőmérséklettel. Ennélfogva így is kifejezhető az entalpia: $H=c_{p}T\,$ és a belső energia: $U=c_{v}T\,$ . Így mondható, hogy az adiabatikus kitevő az entalpia és a belső energia hányadosa:

\kappa ={\frac {H}{U}}

A hőkapacitások pedig felírhatók a $\kappa$ adiabatikus kitevő és az $R$ egyetemes gázállandó függvényeként:

c_{p}={\frac {\kappa R}{\kappa -1}}

és

c_{v}={\frac {R}{\kappa -1}}

Általában nehéz a $c_{v}$ állandó térfogat mellett mért fajhőre az irodalomban konkrét értékeket találni, ezért értékét célszerű az alábbi összefüggésből számítani:

c_{v}=c_{p}-R\,

Kapcsolata a szabadságfokokkal

Az adiabatikus kitevő ideális gázok esetén kifejezhető a gázmolekulák fizikai-kémiai $f\,$ szabadságfokával:

\kappa ={\frac {f+2}{f}}\,

Így egyatomos gázokra

\kappa \ ={\frac {5}{3}}=1,67

kétatomos gázokra pedig (szobahőmérsékleten):

\kappa ={\frac {7}{5}}=1,4

Például a levegő nagyrészt kétatomos gázokból áll, ~78% nitrogénből (N₂) és ~21% oxigénből (O₂) és standard viszonyok mellett gyakorlatilag ideális gáznak tekinthető. A kétatomos gáz molekuláinak öt szabadságfoka van (három transzlációs és két rotációs), a vibrációs szabadságfok csak magas hőmérsékletek esetén jön számításba. Ez

\kappa ={\frac {5+2}{5}}={\frac {7}{5}}=1,4

értéket ad ki, amely jól egyezik a mérések 1,403 eredményével.

Jegyzetek

↑ White, Frank M.: Fluid Mechanics 4th ed. McGraw Hill
↑ Lange's Handbook of Chemistry, 10th ed. page 1524

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