# 绝热指数

（重定向自比热比

${\displaystyle \gamma ={\frac {C_{P}}{C_{v}}}={\frac {c_{P}}{c_{v}}}}$

${\displaystyle \gamma ={\frac {C_{P,m}}{C_{v,m}}}}$

${\displaystyle pV^{\gamma }={\text{const}}}$

## 理想氣體的绝热指數

${\displaystyle \gamma ={\frac {H}{U}}}$

${\displaystyle C_{p,m}-C_{V,m}=R\!}$

${\displaystyle C_{p,m}={\frac {\gamma R}{\gamma -1}}\qquad {\mbox{,}}\qquad C_{V,m}={\frac {R}{\gamma -1}}}$

### 和自由度的關係

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

${\displaystyle \gamma \ ={\frac {f+2}{f}}\qquad {\mbox{, }}\qquad f={\frac {2}{\gamma -1}}}$

${\displaystyle \gamma \ ={\frac {5}{3}}\approx 1.67}$ ,

${\displaystyle \gamma ={\frac {7}{5}}=1.4}$ .

${\displaystyle \gamma ={\frac {5+2}{5}}={\frac {7}{5}}=1.4}$ .

## 熱力學的關係

${\displaystyle C_{p}-C_{v}\ =\ -T{\frac {\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}}{\left({\frac {\partial V}{\partial P}}\right)_{T}}}\ =\ -T{\frac {{\left({\frac {\partial P}{\partial T}}\right)}^{2}}{\frac {\partial P}{\partial V}}}}$

## 各種氣體的绝热指数

−181°C H2 1.597 200°C 乾空氣 1.398 20°C NO 1.400
−76°C 1.453 400°C 1.393 20°C N2O 1.310
20°C 1.410 1000°C 1.365 −181°C N2 1.470
100°C 1.404 2000°C 1.088 15°C 1.404
400°C 1.387 0°C CO2 1.310 20°C Cl2 1.340
1000°C 1.358 20°C 1.300 −115°C CH4 1.410
2000°C 1.318 100°C 1.281 −74°C 1.350
20°C He 1.660 400°C 1.235 20°C 1.320
20°C H2O 1.330 1000°C 1.195 15°C NH3 1.310
100°C 1.324 20°C CO 1.400 19°C Ne 1.640
200°C 1.310 −181°C O2 1.450 19°C Xe 1.660
−180°C Ar 1.760 −76°C 1.415 19°C Kr 1.680
20°C 1.670 20°C 1.400 15°C SO2 1.290
0°C 濕空氣 1.403 100°C 1.399 360°C Hg 1.670
20°C 1.400 200°C 1.397 15°C C2H6 1.220
100°C 1.401 400°C 1.394 16°C C3H8 1.130

## 參考

1. ^ Robert W. Fox, Philip J. Pritchard and Alan T. McDonald. Introduction to Fluid Mechanics 6th. Wiley. 2008. ISBN 9780471202318.
2. ^ White, Frank M.: Fluid Mechanics 4th ed. McGraw Hill
3. ^ Lange's Handbook of Chemistry, 10th ed. page 1524