# 绝热指数

$\gamma = \frac{C_P}{C_v} = \frac{c_P}{c_v}$

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

$pV^\gamma = \text{常數}$

## 理想氣體的绝热指數

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

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

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

### 和自由度的關係

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

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

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

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

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

## 熱力學的關係

$C_p - C_v \ = \ -T \frac{{\left( {\frac{\part V}{\part T}} \right)_P^2 }} {\left(\frac{\part V}{\part P}\right)_T} \ = \ -T \frac{{ \left( {\frac{\part P}{\part T}} \right) }^2} {\frac{\part P}{\part V}}$

## 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