Heat capacity ratio

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Heat Capacity Ratio for various gases<ref>White, Frank M.: Fluid Mechanics 4th ed. McGraw Hill</ref><ref>Lange's Handbook of Chemistry, 10th ed. page 1524</ref>
Temp.	Gas	γ	Temp.	Gas	γ	Temp.	Gas	γ
–181°C	H₂	1.597	200°C	Dry Air	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</td>	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	Dry Air	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

The heat capacity ratio, γ, is simply the ratio of the heat capacity at constant pressure to that at constant volume

<math> \gamma\ =\ \frac{C_P}{C_V}</math>

It should be noted that chemical engineers and many others commonly refer to the heat capacity ratio as κ rather than γ.

For a monoatomic ideal gas, <math>\scriptstyle \gamma\ =\ \frac{5}{3}</math>, while a diatomic ideal gas has <math>\scriptstyle \gamma\ =\ \frac{7}{5}</math>.

For a first approximation assuming ideal gas and C_P, C_V, and γ are constants, it can be written:

<math> C_P\ =\ \frac{\gamma R}{\gamma - 1}</math>

<math> C_V\ =\ \frac{R}{\gamma - 1}</math>

Another interesting relationship between these two is:

This can help determine C_V as usually only C_P is tabulated.

C_P and C_V increase with increasing temperature and γ decreases. Some correlations exist to provide values of γ as a function of the temperature.

Additionally the heat capacity ratio γ can be determined theoretically over the degrees of freedom f of one molecule:

<math> \gamma\ =\ \frac{f+2}{f}</math>

This ratio also gives the important relation for a quasistatic, adiabatic process:

<math> pV^\gamma\ =\ p_0V^\gamma_0\ =\ \emph{constant}</math>

That is, the pressure before the change times the volume before the change raised to the power of γ equals the pressure after the change times the volume after the change raised to the power of γ.