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Specific heat ratio
From Citizendium, the Citizens' Compendium
|Specific heat ratio of various gases|
The specific heat ratio of a gas is the ratio of the specific heat at constant pressure, , to the specific heat at constant volume, . It is sometimes referred to as the adiabatic index or the heat capacity ratio or the isentropic expansion factor or the adiabatic exponent or the isentropic exponent.
- = the specific heat of a gas
- = refers to constant pressure conditions
- = refers to constant volume conditions
Ideal gas relations
In thermodynamic terminology, and may be expressed as:
where stands for temperature, for the enthalpy and for the internal energy. For an ideal gas, the heat capacity is constant with temperature. Accordingly, we can express the enthalpy as and the internal energy as . Thus, it can also be said that the specific heat ratio of an ideal gas is the ratio of the enthalpy to the internal energy:
The specific heats at constant pressure, , of various gases are relatively easy to find in the technical literature. However, it can be difficult to find values of the specific heats at constant volume, . When needed, given , the following equation can be used to determine :
Relation with degrees of freedom
The specific heat ratio ( ) for an ideal gas can be related to the degrees of freedom ( ) of a molecule by:
Thus for a monatomic gas, with three degrees of freedom:
and for a diatomic gas, with five degrees of freedom (three translational and two rotational degrees of freedom, the vibrational degree of freedom is not involved except at high temperatures):
Earth's atmospheric air is primarily made up of diatomic gases with a composition of ~78% nitrogen (N2) and ~21% oxygen (O2). At 20 °C and an absolute pressure of 101.325 kPa, the atmospheric air can be considered to be an ideal gas. This results in a value of:
The specific heats of real gases (as differentiated from ideal gases) are not constant with temperature. As temperature increases, higher energy rotational and vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering .
For a real gas, and usually increase with increasing temperature and decreases. Some correlations exist to provide values of as a function of the temperature.
Isentropic compression or expansion of ideal gases
where, is the absolute pressure and is the volume. The subscripts 1 and 2 refer to conditions before and after the process, or at any time during that process.
Using the ideal gas law, , equation (1) can be re-arranged to:
where is the absolute temperature. Re-arranging further:
Determination of values
Values for are readily available, but values for are not as available and often need to be determined. Values based on the ideal gas relation of are in many cases not sufficiently accurate for practical engineering calculations. If at all possible, an experimental value should be used.
- ↑ Frank M. White (1999). Fluid Mechanics, Fourth Edition. McGraw-Hill. ISBN 0-07-0697167.
- ↑ Norbert A. Lange (Editor) (1969). Lange's Handbook of Chemistry, 10th Edition. McGraw-Hill, page 1524.
- ↑ 3.0 3.1 3.2 3.3 3.4 Stephan R. Turns (2006). Thermodynamics: Concepts and Application, First Edition. Cambridge University Press. ISBN 0-521-85042-8.
- ↑ Richard W. Miller (1996). Flow Measurement Engineering Handbook, 3rd Edition. McGraw-Hill, pp 2.110-2.115. ISBN 0-07-042366-0.
- ↑ Don. W. Green, James O Maloney and Robert H. Perry (Editors) (1984). Perry's Chemical Engineers' Handbook, Sixth Edition. McGraw-Hill, page 6-17. ISBN 0-07-049479-7.
- ↑ K.Y. Narayanan (2001). A Textbook of Chemical Engineering Thermodynamics. Prentice-Hall India. ISBN 81-203-1732-7.
- ↑ Y.V.C. Rao (1997). Chemical Engineering Thermodynamics, First Edition. Universities Press. ISBN 81-7371-048-1.
- ↑ Thermodynamics of Pure Substances Lecture by Mark Gibbs, University of Edinburgh, Scotland.
- ↑ Isochoric heat capacity (pdf page 61 of 308)