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Heat Capacity Ratio for various gases[1][2]

Temp. Gas γ Temp. Gas γ Temp. Gas γ−181 °C H2 1.597 200 °C Dry Air 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 1.470100 °C 1.404 2000 °C 1.088 15 °C 1.404400 °C 1.387 0 °C CO2 1.310 20 °C Cl2 1.340

1000 °C 1.358 20 °C 1.300 −115 °C CH4 1.4102000 °C 1.318 100 °C 1.281 −74 °C 1.35020 °C He 1.660 400 °C 1.235 20 °C 1.32020 °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 1.640200 °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 1.68020 °C 1.670 20 °C 1.400 15 °C SO2 1.290

0 °C Dry Air 1.403 100 °C 1.399 360 °C Hg 1.67020 °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

Heat capacity ratioFrom Wikipedia, the free encyclopedia

In thermal physics and thermodynamics,the heat capacity ratio or adiabaticindex or ratio of specific heats orPoisson constant, is the ratio of the heatcapacity at constant pressure ( ) to heatcapacity at constant volume ( ). It issometimes also known as the isentropicexpansion factor and is denoted by (gamma)(for ideal gas) or (kappa)(isentropic exponent, for real gas). Theformer symbol gamma is primarily usedby chemical engineers. Mechanicalengineers use the Roman letter .[3]

where, is the heat capacity and thespecific heat capacity (heat capacity perunit mass) of a gas. Suffix and referto constant pressure and constant volumeconditions respectively.

To understand this relation, consider thefollowing thought experiment. A closed pneumatic cylinder contains air. The piston is locked. The pressureinside is equal to atmospheric pressure. This cylinder is heated to a certain target temperature. Since thepiston cannot move, the volume is constant. The temperature and pressure will rise. When the targettemperature is reached, the heating is stopped. The amount of energy added equals: , with representing the change in temperature. The piston is now freed and moves outwards, stopping as thepressure inside the chamber equilibrates to atmospheric pressure. We are free to assume the expansionhappens fast enough to occur without exchange of heat (adiabatic expansion). Doing this work, air insidethe cylinder will cool to below the target temperature. To return to the target temperature (still with a freepiston), the air must be heated. This extra heat amounts to about 40% more than the previous amount added.In this example, the amount of heat added with a locked piston is proportional to , whereas the totalamount of heat added is proportional to . Therefore, the heat capacity ratio in this example is 1.4.

Another way of understanding the difference between and is that applies if work is done to thesystem which causes a change in volume (e.g. by moving a piston so as to compress the contents of acylinder), or if work is done by the system which changes its temperature (e.g. heating the gas in a cylinderto cause a piston to move). applies only if ­ that is, the work done ­ is zero. Consider thedifference between adding heat to the gas with a locked piston, and adding heat with a piston free to move,so that pressure remains constant. In the second case, the gas will both heat and expand, causing the pistonto do mechanical work on the atmosphere. The heat that is added to the gas goes only partly into heating the

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gas, while the rest is transformed into the mechanical work performed by the piston. In the first, constant­volume case (locked piston) there is no external motion, and thus no mechanical work is done on theatmosphere; is used. In the second case, additional work is done as the volume changes, so the amountof heat required to raise the gas temperature (the specific heat capacity) is higher for this constant pressurecase.

Contents

1 Ideal gas relations1.1 Relation with degrees of freedom

2 Real gas relations3 Thermodynamic expressions4 Adiabatic process5 See also6 References

Ideal gas relations

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 heat capacity ratio is

the ratio between the enthalpy to the internal energy:

Furthermore, the heat capacities can be expressed in terms of heat capacity ratio ( ) and the gas constant ( ):

,

where is the amount of substance in moles.

It can be rather difficult to find tabulated information for , since is more commonly tabulated. Thefollowing relation, can be used to determine :

Relation with degrees of freedom

The heat capacity ratio ( ) for an ideal gas can be related to the degrees of freedom ( ) of a molecule by:

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Thus we observe that for a monatomic gas, with three degrees of freedom:

,

while for a diatomic gas, with five degrees of freedom (at room temperature: three translational and tworotational degrees of freedom; the vibrational degree of freedom is not involved except at hightemperatures):

.

E.g.: The terrestrial air is primarily made up of diatomic gases (~78% nitrogen (N2) and ~21% oxygen (O2))and at standard conditions it can be considered to be an ideal gas. The above value of 1.4 is highlyconsistent with the measured adiabatic indices for dry air within a temperature range of 0 to 200 °C,exhibiting a deviation of only 0.2% (see tablation above).

Real gas relations

As temperature increases, higher energy rotational and vibrational states become accessible to moleculargases, thus increasing the number of degrees of freedom and lowering . For a real gas, both and increase with increasing temperature, while continuing to differ from each other by a fixed constant (asabove, = ) which reflects the relatively constant difference in work done duringexpansion, for constant pressure vs. constant volume conditions. Thus, the ratio of the two values, ,decreases with increasing temperature. For more information on mechanisms for storing heat in gases, seethe gas section of specific heat capacity.

Thermodynamic expressions

Values based on approximations (particularly ) are in many cases not sufficientlyaccurate for practical engineering calculations such as flow rates through pipes and valves. An experimentalvalue should be used rather than one based on this approximation, where possible. A rigorous value for the

ratio can also be calculated by determining from the residual properties expressed as:

Values for are readily available and recorded, but values for need to be determined via relationssuch as these. See here for the derivation of the thermodynamic relations between the heat capacities.

The above definition is the approach used to develop rigorous expressions from equations of state (such asPeng–Robinson), which match experimental values so closely that there is little need to develop a databaseof ratios or values. Values can also be determined through finite difference approximation.

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Adiabatic process

See also: Adiabatic process and polytropic process

This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process ofa simple compressible calorically perfect ideal gas:

where is the pressure and is the volume.

See also

Heat capacitySpecific heat capacitySpeed of soundThermodynamic equationsThermodynamicsVolumetric heat capacity

References1. White, Frank M.: Fluid Mechanics 4th ed. McGraw Hill2. Lange's Handbook of Chemistry, 10th ed. page 15243. Fox, R., A. McDonald, P. Pritchard: Introduction to Fluid Mechanics 6th ed. Wiley

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