Heat capacity ratio
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure (C_{P}) to heat capacity at constant volume (C_{V}). It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma) for an ideal gas^{[note 1]} or κ (kappa), the isentropic exponent for a real gas. The symbol gamma is used by aerospace and chemical engineers.
Temp. | Gas | γ | Temp. | Gas | γ | Temp. | Gas | γ | ||
---|---|---|---|---|---|---|---|---|---|---|
−181 °C | H_{2} | 1.597 | 200 °C | Dry air | 1.398 | 20 °C | NO | 1.400 | ||
−76 °C | 1.453 | 400 °C | 1.393 | 20 °C | N_{2}O | 1.310 | ||||
20 °C | 1.410 | 1000 °C | 1.365 | −181 °C | N_{2} | 1.470 | ||||
100 °C | 1.404 | 15 °C | 1.404 | |||||||
400 °C | 1.387 | 0 °C | CO_{2} | 1.310 | 20 °C | Cl_{2} | 1.340 | |||
1000 °C | 1.358 | 20 °C | 1.300 | −115 °C | CH_{4} | 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 | H_{2}O | 1.330 | 1000 °C | 1.195 | 15 °C | NH_{3} | 1.310 | |||
100 °C | 1.324 | 20 °C | CO | 1.400 | 19 °C | Ne | 1.640 | |||
200 °C | 1.310 | −181 °C | O_{2} | 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 | SO_{2} | 1.290 | ||||
0 °C | Dry air | 1.403 | 100 °C | 1.399 | 360 °C | Hg | 1.670 | |||
20 °C | 1.400 | 200 °C | 1.397 | 15 °C | C_{2}H_{6} | 1.220 | ||||
100 °C | 1.401 | 400 °C | 1.394 | 16 °C | C_{3}H_{8} | 1.130 |
where C is the heat capacity, the molar heat capacity (heat capacity per mol of a gas) and c the specific heat capacity (heat capacity per unit mass) of a gas. The suffixes P and V refer to constant pressure and constant volume conditions respectively.
The heat capacity ratio is important for its applications in thermodynamical reversible processes, especially involving ideal gases; the speed of sound depends on that factor.
To understand this relation, consider the following thought experiment. A closed pneumatic cylinder contains air. The piston is locked. The pressure inside is equal to atmospheric pressure. This cylinder is heated to a certain target temperature. Since the piston cannot move, the volume is constant. The temperature and pressure will rise. When the target temperature is reached, the heating is stopped. The amount of energy added equals C_{V}ΔT, with ΔT representing the change in temperature. The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. We assume the expansion occurs without exchange of heat (adiabatic expansion). Doing this work, air inside the cylinder will cool to below the target temperature. To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. 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 C_{V}, whereas the total amount of heat added is proportional to C_{P}. Therefore, the heat capacity ratio in this example is 1.4.
Another way of understanding the difference between C_{P} and C_{V} is that C_{P} applies if work is done to the system, which causes a change in volume (such as by moving a piston so as to compress the contents of a cylinder), or if work is done by the system, which changes its temperature (such as heating the gas in a cylinder to cause a piston to move). C_{V} applies only if P dV – that is, the work done – is zero. Consider the difference 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 piston to do mechanical work on the atmosphere. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. In the first, constant-volume case (locked piston) there is no external motion, and thus no mechanical work is done on the atmosphere; C_{V} is used. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant pressure case.
Ideal gas relationsEdit
For an ideal gas, the heat capacity is constant with temperature. Accordingly, we can express the enthalpy as H = C_{P}T and the internal energy as U = C_{V}T. 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 (γ) and the gas constant (R):
where n is the amount of substance in moles.
Mayer's relation allows to deduce the value of C_{V} from the more commonly tabulated value of C_{P}:
Relation with degrees of freedomEdit
The heat capacity ratio (γ) for an ideal gas can be related to the degrees of freedom ( f ) of a molecule by
Thus we observe that for a monatomic gas, with 3 degrees of freedom:
while for a diatomic gas, with 5 degrees of freedom (at room temperature: 3 translational and 2 rotational degrees of freedom; the vibrational degree of freedom is not involved, except at high temperatures):
For example, the terrestrial air is primarily made up of diatomic gases (around 78% nitrogen, N_{2}, and 21% oxygen, O_{2}), and at standard conditions it can be considered to be an ideal gas. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above).
Real-gas relationsEdit
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As temperature increases, higher-energy rotational and vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering γ. For a real gas, both C_{P} and C_{V} increase with increasing temperature, while continuing to differ from each other by a fixed constant (as above, C_{P} = C_{V} + nR), which reflects the relatively constant PV difference in work done during expansion for constant pressure vs. constant volume conditions. Thus, the ratio of the two values, γ, decreases with increasing temperature. For more information on mechanisms for storing heat in gases, see the gas section of specific heat capacity. While at 273 K (0 °C), Monatomic gases such as the noble gases He, Ne, and Ar all have the same value of γ, that being 1.664. However, once you start getting in to diatomic gases and gas compounds, the values for γ change quite often.
Thermodynamic expressionsEdit
Values based on approximations (particularly C_{P} − C_{V} = nR) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves. An experimental value should be used rather than one based on this approximation, where possible. A rigorous value for the ratio C_{P}/C_{V} can also be calculated by determining C_{V} from the residual properties expressed as
Values for C_{P} are readily available and recorded, but values for C_{V} need to be determined via relations such as these. See relations between specific heats 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 as Peng–Robinson), which match experimental values so closely that there is little need to develop a database of ratios or C_{V} values. Values can also be determined through finite-difference approximation.
Adiabatic processEdit
This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas:
- is constant
Using the ideal gas law, :
- is constant
- is constant
where P is pressure in Pa, V is the volume of the gas in and T is the temperature in K.
In gas dynamics we are interested in the local relations between pressure, density and temperature, rather than considering a fixed quantity of gas. By considering the density as the inverse of the volume for a unit mass, we can take in these relations. Since for constant entropy, , we have , or , it follows that
For an imperfect or non-ideal gas, Chandrasekhar ^{[3]} defined three different adiabatic indices so that the adiabatic relations can be written in the same form as above; these are used in the theory of stellar structure:
All of these are equal to in the case of a perfect gas.
See alsoEdit
ReferencesEdit
This article lacks ISBNs for the books listed in it. (July 2017) |
- ^ White, Frank M. Fluid Mechanics (4th ed.). McGraw Hill. ISBN 978-0072281927.
- ^ Lange, Norbert A. Lange's Handbook of Chemistry (10th ed.). p. 1524.
- ^ Chandrasekhar, S. (1939). An Introduction to the Study of Stellar Structure. University of Chicago Press. p. 56. ISBN 978-0-486-60413-8.
NotesEdit
- ^ γ first appeared in a paper by the French mathematician, engineer, and physicist Siméon Denis Poisson:
- Poisson (1808). "Mémoire sur la théorie du son" [Memoir on the theory of sound]. Journal de l'Ecole Polytechnique (in French). 7 (14): 319–392. On p. 332, Poisson defines γ merely as a small deviation from equilibrium which causes small variations of the equilibrium value of the density ρ.
- Poisson (1823). "Sur la vitesse du son" [On the speed of sound]. Annales de chimie et de physique. 2nd series (in French). 23: 5–16.
Meanwhile, in 1816 the French mathematician and physicist Pierre-Simon Laplace had found that the speed of sound depends on the ratio of the specific heats.- Laplace (1816). "Sur la vitesse du son dans l'air et dans l'eau" [On the speed of sound in air and in water]. Annales de chimie et de physique. 2nd series (in French). 3: 238–241.
In 1825, Laplace stated that the speed of sound is proportional to the square root of the ratio of the specific heats:- Laplace, P.S. (1825). Traité de mecanique celeste [Treatise on celestial mechanics] (in French). vol. 5. Paris, France: Bachelier. pp. 127–137. On p. 127, Laplace defines the symbols for the specific heats, and on p. 137 (at the bottom of the page), Laplace presents the equation for the speed of sound in a perfect gas.
- Rankine, William John Macquorn (1851). "On Laplace's Theory of Sound". Philosophical Magazine. 4th series. 1 (3): 225–227.
- See also: Krehl, Peter O. K. (2009). History of Shock Waves, Explosions and Impact: A Chronological and Biographical Reference. Berlin and Heidelberg, Germany: Springer Verlag. p. 276. ISBN 9783540304210.