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Heat capacity ratio

From Wikipedia, the free encyclopedia
(Redirected fromAdiabatic index)
Thermodynamic quantity
Thermodynamics
The classicalCarnot heat engine
Heat capacity ratio for various gases[1][2]
GasTemp. [°C]γ
H2−1811.597
−761.453
201.410
1001.404
4001.387
10001.358
20001.318
He201.66
Ar−1801.760
201.670
O2−1811.450
−761.415
201.400
1001.399
2001.397
4001.394
N2−1811.470
Cl2201.340
Ne191.640
Xe191.660
Kr191.680
Hg3601.670
H2O201.330
1001.324
2001.310
CO201.310
201.300
1001.281
4001.235
10001.195
CO201.400
NO201.400
N2O201.310
CH4−1151.410
−741.350
201.320
NH3151.310
SO2151.290
C2H6151.220
C3H8161.130
Dryair-151.404
01.403
201.400
2001.398
4001.393
10001.365

Inthermal physics andthermodynamics, theheat capacity ratio, also known as theadiabatic index, theratio of specific heats, orLaplace's coefficient, is the ratio of theheat capacity at constant pressure (CP) to heat capacity at constant volume (CV). It is sometimes also known as theisentropic expansion factor and is denoted byγ (gamma) for an ideal gas[note 1] orκ (kappa), the isentropic exponent for a real gas. The symbolγ is used by aerospace and chemical engineers.γ=CPCV=C¯PC¯V=cPcV,{\displaystyle \gamma ={\frac {C_{P}}{C_{V}}}={\frac {{\bar {C}}_{P}}{{\bar {C}}_{V}}}={\frac {c_{P}}{c_{V}}},}whereC is the heat capacity,C¯{\displaystyle {\bar {C}}} themolar heat capacity (heat capacity per mole), andc thespecific heat capacity (heat capacity per unit mass) of a gas. The suffixesP andV refer to constant-pressure and constant-volume conditions respectively.

The heat capacity ratio is important for its applications inthermodynamical reversible processes, especially involvingideal gases; thespeed of sound depends on this factor.

Thought experiment

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To understand this relation, consider the followingthought experiment. A closedpneumatic cylinder contains air. Thepiston 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 equalsCVΔT, withΔT 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 thiswork, 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 toCV, whereas the total amount of heat added is proportional toCP. Therefore, the heat capacity ratio in this example is 1.4.

Another way of understanding the difference betweenCP andCV is thatCP 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).CV applies only ifPdV=0{\displaystyle P\,\mathrm {d} V=0}, that is, no work is done. 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;CV 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 relations

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For an ideal gas, the molar heat capacity is at most a function of temperature, since theinternal energy is solely a function of temperature for aclosed system, i.e.,U=U(n,T){\displaystyle U=U(n,T)}, wheren is theamount of substance in moles. In thermodynamic terms, this is a consequence of the fact that theinternal pressure of an ideal gas vanishes.

Mayer's relation allows us to deduce the value ofCV from the more easily measured (and more commonly tabulated) value ofCP:CV=CP−nR.{\displaystyle C_{V}=C_{P}-nR.}

This relation may be used to show the heat capacities may be expressed in terms of the heat capacity ratio (γ) and thegas constant (R):CP=γnRγ−1andCV=nRγ−1,{\displaystyle C_{P}={\frac {\gamma nR}{\gamma -1}}\quad {\text{and}}\quad C_{V}={\frac {nR}{\gamma -1}},}

Relation with degrees of freedom

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The classicalequipartition theorem predicts that the heat capacity ratio (γ) for an ideal gas can be related to the thermally accessibledegrees of freedom (f) of a molecule byγ=1+2f,orf=2γ−1.{\displaystyle \gamma =1+{\frac {2}{f}},\quad {\text{or}}\quad f={\frac {2}{\gamma -1}}.}

Thus we observe that for amonatomic gas, with 3 translational degrees of freedom per atom:γ=53=1.6666…,{\displaystyle \gamma ={\frac {5}{3}}=1.6666\ldots ,}

As an example of this behavior, at 273 K (0 °C) the noble gases He, Ne, and Ar all have nearly the same value ofγ, equal to 1.664.

For adiatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted byquantum statistical mechanics. Thus we haveγ=75=1.4.{\displaystyle \gamma ={\frac {7}{5}}=1.4.}

For example, terrestrial air is primarily made up ofdiatomic gases (around 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 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).

For a linear triatomic molecule such as CO2, there are only 5 degrees of freedom (3 translations and 2 rotations), assuming vibrational modes are not excited. However, as mass increases and the frequency of vibrational modes decreases, vibrational degrees of freedom start to enter into the equation at far lower temperatures than is typically the case for diatomic molecules. For example, it requires a far larger temperature to excite the single vibrational mode forH2, for which one quantum of vibration is a fairly large amount of energy, than for the bending or stretching vibrations of CO2.

For a non-lineartriatomic gas, such as water vapor, which has 3 translational and 3 rotational degrees of freedom, this model predictsγ=86=1.3333….{\displaystyle \gamma ={\frac {8}{6}}=1.3333\ldots .}

Real-gas relations

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As noted above, as temperature increases, higher-energy vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and loweringγ. Conversely, as the temperature is lowered, rotational degrees of freedom may become unequally partitioned as well. As a result, bothCP andCV increase with increasing temperature.

Despite this, if the density is fairly low andintermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above,CP =CV +nR), which reflects the relatively constantPV 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.

However, when the gas density is sufficiently high and intermolecular forces are important, thermodynamic expressions may sometimes be used to accurately describe the relationship between the two heat capacities, as explained below. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out otherchemical reactions, in which case thermodynamic expressions arising from simpleequations of state may not be adequate.

Thermodynamic expressions

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Values based on approximations (particularlyCP −CV =nR) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. An experimental value should be used rather than one based on this approximation, where possible. A rigorous value for the ratio⁠CP/CV⁠ can also be calculated by determiningCV from the residual properties expressed asCP−CV=−T(∂V∂T)P2(∂V∂P)T=−T(∂P∂T)V2(∂P∂V)T.{\displaystyle C_{P}-C_{V}=-T{\frac {\left({\frac {\partial V}{\partial T}}\right)_{P}^{2}}{\left({\frac {\partial V}{\partial P}}\right)_{T}}}=-T{\frac {\left({\frac {\partial P}{\partial T}}\right)_{V}^{2}}{\left({\frac {\partial P}{\partial V}}\right)_{T}}}.}

Values forCP are readily available and recorded, but values forCV need to be determined via relations such as these. Seerelations 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 asPeng–Robinson), which match experimental values so closely that there is little need to develop a database of ratios orCV values. Values can also be determined throughfinite-difference approximation.

Adiabatic process

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See also:adiabatic process andpolytropic process

This ratio gives the important relation for anisentropic (quasistatic,reversible,adiabatic process) process of a simple compressible calorically-perfectideal gas:

PVγ{\displaystyle PV^{\gamma }} is constant

Using the ideal gas law,PV=nRT{\displaystyle PV=nRT}:

P1−γTγ{\displaystyle P^{1-\gamma }T^{\gamma }} is constant
TVγ−1{\displaystyle TV^{\gamma -1}} is constant

whereP is the pressure of the gas,V is the volume, andT is thethermodynamic temperature.

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 densityρ=M/V{\displaystyle \rho =M/V} as the inverse of the volume for a unit mass, we can takeρ=1/V{\displaystyle \rho =1/V} in these relations.Since for constant entropy,S{\displaystyle S}, we haveP∝ργ{\displaystyle P\propto \rho ^{\gamma }}, orln⁡P=γln⁡ρ+constant{\displaystyle \ln P=\gamma \ln \rho +\mathrm {constant} }, it follows thatγ=∂ln⁡P∂ln⁡ρ|S.{\displaystyle \gamma =\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S}.}

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 ofstellar structure:Γ1=∂ln⁡P∂ln⁡ρ|S,Γ2−1Γ2=∂ln⁡T∂ln⁡P|S,Γ3−1=∂ln⁡T∂ln⁡ρ|S.{\displaystyle {\begin{aligned}\Gamma _{1}&=\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S},\\[2pt]{\frac {\Gamma _{2}-1}{\Gamma _{2}}}&=\left.{\frac {\partial \ln T}{\partial \ln P}}\right|_{S},\\[2pt]\Gamma _{3}-1&=\left.{\frac {\partial \ln T}{\partial \ln \rho }}\right|_{S}.\end{aligned}}}

All of these are equal toγ{\displaystyle \gamma } in the case of an ideal gas.

See also

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Notes

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  1. ^γ first appeared in an article by the French mathematician, engineer, and physicistSiméon Denis Poisson:
    • Poisson (1808)."Mémoire sur la théorie du son" [Memoir on the theory of sound].Journal de l'École 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.
    In Poisson's article of 1823 –
    • 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.
    γ was expressed as a function of density D (p. 8) or of pressure P (p. 9).
    Meanwhile, in 1816 the French mathematician and physicistPierre-Simon Laplace had found that the speed of sound depends on the ratio of the specific heats.However, he didn't denote the ratio as γ.
    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.
    In 1851, the Scottish mechanical engineerWilliam Rankine showed that the speed of sound is proportional to the square root of Poisson's γ:It follows that Poisson's γ is the ratio of the specific heats — although Rankine didn't state that explicitly.

References

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  1. ^White, Frank M. (October 1998).Fluid Mechanics (4th ed.). New York:McGraw Hill.ISBN 978-0-07-228192-7.
  2. ^Lange, Norbert A. (1967).Lange's Handbook of Chemistry (10th ed.). New York:McGraw Hill. p. 1524.ISBN 978-0-07-036261-1.
  3. ^Chandrasekhar, S. (1939).An Introduction to the Study of Stellar Structure. Chicago:University of Chicago Press. p. 56.ISBN 978-0-486-60413-8.
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