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 $γ$ is used by aerospace and chemical engineers.

\gamma ={\frac {C_{P}}{C_{V}}}={\frac {{\bar {C}}_{P}}{{\bar {C}}_{V}}}={\frac {c_{P}}{c_{V}}},

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	1.470
100 °C		1.404	15 °C		1.404		N₂
400 °C		1.387	0 °C	CO₂	1.310	20 °C	Cl₂	1.340
1000 °C		1.358	20 °C		1.300	−115 °C	CH₄	1.410
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	H₂O	1.330	1000 °C		1.195	15 °C	NH₃	1.310
100 °C		1.324	20 °C	CO	1.400	19 °C	Ne	1.640
200 °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	1.680
20 °C	Ar	1.670	20 °C		1.400	15 °C	SO₂	1.290
0 °C	Dry air	1.403	100 °C		1.399	360 °C	Hg	1.670
20 °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

where $C$ is the heat capacity, ${\bar {C}}$ the molar heat capacity (heat capacity per mole), 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 this 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 $C V Δ T$ , with $Δ T$ 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 $PdV=0$ , 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; $C V$ 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 relations

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:

\gamma ={\frac {H}{U}}.

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

C_{P}={\frac {\gamma nR}{\gamma -1}}\quad {\text{and}}\quad C_{V}={\frac {nR}{\gamma -1}},

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 $C P$ :

C_{V}=C_{P}-nR.

Relation with degrees of freedom

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

\gamma =1+{\frac {2}{f}},\quad {\text{or}}\quad f={\frac {2}{\gamma {\text{ - 1}}}}.

Thus we observe that for a monatomic gas, with 3 degrees of freedom:

\gamma ={\frac {5}{3}}=1.6666\ldots ,

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):

\gamma ={\frac {7}{5}}=1.4.

For example, the terrestrial air is primarily made up of diatomic 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 triatomic gas as water vapor having 6 degrees of freedom :

\gamma ={\frac {8}{6}}=1.3333\ldots .

Real-gas relations

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.

Thermodynamic expressions

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 $.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap} C P / C V$ can also be calculated by determining $C V$ from the residual properties expressed as

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 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 process

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

PV^{\gamma }

is constant

Using the ideal gas law, $PV=nRT$ :

P^{1-\gamma }T^{\gamma }

is constant

TV^{\gamma -1}

is constant

where $P$ is pressure in Pa, $V$ is the volume of the gas in $m^{3}$ 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 $\rho =M/V$ as the inverse of the volume for a unit mass, we can take $\rho =1/V$ in these relations. Since for constant entropy, $S$ , we have $P\propto \rho ^{\gamma }$ , or $\ln P=\gamma \ln \rho +constant$ , it follows that

\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 of stellar structure:

\Gamma _{1}=\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S},

{\frac {\Gamma _{2}-1}{\Gamma _{2}}}=\left.{\frac {\partial \ln T}{\partial \ln P}}\right|_{S},

\Gamma _{3}-1=\left.{\frac {\partial \ln T}{\partial \ln \rho }}\right|_{S}.

All of these are equal to $\gamma$ in the case of a perfect gas.

References

White, Frank M. (October 1998). 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.

Notes

γ first appeared in an article 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'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 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.
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 engineer William Rankine showed that the speed of sound is proportional to the square root of Poisson's γ:
- Rankine, William John Macquorn (1851). "On Laplace's Theory of Sound". Philosophical Magazine. 4th series. 1 (3): 225–227.
It follows that Poisson's γ is the ratio of the specific heats — although Rankine didn't state that explicitly.
- 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.

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[1] White, Frank M. (October 1998). Fluid Mechanics (4th ed.). McGraw Hill. ISBN 978-0072281927.

[2] Lange, Norbert A. Lange's Handbook of Chemistry (10th ed.). p. 1524.

[4] Chandrasekhar, S. (1939). An Introduction to the Study of Stellar Structure. University of Chicago Press. p. 56. ISBN 978-0-486-60413-8.

[3] γ first appeared in an article 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'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 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.
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 engineer William Rankine showed that the speed of sound is proportional to the square root of Poisson's γ:
Rankine, William John Macquorn (1851). "On Laplace's Theory of Sound". Philosophical Magazine. 4th series. 1 (3): 225–227.
It follows that Poisson's γ is the ratio of the specific heats — although Rankine didn't state that explicitly.
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.

[5] 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.

[6] 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.

[7] 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.

[8] 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.

[9] Rankine, William John Macquorn (1851). "On Laplace's Theory of Sound". Philosophical Magazine. 4th series. 1 (3): 225–227.

[10] 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.

Heat capacity ratio

Ideal-gas relations

Relation with degrees of freedom

Real-gas relations

Thermodynamic expressions

Adiabatic process

See also

References

Notes