Specific heat ratio: Difference between revisions

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== Isentropic compression or expansion of ideal gases ==
== Isentropic compression or expansion of ideal gases ==


The specific heat ratio plays an important part in the [[isentropic]] process of an ideal gas (i.e., a process that occurs at constant [[entropy (thermodynamics)|entropy]]):<ref name=Turns/>
The specific heat ratio plays an important part in the [[isentropic process]] of an ideal gas (i.e., a process that occurs at constant [[entropy (thermodynamics)|entropy]]):<ref name=Turns/>


:(1) &nbsp; &nbsp;<math> p_1{V_1}^\kappa = p_2{V_2}^\kappa </math>
:(1) &nbsp; &nbsp;<math> p_1{V_1}^\kappa = p_2{V_2}^\kappa </math>
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== References ==
== References ==
<references/>
<references/>[[Category:Suggestion Bot Tag]]

Latest revision as of 06:00, 21 October 2024

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Specific heat ratio of various gases[1][2][3]
Gas °C k   Gas °C k
H2 −181 1.597 Dry
Air 		20 1.40
−76 1.453 100 1.401
20 1.41 200 1.398
100 1.404 400 1.393
400 1.387 CO2 0 1.310
1000 1.358 20 1.30
2000 1.318 100 1.281
He 20 1.66 400 1.235
N2 −181 1.47 NH3 15 1.310
15 1.404 CO 20 1.40
Cl2 20 1.34 O2 −181 1.45
Ar −180 1.76 −76 1.415
20 1.67 20 1.40
CH4 −115 1.41 100 1.399
−74 1.35 200 1.397
20 1.32 400 1.394

The specific heat ratio of a gas is the ratio of the specific heat at constant pressure, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p} , to the specific heat at constant volume, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} . 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.

Either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} (kappa), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} (Roman letter k) or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} (gamma) may be used to denote the specific heat ratio:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa = k = \gamma = \frac{C_p}{C_v}}

where:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} = the specific heat of a gas
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} = refers to constant pressure conditions
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} = refers to constant volume conditions



Ideal gas relations

In thermodynamic terminology, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_V} may be expressed as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p=\left(\frac{\partial H}{\partial T}\right)_p }     and     Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_V=\left(\frac{\partial U}{\partial T}\right)_V}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} stands for temperature, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} for the enthalpy and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} for the internal energy. For an ideal gas, the heat capacity is constant with temperature. Accordingly, we can express the enthalpy as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = C_p\, T} and the internal energy as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = C_v\, T} . 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:[3]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa = \frac{H}{U}}

The specific heats at constant pressure, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p} , 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} . When needed, given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p} , the following equation can be used to determine Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v}  :[3]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v = C_p - R}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is the molar gas constant (also known as the Universal gas constant). This equation can be re-arranged to obtain:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p = \frac{\kappa R}{\kappa - 1} \qquad \mbox{and} \qquad C_v = \frac{R}{\kappa - 1}}

Relation with degrees of freedom

The specific heat ratio ( Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} ) for an ideal gas can be related to the degrees of freedom ( Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} ) of a molecule by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa = \frac{f+2}{f}}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa = \frac{5}{3} = 1.67}

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):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa = \frac{7}{5} = 1.4} .

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa = \frac{5 + 2}{5} = \frac{7}{5} = 1.4}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} .

For a real gas, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} usually increase with increasing temperature and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} decreases. Some correlations exist to provide values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} as a function of the temperature.[4]

Isentropic compression or expansion of ideal gases

The specific heat ratio plays an important part in the isentropic process of an ideal gas (i.e., a process that occurs at constant entropy):[3]

(1)    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1{V_1}^\kappa = p_2{V_2}^\kappa }

where, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is the absolute pressure and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle pV = nRT} , equation (1) can be re-arranged to:

(2)    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{p_1}{p_2} = \left(\frac{V_2}{V_1}\right)^\kappa = \left(\frac{T_2}{T_1}\right)^\kappa \left(\frac{p_1}{p_2}\right)^\kappa}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the absolute temperature. Re-arranging further:

(3)    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{T_2}{T_1}\right)^\kappa = \left(\frac{p_1}{p_2}\right)\left(\frac{p_2}{p_1}\right)^\kappa = \left(\frac{p_2}{p_1}\right)^{\kappa-1}}

we obtain the equation for the temperature change that occurs when an ideal gas is isentropically compressed or expanded:[3][5]

(4)     Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{T_2}{T_1} = \left(\frac{p_2}{p_1}\right)^{(\kappa-1)/\kappa}}

Equation (4) is widely used to model ideal gas compression or expansion processes in internal combustion engines, gas compressors and gas turbines.

Determination of  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v}  values

Values for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p} are readily available, but values for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} are not as available and often need to be determined. Values based on the ideal gas relation of  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_p - C_v = R} are in many cases not sufficiently accurate for practical engineering calculations. If at all possible, an experimental value should be used.

A rigorous value can be calculated by determining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_v} from the residual property functions (also referred to as departure functions)[6][7][8] using this relation:[9]

Equations of state (EOS) (such as the Peng-Robinson equation of state) can be used to solve this relation and to provide values of that match experimental values very closely.

References

  1. ↑ Frank M. White (1999). Fluid Mechanics, Fourth Edition. McGraw-Hill. ISBN 0-07-0697167. 
  2. ↑ Norbert A. Lange (Editor) (1969). Lange's Handbook of Chemistry, 10th Edition. McGraw-Hill, page 1524. 
  3. ↑ 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. 
  4. ↑ Richard W. Miller (1996). Flow Measurement Engineering Handbook, 3rd Edition. McGraw-Hill, pp 2.110-2.115. ISBN 0-07-042366-0. 
  5. ↑ 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. 
  6. ↑ K.Y. Narayanan (2001). A Textbook of Chemical Engineering Thermodynamics. Prentice-Hall India. ISBN 81-203-1732-7. 
  7. ↑ Y.V.C. Rao (1997). Chemical Engineering Thermodynamics, First Edition. Universities Press. ISBN 81-7371-048-1. 
  8. ↑ Thermodynamics of Pure Substances Lecture by Mark Gibbs, University of Edinburgh, Scotland.
  9. ↑ Isochoric heat capacity (pdf page 61 of 308)
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