User:Milton Beychok/Sandbox: Difference between revisions

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{| class="toccolours" border="1" style="float: right; clear: right; margin: 0 0 1em 1em; border-collapse: collapse;" cellpadding=3
! colspan="7" align="center" style="background:#cccccc;"| Specific heat ratio of various gases<ref>{{cite book|author=Frank M. White|title=Fluid Mechanics|edition= Fourth Edition|publisher=McGraw-Hill|year=1999|id=ISBN 0-07-0697167}}</ref><ref>{{cite book|author=Norbert A. Lange (Editor)|title=Lange's Handbook of Chemistry|edition=10th Edition|publisher=McGraw-Hill|pages=page 1524|year=1969|id=}}</ref><ref name=Turns>{{cite book|author=Stephan R. Turns|title=Thermodynamics: Concepts and Application|edition=First Edition|publisher=Cambridge University Press|year=2006|id=ISBN 0-521-85042-8}}</ref>
|-
| bgcolor="#E0E0E0" align="center" | Gas
| bgcolor="#E0E0E0" align="center" | °C
| bgcolor="#E0E0E0" align="center" | '''''k'''''
| bgcolor="#E0E0E0" width="2" rowspan="17"| &nbsp;
| bgcolor="#E0E0E0" align="center" | Gas
| bgcolor="#E0E0E0" align="center" | °C
| bgcolor="#E0E0E0" align="center" | '''''k'''''
|-
| rowspan="7" align="center" | H<sub>2</sub>
| align="center" |−181
| 1.597
| rowspan="4" align="center" | Dry<BR>Air
| align="center" |20
| 1.40
|-
| align="center" |−76
| 1.453
| align="center" |100
| 1.401
|-
| align="center" |20
| 1.41
| align="center" |200
| 1.398
|-
| align="center" |100
| 1.404
| align="center" |400
| 1.393
|-
| align="center" |400
| 1.387
| rowspan="4" align="center" | CO<sub>2</sub>
| align="center" |0
| 1.310
|-
| align="center" |1000
| 1.358
| align="center" |20
| 1.30
|-
| align="center" |2000
| 1.318
| align="center" |100
| 1.281
|-
| align="center" | He
| align="center" |20
| 1.66
| align="center" |400
| 1.235
|-
|rowspan="2" align="center"|N<sub>2</sub>
| align="center" |−181
| 1.47
| align="center" |NH<sub>3</sub>
| align="center" |15
| 1.310
|-
| align="center" |15
| 1.404
| align="center" | CO
| align="center" |20
| 1.40
|-
| align="center"|Cl<sub>2</sub>
| align="center" |20
| 1.34
| rowspan="6" align="center" | O<sub>2</sub>
| align="center" |−181
| 1.45
|-
| rowspan="2" align="center" | Ar
| align="center" |−180
| 1.76
| align="center" |−76
| 1.415
|-
| align="center" |20
| 1.67
| align="center" |20
| 1.40
|-
| rowspan="3" align="center" | CH<sub>4</sub>
| align="center" |−115
| 1.41
| align="center" |100
| 1.399
|-
| align="center" |−74
| 1.35
| align="center" |200
| 1.397
|-
| align="center" |20
| 1.32
| align="center" |400
| 1.394
|}


The '''specific heat ratio''' of a [[gas]] is the ratio of the [[specific heat]] at constant [[pressure]], <math>C_p</math>, to the specific heat at constant volume, <math>C_v</math>. It is sometimes  referred to as the '''adiabatic index''' or the '''heat capacity ratio''' or the '''isentropic expansion factor'''.
Either <math>k</math> ([[K (letter)|Roman letter k]]), <math>\gamma</math> ([[gamma]]) or <math>\kappa</math> ([[kappa]]) may be used to denote the specific heat ratio:
:<math>k = \gamma = \kappa = \frac{C_p}{C_v}</math>
where:
:<math>C</math> = the specific heat of a gas
:<math>p</math> = refers to constant pressure conditions
:<math>v</math> = refers to constant volume conditions
<BR><BR>
== Ideal gas relations ==
For an [[Ideal gas law|ideal gas]], the [[heat capacity]] is constant with [[temperature]]. Accordingly we can express the [[enthalpy]]  as <math>H = C_p T</math> and the [[internal energy]] as <math>U = C_v T</math>. Thus, it can also be said that the specific heat ratio of an ideal gas is the ratio between the enthalpy to the internal energy:<ref name=Turns/>
:<math> k = \frac{H}{U}</math>
The specific heats at constant pressure, <math>C_p</math>, 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, <math>C_v</math>. When needed,  given <math>C_p</math>, the following equation can be used to determine <math>C_v</math> :<ref name=Turns/>
:<math>C_v = C_p - R</math>
where <math>R</math> is the [[molar gas constant]] (also known as the [[Universal gas constant]]). This equation can be re-arranged to obtain:
:<math> C_p = \frac{k R}{k - 1} \qquad \mbox{and} \qquad C_v = \frac{R}{k - 1}</math>
=== Relation with degrees of freedom ===
The specific heat ratio ( <math>k</math> ) for an ideal gas can be related to the [[degrees of freedom (physics)|degrees of freedom]] ( <math>f</math> ) of a molecule by:
:<math> k = \frac{f+2}{f}</math>
Thus for a [[monatomic]] gas, with three degrees of freedom:
:<math> k = \frac{5}{3} = 1.67</math>
and for a [[diatomic]] gas, with five degrees of freedom (at room temperature):
:<math> k = \frac{7}{5} = 1.4</math>.
Earth's atmospheric [[air]] is primarily made up of [[diatomic]] gases with a composition of ~78% [[nitrogen]] (N<sub>2</sub>) and ~21% [[oxygen]] (O<sub>2</sub>). At 20 °C and an absolute pressure of 101.325 kPa, the atmospheric air can be considered to be an ideal gas. A diatomic molecule has five degrees of freedom (three translational and two rotational degrees of freedom, the vibrational degree of freedom is not involved except at high temperatures). This results in a value of:
: <math> k = \frac{5 + 2}{5} = \frac{7}{5} = 1.4</math>
which is consistent with the value of 1.40 listed for oxygen in the above table.
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 <math>k</math>.
For a real gas, <math>C_p</math> and <math>C_v</math> usually increase with increasing temperature and <math>k</math> decreases. Some correlations exist to provide values of <math>k</math> as a function of the temperature.
== 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]]):<ref name=Turns/>
:(1) &nbsp; &nbsp;<math> p_1{V_1}^k = p_2{V_2}^k </math>
where, <math>p</math> is the absolute pressure and <math>V</math> 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, <math>pV = nRT</math>, equation (1) can be re-arranged to:
:(2) &nbsp; &nbsp;<math>\frac{p_1}{p_2} = \left(\frac{V_2}{V_1}\right)^k = \left(\frac{T_2}{T_1}\right)^k  \left(\frac{P_1}{P_2}\right)^k</math>
where <math>T</math> is the absolute temperature. Re-arranging further:
:(3) &nbsp; &nbsp;<math>\left(\frac{T_2}{T_1}\right)^k = \left(\frac{P_1}{P_2}\right)\left(\frac{P_2}{P_1}\right)^k = \left(\frac{P_2}{P_1}\right)^{k-1}</math>
we obtain the equation for the temperature change that occurs when an ideal gas is isentropically compressed or expanded:<ref name=Turns/><ref>{{cite book|author=Don. W. Green, James O Maloney and Robert H. Perry (Editors)|title=Perry's Chemical Engineers' Handbook|edition=Sixth Edition|publisher=McGraw-Hill|pages=page 6-17|year=1984|id=ISBN 0-07-049479-7}}</ref>
:(4) &nbsp; &nbsp; <math>\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(k-1)/k}</math>
Equation (4) is widely used to model ideal gas compression or expansion processes in [[internal combustion engine]]s, [[gas compressor]]s and [[gas turbines]].
== Determination of &nbsp;<math>C_v</math> &nbsp;values ==
Values for <math>C_p</math> are readily available, but values for <math>C_v</math> are not as available and often need to be determined. Values based on the ideal gas relation of &nbsp;<math>C_p - C_v = R</math> 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 <math>C_v</math> from the [[residual property function]]s (also referred to as [[departure functions]])<ref>{{cite book|author=K.Y. Narayanan|title=A Textbook of Chemical Engineering Thermodynamics|edition=|publisher=Prentice-Hall India|year=2001|id=ISBN 81-203-1732-7}}</ref><ref>{{cite book|author=Y.V.C. Rao|title=Chemical Engineering Thermodynamics|edition=First Edition|publisher=Universities Press|year=1997|id=ISBN 81-7371-048-1}}</ref><ref>[http://www.chemeng.ed.ac.uk/~mbiggs/mjbc_teaching/thermodynamics4/2004-05/ohs/Thermodynamic%20relationships%20for%20pure%20substances%20II.pdf Thermodynamics of Pure Substances] Lecture by Mark Gibbs, University of Edinburgh, Scotland.</ref> using this relation:<ref>[http://itll.colorado.edu/HYSYSDoc/Doc/COMThermo/RefGuide.pdf Isochoric heat capacity] (pdf page 61 of 308)</ref>
:<math>C_v = C_p + T \frac{{\; \left( {\frac{\part P}{\part T}} \right) }^2_V} {\left( \frac{\part P}{\part V} \right)_T} </math>
[[Equations of state]] (EOS) (such as the Peng-Robinson equation of state) can be used to solve this relation and to provide values of <math>C_v</math> that match experimental values very closely.
== Definitions of specific heat and heat capacity ==
The ''specific heat'' (or ''specific heat capacity''), is the amount of heat energy required to increase the temperature of a unit quantity of a substance by a certain temperature interval. For example, the heat required to raise the temperature of water by 1 [[Kelvin (unit)|kelvin]] is 4.184 [[Joule (unit)|joule]]s. The specific heat capacity is usually expressed as Jg<sup>-1</sup>K<sup>-1</sup>. It may also be expressed on a molar basis as Jmol<sup>-1</sup>K<sup>-1</sup>.
The ''heat capacity'' (as distinct from ''specific heat'') is the amount of heat energy required to increase the temperature of a substance by a certain temperature interval. Heat capacity is an extensive property because its value is proportional to the amount of the substance. For example, a kilogram of water has a greater heat capacity than 100 grams of water. The heat capacity is usually expressed as JK<sup>-1</sup>.
''Specific heat capacities'' and ''heat capacities'' have the same symbols of <math>C_p</math> and <math>C_v</math>. The specific heat ratio, <math>k</math>, has the same numeric value whether based on ''specific heats''' or ''heat capacities''.
== References ==
<references/>

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