Poincaré–Hopf theorem: Difference between revisions

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{{Thermodynamics|cTopic=[[Thermodynamic system|Systems]]}}
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'''Real gases''' – as opposed to a perfect or [[ideal gas]] – exhibit properties that cannot be explained entirely using the [[ideal gas law]]. To understand the behaviour of real gases, the following must be taken into account:
 
* compressibility effects;
* variable [[specific heat capacity]];
* [[van der Waals force]]s;<!-- "van", see [[Talk:Van der Waals#Van should be capitalized unless preceded by first name]] rebuttal -->
* non-equilibrium thermodynamic effects;
* issues with molecular dissociation and elementary reactions with variable composition.
 
For most applications, such a detailed analysis is unnecessary, and the ideal gas approximation can be used with reasonable accuracy. On the other hand, real-gas models have to be used near the [[condensation]] point of gases, near [[Critical point (thermodynamics)|critical point]]s, at very high pressures, to explain the [[Joule–Thomson effect]] and in other less usual cases.
 
== Models ==
[[File:Real Gas Isotherms.svg|thumb|400px|[[Isotherm]]s of real gas (sketchy)<br />
<br />
Dark blue curves – isotherms below the critical temperature. Green sections – [[Metastability|metastable states]].<br />
<br />
The section to the left of point F – normal liquid.<br />
Point F –  [[boiling point]].<br />
Line FG – [[Vapor-liquid equilibrium|equilibrium]] of liquid and gaseous phases.<br />
Section FA – [[Superheating|superheated liquid]].<br />
Section F′A – [[stretched liquid]] (p<0).<br />
Section AC – [[analytic continuation]] of isotherm, physically impossible.<br />
Section CG – [[supercooled vapor]].<br />
Point G – [[dew point]].<br />
The plot to the right of point G – normal gas.<br />
Areas FAB and GCB are equal.<br />
<br />
Red curve – Critical isotherm.<br />
Point K – [[Critical point (thermodynamics)|critical point]].<br />
<br />
Light blue curves – supercritical isotherms]]
<!--Add a description as to what this is and what all the following stuff is : Katanada-->
{{main|Equation of state}}
 
===van der Waals model===
{{main|van der Waals equation}}
 
Real gases are often modeled by taking into account their molar weight and molar volume
 
:<math>RT=\left(P+\frac{a}{V_m^2}\right)(V_m-b)</math>
 
Where P is the pressure, T is the temperature, R the ideal gas constant, and V<sub>m</sub> the [[molar volume]]. a and b are parameters that are determined empirically for each gas, but are sometimes estimated from their [[critical temperature]] (T<sub>c</sub>) and [[critical pressure]] (P<sub>c</sub>) using these relations:
 
:<math>a=\frac{27R^2T_c^2}{64P_c}</math>
 
:<math>b=\frac{RT_c}{8P_c}</math>
 
===Redlich–Kwong model===
 
The [[Redlich–Kwong equation of state|Redlich–Kwong equation]] is another two-parameter equation that is used to model real gases. It is almost always more accurate than the [[van der Waals equation]], and often more accurate than some equations with more than two parameters. The equation is
 
<math>RT=P(V_m-b)+\frac{a}{V_m(V_m+b)T^\frac{1}{2}}(V_m-b)</math>
 
where a and b two empirical parameters that are '''not''' the same parameters as in the van der Waals equation. These parameters can be determined:
 
<math>a=0.4275\frac{R^2T_c^{2.5}}{P_c}</math>
 
<math>b=0.0867\frac{RT_c}{P_c}</math>
 
===Berthelot and modified Berthelot model===
 
The Berthelot equation (named after D. Berthelot<ref>D. Berthelot in ''Travaux et Mémoires du Bureau international des Poids et Mesures – Tome XIII'' (Paris: Gauthier-Villars, 1907)</ref> is very rarely used,
 
<math>P=\frac{RT}{V_m-b}-\frac{a}{TV_m^2}</math>
 
but the modified version is somewhat more accurate
 
<math>P=\frac{RT}{V_m}\left[1+\frac{9P/P_c}{128T/T_c}\left(1-\frac{6}{(T/T_c)^2}\right)\right]</math>
 
===Dieterici model===
 
This model (named after C. Dieterici<ref>C. Dieterici, Ann. Phys. Chem. Wiedemanns Ann. 69, 685 (1899)</ref>) fell out of usage in recent years
 
<math>P=RT\frac{\exp{(\frac{-a}{V_mRT})}}{V_m-b}</math>.
 
===Clausius model===
 
The Clausius equation (named after [[Rudolf Clausius]]) is a very simple three-parameter equation used to model gases.
 
<math>RT=\left(P+\frac{a}{T(V_m+c)^2}\right)(V_m-b)</math>
 
where
 
<math>a=\frac{27R^2T_c^3}{64P_c}</math>
 
<math>b=V_c-\frac{RT_c}{4P_c}</math>
 
<math>c=\frac{3RT_c}{8P_c}-V_c</math>
 
where V<sub>c</sub> is critical volume.
 
===Virial model===
 
The [[Virial coefficient|Virial]] equation derives from a [[Perturbation theory|perturbative treatment]] of statistical mechanics.
 
<math>PV_m=RT\left(1+\frac{B(T)}{V_m}+\frac{C(T)}{V_m^2}+\frac{D(T)}{V_m^3}+...\right)</math>
 
or alternatively
 
<math>PV_m=RT\left(1+\frac{B^\prime(T)}{P}+\frac{C^\prime(T)}{P^2}+\frac{D^\prime(T)}{P^3}+...\right)</math>
 
where A, B, C, A&prime;, B&prime;, and C&prime; are temperature dependent constants.
 
===Peng–Robinson model===
 
[[Peng-Robinson equation of state]] (named after [[D.-Y. Peng]] and D. B. Robinson<ref>{{cite journal | title = A New Two-Constant Equation of State | journal = Industrial and Engineering Chemistry: Fundamentals | volume = 15 | year = 1976 | pages = 59–64 | author = Peng, D. Y., and Robinson, D. B. | doi = 10.1021/i160057a011}}</ref>) has the interesting property being useful in modeling some liquids as well as real gases.
 
<math>P=\frac{RT}{V_m-b}-\frac{a(T)}{V_m(V_m+b)+b(Vm-b)}</math>
 
===Wohl model===
 
The Wohl equation (named after A. Wohl<ref>A. Wohl "Investigation of the condition equation", Zeitschrift für Physikalische Chemie (Leipzig) 87 pp. 1–39 (1914)</ref>) is formulated in terms of critical values, making it useful when real gas constants are not available.
 
<math>RT=\left(P+\frac{a}{TV_m(V_m-b)}-\frac{c}{T^2V_m^3}\right)(V_m-b)</math>
 
where
 
<math>a=6P_cT_cV_c^2</math>
 
<math>b=\frac{V_c}{4}</math>
 
<math>c=4P_cT_c^2V_c^3</math>.
 
===Beattie–Bridgman model===
 
<ref>Yunus A. Cengel and Michael A. Boles,Thermodynamics: An Engineering Approach  7th Edition, , McGraw-Hill, 2010,ISBN 007-352932-X</ref>
 
This equation is based on five experimentally determined constants. It is expressed as
 
<math>P=\frac{RT}{v^2}\left(1-\frac{c}{vT^3}\right)(v+B)-\frac{A}{v^2}</math> 
 
where
:<math>A = A_0 \left(1 - \frac{a}{v} \right)</math>
:<math>B = B_0 \left(1 - \frac{b}{v} \right)</math>
This equation is known to be reasonably accurate for densities up to about 0.8&nbsp;ρ<sub>cr</sub>, where ρ<sub>cr</sub> is the density of the substance at its critical point. The constants appearing in the above equation are available in following table
when P is in KPa, v is in <math>\frac{m^3}{Kmol}</math>, T is in K and R=8.314<math>\frac{kPa.m^3}{Kmol.K}</math><ref>Gordan J.Van Wylen and Richard E.Sonntage, Fundamental of classical Thermodynamics,3rd ed,new york,John Wiley &Sons,1986 P46 table 3.3</ref>
{| class="wikitable"
|-
! Gas !! A<sub>0</sub>!! a !! B<sub>0</sub>!! b !! c
|-
| Air|| 131.8441 || 0.01931||0.04611|| -0.001101 || 4.34×10^4
|-
| Argon, Ar|| 130.7802 || 0.02328 || 0.03931 || 0.0 || 5.99×10^4
|-
| Carbon Dioxide, CO<sub>2</sub> ||507.2836 || 0.07132|| 0.10476 || 0.07235 || 6.60×10^5
|-
| Helium, He || 2.1886 || 0.05984 || 0.01400 || 0.0 || 40
|-
| Hydrogen, H<sub>2</sub> || 20.0117 || -0.00506 || 0.02096 || -0.04359 || 504
|-
|Nitrogen, N<sub>2</sub> || 136.2315 || 0.02617 || 0.05046|| -0.00691|| 4.20×10^4
|-
| Oxygen, O<sub>2</sub>|| 151.0857 || 0.02562 || 0.04624 || 0.004208 || 4.80×10^4
|}
 
===Benedict–Webb–Rubin model===
{{Main|Benedict–Webb–Rubin equation}}
 
The BWR equation, sometimes referred to as the BWRS equation,
 
<math>P=RTd+d^2\left(RT(B+bd)-(A+ad-a{\alpha}d^4)-\frac{1}{T^2}[C-cd(1+{\gamma}d^2)\exp(-{\gamma}d^2)]\right)</math>
 
where d is the molar density and where a, b, c, A, B, C, α, and γ are empirical constants. Note that the γ constant is a derivative of constant α and therefore almost identical to 1.
 
==See also==
* [[Gas laws]]
* [[Ideal gas law]] by [[Boyle's law|Boyle]] & [[Gay-Lussac's law|Gay-Lussac]]
* [[Compressibility factor]]
* [[Equation of state]]
 
==References==
{{Reflist}}
*Dilip Kondepudi, Ilya Prigogine, Modern Thermodynamics, John Wiley & Sons, 1998, ISBN 0-471-97393-9
*Hsieh, Jui Sheng, Engineering Thermodynamics, Prentice-Hall Inc., Englewood Cliffs, New Jersey 07632, 1993. ISBN 0-13-275702-8
*Stanley M. Walas, Phase Equilibria in Chemical Engineering, Butterworth Publishers, 1985. ISBN 0-409-95162-5
*M. Aznar, and A. Silva Telles, A Data Bank of Parameters for the Attractive Coefficient of the Peng-Robinson Equation of State, Braz. J. Chem. Eng. vol. 14 no. 1 São Paulo Mar. 1997, ISSN 0104-6632
*[http://books.google.com/books?id=iYWiCXziWsEC&pg=PA55&dq=Beattie-Bridgman+%3D&hl=el&ei=qugCTtLnFo30sgbD3ZDtDQ&sa=X&oi=book_result&ct=book-thumbnail&resnum=1&ved=0CCwQ6wEwAA#v=onepage&q=Beattie-Bridgeman%20%3D&f=false An introduction to thermodynamics by Y. V. C. Rao]
*[http://books.google.gr/books?id=DWRkfjIFdOIC&pg=PA52&lpg=PA52&dq=Harmens%E2%80%93Knapp+equation+real+gas&source=bl&ots=CtwG9W9gv5&sig=rAHgTqR1Jzq1DU-PPPgSkjpjRYk&hl=el&ei=k-QCTrexAoftOY_k7ewN&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCwQ6AEwAg#v=onepage&q=Knapp&f=false The corresponding-states principle and its practice: thermodynamic, transport and surface properties of fluids by Hong Wei Xiang]
 
==External links==
*http://www.ccl.net/cca/documents/dyoung/topics-orig/eq_state.html
 
[[Category:Gases]]

Latest revision as of 02:12, 30 December 2014

SingaporePropertyCycle.com.sg provides you with the graphs and charts to trace the property market cycle and make better actual estate funding selections. URA). In distinction, when adjusted for inflation, the property worth index actually fell by round 1.1% over the identical interval. Frasers Property includes the worldwide property activities of Singapore-based Frasers Centrepoint Limited. Mobile Apps FREE Signal Up Log in Property Agents Suggestions JLL acquires YY Property Options Sdn Bhd in Malaysia

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