State and equal-intrinsic energy curved surfaces of ideal gas

Transcription

1 Interdiscilinary Journal of Chemistry esearch Article ISSN: State and equal-intrinsic energy curved surfaces of ideal gas Xiaowu Yao 1 and Haoyun An* 1 1 Zhengzhou Granlen Pharmaech Ltd 13 Eastern Hanghai oad Zhengzhou Henan China 4516 Abstract he state and equal-intrinsic energy curved surfaces of the ideal gas were investigated utilizing analytic geometrical method he results indicated that the derived state curved surface is the ruled quadric surface and the curved surface of equal-intrinsic energy in the system is the inverse surface he results thus obtained can be used to exlain the basic inter-comarison of intrinsic energies among all systems Finally the verifiability and falsifiability of Joule s Law were analyzed based on the twelve artial differential equations of the continued equality thus derived able of contents image (OC Grahic) Investigation of state and equal-intrinsic energy curved surfaces of the ideal gas using analytic geometrical method generated intrinsic energy equation U n continued equality covering all the basic theories of ideal gas Highlights Ideal gas state curved surface P n is a ruled quadric surface he curved surface equation of a state is n welve differential equations were obtained through theoretical derivation from various resects Curved surface equation of intrinsic energy U n the continued equality derived herein includes all the basic theories of ideal gas Joule s law has both verifiability and falsifiability It includes four of the twelve artial differential equations derived from continued equality but excludes the rest eight Introduction In the free exansion exeriment Gay-Lussac and Joule roved that both gas ressure and volume changed before and after the vacuum exansion while temerature and intrinsic energy are ket unchanged hus Joule s Law concluded that the intrinsic energy of the ideal gas was only correlated with temerature of the system while indeendent of its ressure and volume [1] Even though Joule s free exansion exeriment was not necessarily the most accurate it significantly imacted the develoment of modern thermodynamics theory gaining unanimous recognition hereafter tremendous amount of research was done to imrove Joule law U to the late th century eole just began to change their oinions about free exansion Klotz and osenberg in 197 [] argued that temerature during the free exansion rocess was not a definite value and it restored automatically after the system rebalanced In 1989 Xiancai Fu and co-workers [1] described that the temerature remained unchanged during the free exansion rocess because of the large heat caacity of water thus the temerature change could not be measured therefore the conclusion could be correct under the condition of P Around 1997 while debating with James J obinson A G Guy [34] derived U 3/ P [5] which illustrated that the internal energy U is the comosite function of both P and instead of only In 1997 Yan and Gao [6] also did further free exansion studies in this regard resulting in the same conclusion Liu and coworkers [7] then verified the same conclusion from the quantum statistical hysics oint of view in In 1998 Zhao and Luo [8] exlicitly stated in their book hermotics that the intrinsic energy of ideal gas is not related with volume concluded from the exeriment conducted by both Gay Lussac and Joule cannot be considered as the final conclusion he authors further stated that the Joule law relied on the exeriment erformed by ossini Frandsen [9] in 193 It is clear that the classical free exansion Joule s law and the concet of ideal gas fell into crisis under such a situation First free exansion got into the following dilemma: if the temerature varies in the free exansion rocess is acceted it would then be a counter examle to Joule s law; If it was aroved that the temerature of ideal gas in thermodynamics could change before and after free exansion then it could be hard to distinguish ideal gas and non-deal gas because the free exansion system belongs to non-deal gas in the statistic hysics when ressure P >> in the classical free exansion system If one insisted that the temerature remained unchanged before and after free exansion namely the temerature of non-ideal gas in statistical hysics didn t change before and after free exansion and then it could not be distinguished from the ideal gas Secondly Joule s law faces the following challenge: theoretically when ideal gaseous equation is utilized in the comosite function in molecular kinetic theory and mathematics the intrinsic energy of the deal gas system is still the function of P and On the exeriment basis Joule s law deends on the success of ossini Frandsen s exeriment after losing suort from free exansion [9] Washburn once revised and develoed the bomb calorimetry based on this exeriment However the fundamental base Corresondence to: Haoyun An 16 Eastern Hanghai d Zhengzhou Henan China 4616 el: han@granlencom Key words: continued equality state curved surface equal-intrinsic energy surface free exansion Joule s law eceived: January 17 17; Acceted: February 14 17; Published: February Interdisci J Chem 17 doi: /IJC1111 olume 1(): 66-71

2 An H (16) State and equal-intrinsic energy curved surfaces of ideal gas for this exeriment was that the intrinsic energy and ressure of gas system are in a linear relationshi namely U (P ) f () P + g () When this linear relationshi is extraolated to P one can only say U g () However if thus reached the conclusion that the intrinsic energy had no relationshi with ressure it would be obviously contradictory with the remise that the intrinsic energy and ressure of gas system are in a linear relationshi P was only a value used instead of omitting the influence of ressure on intrinsic energy Unless the statement as the ressure changes the data of each exerimental oint near P forms divergent sequence around intrinsic energy axile stands the Joule law couldn t be suorted he trend for the theoretical exloration in the field would certainly derive the exerimental base of Joule s law he real key here is that Joule s law and the first law of thermodynamics are incomatible he first law clearly determines that the intrinsic energy is the univalent function of states while the Joule s law determines the intrinsic energy of a system only relying on temerature roerty even under the circumstance of the undetermined system state he roerties of ressure and volume could be used to determine the state of an enclosed system while Joule law claimed that they have nothing to do with the intrinsic energy he equation U 3/ P derived by A G Guy [4] was the conclusion of statistical hysical axiom but it could not be used as the final argument for the thermodynamic axiom We reorted in 11 [1] that the internal energy of an ideal-gas can not only be exressed as the roduct of its volume and ressure but can also be exressed as the roduct of its temerature and quantity of gas herefore the statement of the internal energy of an ideal gas is indeendent of its ressure and volume is an inaccurate statement In 14 Steanov [11] described the ideal gas aradox and declared that it was the wrong aroach to exress the internal energy of a simle gas as the only function of temerature Instead multivariable equations of state derived from comosite functions must be used In 16 we [1] reorted the studies on the relationshi between internal energy and state of ideal gas in which the Continued Equality deicted as he internal energy of the ideal gas has interrelationshi with each state function was derived for the first time he historical contributions of Joule s law and its contradictions in thermodynamics indicated that it should have both verifiability and falsifiability in theory [1] In this aer we describe the curved surfaces of both state and equal-intrinsic energy of ideal gas on the basis of the static and homogeneous states by utilizing analytic geometrical method and by following the first law of thermodynamics We exlored and determined the method to solve the roblems and contradictions mentioned above he continued equality was derived and further exerimentally verified [13] State curved surface of ideal gas In thermodynamic theory system the so-called ideal gas refers to the relationshi among all the system state roerties that has to be consistent with the laws of Boyle Gay-Lussac and Charles In the 19th century eole merged them as the following ideal gas equation: n he ideal gas in statistical hysics is the gas formed by a grou of constantly moving and non-otential mechanical articles With the method dealing with the close indeendent subsystem it can be Nk n obtained: herefore in the classical theory arena studies of ideal gas equation meet the common requirements of the above two theory systems According to the extensive roerty of intrinsic energy it can be exlained that gas volume is also the essential variable to determine intrinsic energy herefore the ideal gas equation obtained from both thermodynamics and statistical hysics ersectives should be a function equation formed by four roerties P n Because the state equation obtained is a quadratic function thus it could be assumed that its standard form is a quaternary quadratic equation: a1 + a + a3n + a4 + a5 + a6n + + a + an+ a + a n + a + a + a n+ a + a he ideal gas equation obtained above is the result when a 5 1 a 1 - and all of the rest coefficients are zero Under the condition of n and the ideal gas equation could be exressed as the form of Charles law: K K n it means in P~~ coordinates all the ossible corresonding oints for temerature and ressure values could be interconnected to a straight line and its linear sloe is K n he closed systems with the same gas volume could have different sloes because of different volumes; therefore all the straight lines may not coexist in the same lane Because could value continuously within the interval of ( + ) thus all P ~ straight lines could be interconnected to a curved surface Under the condition of n each state oint determined by P and would definitely be assed K through by certain straight line determined by and is assed K n through by only one single straight line hus it can be determined that the state curved surface of a closed ideal gas system is a ruled quadric surface Similarly the same result can also be obtained from K Gay Lussac law exressed as: herefore the state surface K n has two grous of straight generatrixes (see Figure 1) Symmetry of curved surface he curved surface shown in Figure 1 has one symmetric lane P he intersecting line of state curved surface and symmetric lane is: n therefore P -n or - n Here is the arabola equation which gives the OA line in Figure 1 Moreover it is also erendicular to the symmetric lane and arallel to the lane of axis ; then the intersecting line between + cand the state curved surface is: n namely + c n + c his is also a arabola equation which gives a line with oosite direction such as Line BC in Figure 1 herefore the state curved surface of ideal gas may be a hyerbolic araboloid It can be verified as follows: otating the ~ ~ coordinate system 45 horizontally around the Axis the substitution equation of the coordinates is: cosθ sinθ sinθ + cosθ Substituting them into the ideal gas equation gives: Interdisci J Chem 17 doi: /IJC1111 olume 1(): 66-71

3 An H (16) State and equal-intrinsic energy curved surfaces of ideal gas Figure 1 State curved surface of the sealed ideal gas system in P three-dimensional sace Figure State curved surface and equal-intrinsic energy of enclosed ideal gases in P three-dimensional sace Figure 3 Curved surface of both state and equal internal energy for ideal gases with oen and constant volume in P three-dimensional sace ( cosθ sin θ)( sinθ + cos θ) n cosθsinθ + cos θ sin θ cosθsinθ ( ) cosθsin θ + (cos θ sin θ) n Substituting θ 45 o o o and sin 45 cos 45 into above equation resulted in: ( ) + ( ) n Arrangement resulted in: n or n n his is the standard equation of hyerbolic araboloid with a shae of saddle his surface has two grous of straight generatrixes having the following equations: λ + µ λ µ n n n n µ λ µ + λ n n n n Substitute the coordinate transformation equations cosθ + sinθ sinθ + cosθ into the above equation: cosθ + sinθ + sinθ cosθ ( cosθ + sinθ) + ( sinθ cosθ) cosθ + sinθ sinθ + cosθ ( cosθ sinθ) + ( sinθ + cosθ) λ λ λ µ n n µ n λ Make: K then: K µ n λ from µ λ obtain: n n µ n n λ K so K n µ n n n K n his result is exactly the form of Gay Lussac s law and also the intersecting line of isobaric lane and state curved surface from λ µ obtain λ µ n n n λ make: K n µ then K from µ + λ obtain: λ n n µ n K λ K n µ n n n K n his is the form of Charles law and also the intersection equation of isosteric lane and state curved surface he intersection standard equation of state curved surface and isothermal lane can be exressed as: 1 ( n n is a constant) then: ( cosθ + sin θ) ( sinθ cos θ) n Interdisci J Chem 17 doi: /IJC1111 olume 1(): 66-71

4 An H (16) State and equal-intrinsic energy curved surfaces of ideal gas cos cos sin sin cos cos sin sin n c θ + θ θ + θ θ + θ θ θ 4 n when is a constant it can be obtained: his is the mathematical form of Boyle law and also the intersection equation of isothermal lane and state curved surface Influence of gas content n on state curved surface In P~~ coordinate system the role of the fourth variable n is when the roduct of and is fixed the greater the volume of gas content n the lower the corresonding temerature he osition of the state curved surface is also relatively low herefore the geometry sace exressed by ideal gas equation is a family of curved surfaces which are overlaing and mutually intersecting with axes P and Coverage of curved surface According to the hysical significance of system roerty domain of definition of n can continuously value within the interval ( + ) So the family of curved surfaces should fill the first octant of three-dimensional sace Exression for intrinsic energy of the ideal gas with roerty coordinates According to the oint of view that intrinsic energy is the univalent function of state when the roerty coordinate system of ideal gas determines the state of the system it also inevitably determines the intrinsic energy of the system For general cases when the gas volume is variable the system state requires at least three roerties to be determined Different combinations corresond to four 3-dimensional coordinates: U f( ) f ( n) f( n ) f ( n ) he coordinate system ~ ~ is considered as an examle for further analysis (see below) Derivation from the coordinate system ~ ~ : intrinsic energy can be exressed as: U f( ) 1 It can be differentiated as: du d + d + d Herein requires d and d his is the rate of intrinsic energy change with ressure in the oen isosteric system with constant temerature he oen isosteric system means the artificially secified sace of a research system he secified sace is considered as the research system and the outside of this sace is the environment he exchange of substance and energy can be realized between the system and environment With the consideration of the influence of heat convection in the oen system the amount of heat exchange of an oen system should be the function of gas amount and temerature of the system according to the definition of heat caacity: δ Q d( n ) nd + dn Because d δ W Pd when d δ QP C Pdn dn d according to the first law it can be obtained: du δ d Q δw so requires d d hat is the changing rate of volume and intrinsic energy for the oen system when the ressure and temerature kee constant because d δ QP C Pdn C dn + dn dn + dn d + d and δ W d du δ QP δw C d + d d namely du d d so d requires d d Namely the rate of intrinsic energy of ideal gas changes with temerature when ressure and volume remain constant in an oen system Because d make δ W d and δ ( ) C C Q Cd n d( n ) d( ) by the first law of thermodynamics it can be determined as: du δ Q δw C d C + d When d d it can be obtained du δ Q δw So In the coordinate system ~ ~ with U f( ) 1 or its differential: du d + d + d he function of the intrinsic energy can be exressed as: du d + d d ( ) he above derivation can be analyzed as the following: C% du d ( ) illustrates that the intrinsic energy of the system can be determined by the roduct of ressure and volume in the coordinate system ~ ~ he intrinsic energy of the system varies with the rectangular lanar area within the lane ~ surrounded by the four straight lines axis P and as well as and When the system state is determined there is only one intrinsic energy value However after the intrinsic energy is determined there may be many corresonding system states All corresonding oints for the and values of the ossible states within the lane ~ are interconnected into an inverse curve: c In coordinate system ~ ~ with the continuous change of gas volume this curve can be develoed into a curved surface arallel to the axis his is the equal-intrinsic energy surface for U U he surface generatrix is namely the geometric significance of he intersecting line of the equal-intrinsic energy surface n and the state curved surface n n is ku ku obtained: (constant) n Interdisci J Chem 17 doi: /IJC1111 olume 1(): 66-71

5 An H (16) State and equal-intrinsic energy curved surfaces of ideal gas in the above equation U stands for the equal-intrinsic energy surface n is the state curved surface of certain enclosed system and is the isothermal surface According to current thermodynamic method if the system always remains as the uniform state during the rocess when the intrinsic energy and temerature kee constant in the free exansion rocess the ressure and volume of this system can only change along with the common intersecting line of the equalintrinsic energy surface isothermal surface and the state surface of enclosed system Derivation of exression forms in other coordinate systems By utilizing the same method mentioned above to analyze the coordinate system ~ ~ n similar result can also be obtained: In the coordinates ~ ~ n equal-intrinsic energy can be exressed as: U f ( n) It can be differentiated as: du d + d + dn n n n artial differential equation can be derived: n n n C% d Substitution gives: du ( ) he result is the same as that obtained from the coordinate system ~ ~ he geometrical attern and analyzing methods used are all the same as those in the coordinate system ~ ~ excet the axis of the coordinate system is changed into axis n In the coordinate system ~ n~ intrinsic energy can be exressed as U f( n ) 3 and it is differentiated as: du d + dn + d n n n the artial differential equation can be derived as: n C nc n n So: du d + dn + d + n n dn + n d n Arrangement results in: du C d( n ) du d( n ) means that the system intrinsic energy can be determined by the roduct of gas volume and temerature in the coordinate system ~ n~ he intrinsic energy value of the system is roortional to the area of the rectangle formed by axis n axis n n and four straight lines in the n ~ lane In coordinate system n~ ~ intrinsic energy can be exressed as U f ( n ) 4 It is differentiated as: du dn + d + d n n n the artial differential equation can be derived as: n C n n nc du dn + d + d n n dn ++ n d n Arrangement results in: du d( n ) his result is consistent with that of coordinate system ~ n~ and the corresonding grahics and analysis method are also similar Comrehensive results Combination of all exressions for the system intrinsic energy of an ideal gas within the four coordinate systems mentioned above results in the following continued equality: du d( ) n d integration gives: U U n n Any selected standard system state can be exressed as: U n So the function relationshi between the intrinsic energy and state of the system can be exressed as: U n his equation can be utilized to determine the intrinsic energy of any ideal gas system when the common reference standard was set based on the base oint such as lim K of the coordination δ system Further reviewing and understanding free exansion and Joule Law Analysis of verifiability and falsifiability of Joule Law he verifiability of Joule s law is that it determines a measure method of system intrinsic energy while its falsifiability is its exclusion for other measure methods for intrinsic energy Using the twelve artial differential equations for the intrinsic energy measurement derived by the analytic geometrical method described above one can determine the uniform measurement of the intrinsic energy for any system including different quality system hrough careful one-byone comarison one can conclude that Joule law statements intrinsic energy only has relationshi with system temerature includes the following four artial differential equations in the closed system: n ; n ; n nc ; n nc However considering Joule s statement system intrinsic energy has nothing to do with ressure and volume along with the relationshi between intrinsic energy and gas volume Joule s law excludes the rest of eight artial differential results as shown below: ; ; ; C ; n C ; ; ; ; n n n n Because Joule law s statement includes four of these equations the difference of intrinsic energy in closed system before and after the state change can be determined based on Joule law he exclusion of the other eight equations by Joule s law leads to its incomatibility with the first law of thermodynamics In all the twelve artial differentials four artial differentials indicate Joule s law s verifiability while eight artial differentials show its falsifiability which also embodies the comlementation of verifiability and falsifiability Free exansion exeriment he origin of the dilemma as to whether the temerature changes before and after free exansion is the difference between ideal gas Interdisci J Chem 17 doi: /IJC1111 olume 1(): 66-71

6 An H (16) State and equal-intrinsic energy curved surfaces of ideal gas concet in the definition of mathematical limit in statistical hysics and the definition of measuring distinguishability in thermodynamics Joule s law gives the rivilege that ideal gas temerature roerty is the only related factor with intrinsic energy which dictates whether the temerature changes before and after free exansion becoming the only factor to define an ideal gas In analyzing this core dilemma the continued equality clearly roved the verifiability and falsifiability of Joule s law; free exansion was no longer the standard to distinguish the ideal and non-ideal gasses he corresonding meaning then naturally turned into when the change of molecular interactions is small enough to be ignored before and after the vacuum exansion the temerature change can also be ignored herefore both challenges the dilemma of free exansion and the relationshi between intrinsic energy of ideal gas with temerature are easily resolved Ideal gas then naturally regressed as a theoretical model Connection of continued equality with the current theory he continued equality obtained above by using analytic geometrical method based on the first law of thermodynamics can also be derived through thermodynamic method and statistical hysics method and it covers the classical gas three laws olytroic rocess and Joule s law as well as all the basic theories of ideal gas in statistical hysics In addition the derivation of Joule s law by the second law of thermodynamics is essentially the results by introducing the inherent roerty of intrinsic energy into Maxwell equation and it has no n n direct relationshi with the second law Based on its containment of thermodynamics content and reasonable connection the continued equality described herein is smoothly incororated into the current theory system Conclusions Ideal gas state curved surface exressed by P n is a ruled quadric surface in P ~ ~ three-dimensional sace and a hyerbolic araboloid in the shae of saddle after horizontally rotating 45 he curved surface equation of a state is he equal intrinsic n energy surface of ideal gas in three-dimensional roerty coordinates is erendicular to P ~ lane or the inverse surface of the n ~ lane Curved surface equation of intrinsic energy is U n known as continued equality which includes all the basic theories of the ideal gas Among the twelve artial differential equations derived from continued equality Joule s law only includes four of them but excludes the rest eight which indicates that Joule s law has both verifiability and falsifiability he detail exerimental verification on the continued equality will be described in the subsequent article [13] Acknowledgements he authors thank Dr Zhiyong en and Chihui An for valuable suggestions comments and encouragement during the rearation of this manuscrit and literature collection9 eferences 1 Fu XC Shen WX Yao Y (1989) Physical Chemistry Higher Education Press Beijing Klotz IM osenberg M (197) Chemical hermodynamics: Basic heory and Methods hird Edition W A Benjamin Inc Menlo Park California 3 Guy AG (1997) More on ideal gas J Miner Met Mater Soc 49: Stracher GB Guy AG (1998) he internal energy of a P gas: a univariable or multivariable function Math Geol 3: obinson JJ rumble KP Gaskell D (1997) More on the ideal gas law J Miner Met Mater Soc 49: 3 6 Yan FL Gao (1997) Function forms of P 3U/ College Physics 16: 1-7 Liu FZ Wu YX Han LY () A relationshi between internal energy and state of ideal gas J Xuzhou Norm Univ (Nat Sci Ed) 18: Zhao KH Luo WY (1998) hermotics Higher Education Press Beijing 9 ossini FD Frandsen M (193) he calorimetric determination of the intrinsic energy of gases as a function of the ressure Data on oxygen and its mixtures with carbon dioxide to 4 atmosheres at 8 C Bur Stand J es 9: Yao XW (11) Mathematics relation between the state function and internal energy of the erfect gas Henan Sci 9: Steanov I (14) Exlanation of an ideal gas aradox Phys Chem: An Indian J 9: An H Yao XW Yang GS Hou JQ Niu S (16) Studies on the relationshi between internal energy and state of ideal gas 51at ACS National Meeting San Diego CA USA March PAPE No PHYS Yao XW An H Exerimental alidation of the elationshi between Internal Energy and State of Ideal Gas his Journal subsequent article Coyright: 17 An H his is an oen-access article distributed under the terms of the Creative Commons Attribution License which ermits unrestricted use distribution and reroduction in any medium rovided the original author and source are credited Interdisci J Chem 17 doi: /IJC1111 olume 1(): 66-71