A. G. Zakharov. Im Neuwerk 5, Stade, Germany

Transcription

1 9 Modern Orani Cheistry Researh, Vol., No., Auust 07 The Copression of Gases in Presene of Condensation Centers in Copressible Volue, the Equation of State and Isother of Polyoleular Adsorption for the Studied Syste A. G. Zakharov I Neuwerk 5, 680 Stade, Gerany Eail: saharowneuwerk@yahoo.de Abstrat. This work investiates the opression of ases in a losed volue with presene of ondensation enters in it. An equation desribin the output oleules fro the as phase when it is opressed was suested. The solution to this equation predited the existene of two reies in the developent of this proess and the possibility of transition fro one to the other at a point. We analyzed the effet of the nuber of ondensation enters on the view of the proposed equation, and, onsequently, on the proess of reoval of oleules fro the as phase. Usin the obtained results it was proposed the equation of state of real ases, whih physially orretly desribes their opression and ondensation on the ative enters. Applied approah to the analysis of the studied syste allowed a loially onsistent way to obtain the isother of polyoleular adsorption, ontainin the ain eleents of the experiental urves and a different ore siple for fro known equations. Keywords: Copression of ases, ondensation enters, equation of state, the isother of polyoleular adsorption Introdution Heteroeneous proesses involvin as phase, inlude a wide rane of industrially iportant reations, whih auses a onstant desire to understand the harateristis of their ourrene. When desribin the behaviour of the as phase in suh proesses typially use so-alled state equation, the siplest of whih is the equation of state of an ideal as: PV = Nk T () where P is the as pressure, V is volue, N is the total nuber of as oleules in the syste, k is oltzann onstant, T is the teperature. An ideal as is as in onditions where is perissible to nelet the size of its oleules and the presene of interation between the. If that nelet is unjustified and to orretly desribe the behaviour of as is required to take aount of these fators, the as is alled real. The ost well known equation of state of real ases, proposed in 87, is the equation of van der Waals [,] an P + ( V Nb) = Nk T () V where, aordin to the oon interpretation, the seond ter in the first parenthesis reflets the additional pressure that exists in the syste due to the presene of attrative fores between as oleules ( a a diensional onstant), and the seond ter in the seond braket is equal to the volue that is inaessible to their oveent. Proeedin fro the latter, the paraeter b an be interpreted as the effetive volue of the as oleules. Several other equations of state of real ases are known [-5]. They are appliable to ertain speifi systes. One of these is the enedit - Webb Rubin equation, whih partiularly well desribes the opression of liht hydroarbons and their ixtures. However, to larify the physial piture of this proess, they do not provide additional inforation opared with the equation of van der Waals, whih allows us in this paper to liit the disussion of the latter. Copyriht 07 Isaa Sientifi Publishin

2 Modern Orani Cheistry Researh, Vol., No., Auust 07 9 The task that was solved in the forulation of equation (), was to desribe with a sinle equation of the proess as opression, his ondensation and opression of the liquid fored as a result of ondensation. It is not surprisin that, beause of the oplexity of this proble, the utual influene of the proesses under onsideration, the van der Waals equation has a nuber of probles. As shown in [6] these inlude:. Its effetiveness lies to the riht of straiht V = Nb, i.e. the equation, beause their strutural features does not and annot desribe the opression of liquid;. The equation does not desribe reions of the siultaneous existene of the as and liquid phase, i.e. ondensation of as. On its raphial representation there are no areas of onstant pressure with hanin volue;. For ertain physially plausible onditions of ondutin the proess of opression, it predits pressure transition in a neative reion, whih annot be justified. The solution to these probles is iven in [6-8], where the author obtained dependenes desribin the output of oleules fro the as phase, naturally solvin the proble of transition at the point with hyperboli plot of as opression at the site of its ondensation with onstant pressure with hanin volue. It also investiated the influene of the size of as oleules on the proess of their reoval fro the as phase and obtained the equation of state, a raphial representation of whih athes qualitatively well with experientally reorded urves. While in [7,8] in a fairly oplete for solved the proble of the desription of the opression of the as to the aepted syste (ideal as) in the is not disussed experientally known fat of the influene of solid ipurities (ondensation entres) in the as phase on the proess of ondensation, and, of ourse, the equations of state do not ontain suh paraeters as the nuber of ondensation entres ( N ). This allows us to set the work task of the study of peuliarities of the proess of opression a heially inert ases in the presene of ondensation entres (solid partiles, dust) in the opressible volue with identifyin features of their fillin by adsorbed as and obtainin a respetive equations of state. Methodoloy Copression To solve this proble we use the ethod of reasonin and analysis suested in [7,8]. In these works it is assued that in a losed volue V is the total nuber of as oleules N. Part of the is in the for of sinle partile we denote the usin N and we assue their as as phase. All the reainin oleules are assoiated in oplexes with the nuber of partiles of or ore representin the iropartiles of the liquid. Reduin the nuber of sinle oleules in the as phase an o throuh double ollisions aon theselves with the foration of assoiated oplexes onsistin of two oleules, and as a result of their adsorption on existin partiles of the fluid. The rise is due to the desorption of sinle oleules fro the oposition of the assoiated oplexes (iro-partiles of the liquid). It is aepted that due to the sallness of the last eah of their onstituent oleules ay partiipate in the ats of adsorption and desorption. With that said, the ondition of equality of the speeds of inrease and derease in N, will have the followin for [6,7] α N α ( N N ) N + = β ( N N ) () V V where α is the rate onstant of foration of the assoiated oplex (iro-partiles of liquid) resultin fro double ollisions of as oleules with eah other, α is the rate onstant of the proess of their adsorption on the liquid phase, β is the rate onstant of the desorption proess of as oleules fro the iro-partiles of the liquid. The diension of α and α is equal to [ h - ], and the diension of β [h - ]. If both parts of equation () are divided by β and are denoted by α α k =, k = (4) β β Copyriht 07 Isaa Sientifi Publishin

3 94 Modern Orani Cheistry Researh, Vol., No., Auust 07 the nuber of onstants in it will be redued to two, and let the be the equilibriu onstants of the relevant proesses. The equation () will take the followin for k N k ( N N ) N + = N N.? (5) V V Physially eaninful solution of this square relative to N equations an be represented in the for N 4 ( ) ( k k ) V + Nk NVk + V Nk = Curves alulated by (6), show that this equation for ertain values of its onstants, satisfatorily desribes the ain experientally known features of the proess of reoval of oleules fro the as phase when it is opressed (Fi., urve ). If the volue of as is lare, all the oleules are in the for of sinle partiles and N = N (setion C of the urve ). When the as is opressed at point the pressure reahes its saturation and further N linearly dereases with dereasin V (plot 0). The inrease in the onstant k shifts the position of a point in the reion of lare volues, and dereasin this settin brins it loser to the axis of N. The derease in the total nuber of oleules in the syste also, in aordane with the physial piture of the proess, lowers the area C of the disussed dependene. Taken upon reeipt of the urve Fi. the value of a onstant k is sinifiantly less than the value of k. Only under this ondition the transition fro the field of onstany N to its linear derease ours at the point or alost point. The latter will our when k =0. If the value k is near to k, the kink at point is soothed, reduin N starts to this point, and this redution inreases with inreasin k (Fi. urves and ). Physially this eans that the saturation pressure as a harateristi of the syste, as a border, peuliar to it, will take plae only if the rate of foration fluid (iropartiles) resultin fro double ollisions of the as oleules will be low. If this ondition is not et, then the foration fluid will be ontinuous, onotoni proess at all levels of the syste. Hidden reak Point This analysis, arried out in [7,8], here an be suppleented by the followin onsiderations. It is obvious that the point on the urve in Fi. is a speial point of the investiated funtional dependene. There is no doubt that this partiular point is also on urves and of the sae fiure, althouh based on raphial representation, it annot be assued that the presene of suh features will be desribed in their proess. Naturally, there is a need to find a ethod to hek any funtional dependene on the presene of suh a sinular point. This is not the point of axiu, not the iniu point and not the infletion point of the funtion. It will probably be onsistent to all it a hidden break point of the orrespondin funtional dependene, thus pointin to the interestin feature that is present in her and in her onotonous harater, if it has a siilar point. If you take the derivative of expression (6), in the ase of urve of Fi. it is lear that the rate of hane of the value of this derivative will have an extreu at the point. The existene of an extreu of the rate of hane of the derivative of a funtion is equivalent to the stateent about existene of extreu of the seond derivative of this funtion. The orretness of this hypothesis for this ase is easy to verify if we take the seond derivative of expression (6) with respet to the variable V. Graphial representation of the resultin dependenies, usin data urves and in Fi., will have respetively the shape of urves 4 and 5 of the sae fiure. It is lear the presene of extrea on these dependenes, the inrease k leads to the dilution urve, the extreu is less pronouned, but it's there (Fi., urve 5). (6) Copyriht 07 Isaa Sientifi Publishin

4 Modern Orani Cheistry Researh, Vol., No., Auust N 0 5 C V, м 5 4 Fiure. The dependene of the nuber of oleules in the as phase fro the volue if possible the proess of ondensation (urves,, ) and seond derivatives fro funtions and (urves 4 and 5). All the urves are alulated by = 0 5, 9 5 = 0, = N k k 0 ; N = 0, k = 5 0, k = 4 0 ; N = 0, k = 0, k = 4 0. It should, apparently, be noted that the Lanuir isother, althouh its raphi representation oinides in for with urves and in Fi., doesn't have a hidden break point and under no values of the eber onstants an be transfored to the for of urve of this fiure. You an find whih volue orresponds to the point soure dependene in the extrees of its seond derivative. Clearly, in this ase, the value of the third derivative aordin to (6) will be zero. Doin the proper onversions, it an be shown that this will only be done if: V = N k k (7) ( ) Substitutin in the last expression the values k, k and N taken upon reeipt of the urve in Fi., we obtain that the point on the urve, i.e. the break point of the funtional dependeny orrespondin to V is equal to.98. On the urve this point is also present and its oordinate is 9. If we aept that k in (7) is very sall, the orrespondin frature funtional dependene will take plae at the point, and up to this point and it an be onsidered that N = N. Then takin into aount (7) to deterine the saturation pressure ( P 0 ), whih as shown in [7], as well kt P = (8) 0 k i.e. it is deterined by two onstants and teperature, and one of the onstants is introdued in [6,7] onstant k. It is lear that this analysis defines a new lass of physially iportant funtions, with hidden speial points (hidden break point of the funtion) and one is able to ake this point lear by a orrespondin hane in the onditions of flow desribin their proess. It is also lear that the study of the properties of this lass of funtions should be the task of a separate researh. 4 Copression with Condensation Centres Havin this approah and the results obtained with its help, one an proeed to an analysis of those features of the proess that will be due to the presene of ondensation entres in the opressible volue. Let us assue that in a losed volue is the total nuber of as oleules N and the Copyriht 07 Isaa Sientifi Publishin

5 96 Modern Orani Cheistry Researh, Vol., No., Auust 07 adsorption entres N. Part of the as oleules is in the as phase as sinle partiles the adsorbed state on adsorption entres N - N, and soe in N. Redution of the oleules in the as phase will o throuh their adsorption on the vaant entres N, and in the adsorbed state the nuber of oleules of the as phase, as noted above, is N - N. Desorption rate of oleules fro the surfae will be proportional to the quantity adsorbed on it, i.e. the sae N - N. If, as the siplest assuption to ake that all adsorbed oleules are equally aessible to adsorption-desorption proesses proeedin with their partiipation, for the ase of equilibriu an be written ( + ) ( ) k N N N N k N N N + = N N (9) V V k the equilibriu onstant for the adsorption proess at adsorption entres and the where k and already adsorbed oleules of the as phase. Equation (9) solves the sae task as equation (5), i.e. it desribes the hane in the nuber of oleules in the as phase when it is opressed, but for the ase of the presene in the opressible volue of the ondensation entres N. Its physially eaninful solution an be represented in the for N ( ) ( ( ) ) ( k + k ) V Nk + Nk + k N 4NV k k + V + Nk + k N + N = Last equation, as and equation (6), predits the existene of two possible types of reularities of output oleules fro the as phase (Fi. ). N 0 5 (0) C V, м Fiure. The dependene of the nuber of oleules in the as phase fro V in the presene of the ondensation enters in the opressible volue. All the urves are alulated by (0) with: N = 0 5, N = 8 0, k = 0 5, k =. 0 5 ; 5 5 N k k N k k 5 N = 0, = 0, = 0, =.8 0 ; N = 0, = 8 0, = 0, =. 0. In one ase, the nuber of oleules in the as reains onstant up to a ertain volue, and then linearly dereases with its derease (Fi., urves and ), another sae output urve of oleules of as has no sinular points and notes N derease with dereasin V in any field of its values (Fi., urve ). Dependenies alulated by (0), have the sae features as the results obtained when Copyriht 07 Isaa Sientifi Publishin

6 Modern Orani Cheistry Researh, Vol., No., Auust alulatin aordin to the equation (6). Curves and in Fi. show that if the volue of as is lare, all the oleules are in the for of sinle partiles and N = N (setion C of the urve ). When the as is opressed at point the pressure is reahed to its saturation and further N linearly dereases with dereasin V (plot 0). The inrease in the onstant k shifts, as in the previous ase, the position of a point in the reion of lare volues, and dereasin this settin brins it loser to the axis N. The derease in the total nuber of oleules in the syste also, in aordane with the physial paintin proess, lowers the sun disussin plot dependene (Fi., urve ). Unlike equation (6) equation (0) allows followin N influene on the patterns of output oleules fro the as phase when it is opressed. Curve in Fi. obtained for the sae values of all peranent and urve, exept N, whih is taken equal to 0 4. It is seen that the inrease in the nuber of ative entres in the volue led to a qualitative hane in the patterns of output oleules fro the as phase when it is opressed. The proess has beoe onotonous harater on the urve there is no point that separates the reions of saturated and unsaturated as. On the basis of equation (0) one an diretly trae the dependene of the nuber of oleules in the as phase ( N ) on the nuber of adsorption entres in the opressible volue ( N ). Four urves obtained by appropriate alulations are shown in Fi.. N N 0 4 Fiure. The dependene of the nuber of oleules in the as phase fro the nuber of the ondensation enters in the opressible volue. All the urves are alulated by (0) with: N = 0 5, V = ; 0 k = 5 0, k = 0 ; 4 5 k = 5 0, k =. 0 ; k = 0 5, k 5 =. 0 ; k = 5 0, k = 4 0. Apparently they reflet physially expeted piture of the proess. For sall k and k, i.e. at a low intensity of adsorption-desorption proesses in the syste, introdution to opressible volue of additional nuber of ative entres has virtually no effet on the value of N (Fi., urve ). The inrease k and k, leads to the fat that N dereases with inreasin N (Fi., urve ). For lare values of k this derease beoes linear, whih suests that it is aused by the adsorption on the ative entres (Fi., urve ). If you save the values k to inrease k, N is redued in the reion of sall N (Fi., urve ), whih, apparently, points to the inreasin role of poly oleular Copyriht 07 Isaa Sientifi Publishin

7 98 Modern Orani Cheistry Researh, Vol., No., Auust 07 adsorption under these onditions. Even reater k leads to even reater redution in N at sall N, what learly testifies to the ative flow in the ase of polyoleular adsorption. It should also be noted that all the alulated variants of the studied dependene indiate the equality of the nuber of oleules in the as phase ( N )and the total nuber of oleules in the syste ( N )if there are no ondensation entres in the opressible volue ( N =0). The sae result an be obtained if you take the liit fro the riht side of equation (0) with N tendin to zero. This liit is equal to N. The Ipleentation of these physially reasonable requireents an, apparently, be rearded as a definite onfiration of the orretness of equation (0). 5 Equation of State In the presene of dependene (0) an be solved for the analysed ase one of the probles of the equation of Van-der-Waals task transition at the point with hyperboli plot of as opression at the site of its ondensation with onstant pressure with hanin volue. For this we write the equation of state of ideal ases () in the for PV = N kt () whih takin into aount that the nuber of oleules in the as phase is not onstant and ay vary durin the opression proess. It is lear that in () N shall be deterined by (0). Calulated usin these two equations the urves are shown in Fi. 4. P 0 4, Па V, м Fiure 4. Isothers of the ideal as in the presene of the enters of ondensation in the opressible volue. 5 Calulated by () with (0) with: k =.6 0, T = 00, k = 0 ; N = 5 0, N = 0, k =.5 0 ; N = 4 0, N = 0, k = ; N = 0, N = 0, k =. 0 ; 4 N = 5 0, N = 5 0, k =.5 0. For reater larity, they are shifted relative to eah other by hanin the paraeter N. Curve (Fi. 4) obtained at low k, qualitatively oinides with the isother of the ideal as (with zero k this oinidene will be of a quantitative nature). There is no plot of onstant pressure with hanin volue. The inrease of the equilibriu onstant of the adsorption proess of as oleules on liquid k leads to a siilar plot to relevant urves (urves and ), i.e. the isother ondition beins to reflet the proess of ondensation of as. It is lear that with ore k this plot orresponds to a lower Copyriht 07 Isaa Sientifi Publishin

8 Modern Orani Cheistry Researh, Vol., No., Auust pressure, whih oinides with the result expeted fro a real proess. The transition point of the funtion with a hyperboli plot to the plot horizontal line oinides with the break point funtion onstruted by (0) with appropriate values of onstants. It is lear that the pressure values on the plot a horizontal line equal to the saturation pressure and ust be deterined by (8). When alulatin the urve (4) in Fi. 4 all onstant values exept N, whih inreased by five orders of anitude, equal to the orrespondin values used to obtain urve. It is seen that the inrease in the nuber of ative entres in the opressible volue leads to the derease in pressure with a orrespondin V - urve 4 in the whole area presented in Fi. 4, lies below urve. It is lear that suh behaviour is physially reasonable. Adsorption of in addition entered entres should lead to the redution of the pressure in the volue. In addition a lare nuber of ative entres in the opressible volue lead to the blurrin of the ondensation area, on urve 4 there is no plot of onstant pressure with hanin volue. The sae effet an be ahieved with urves and, if adopted in the alulations the value N is inreased to 0 5. The output urves of Fi. to zero orrespond to the aepted odel of an ideal as, whose oleules do not oupy any volue. In ase of a real as in equation (9) (as is done in obtainin equation of Van der Waals) an introdue an aendent, takin into aount the effetive volue of oleules ( V ): k N ( N N+ N ) k ( N N ) N + = N N () V NV V NV The prora "Matheatia 0" ives the solution of this equation in the followin a little ubersoe, but produtive for: A+ N = () ( k k ) where A = Nk + k N N ( ) k ( N N ) + ( V + Nk + NV ) k ( + ) ( + ) + ( ) N N V NV Nk N N = ( V NV ) + ( V NV ) The results of alulations based on the latest equation with a relatively sall N (if N =0 5, N is of the order of 0 5 ) ive urves and in Fi. 5. The for of urves is quite unusual. Sine they have both a break point for the funtion and a point for V = NV of an abrupt hane in its value. For all the unusual results of the alulation fro the funtional dependene (), the proress of the urves obtained is physially justified. In the reion less than NV, the fored liquid with the nuber of oleules in N is opressed, and it will not hane under opression. This reflets the results of the alulation done by the prora "Matheatia 0" (Fi. 5). If N is lare (approahes the value N or exeeds) the alulated urves appear distortion et rid of whih is possible iven the physial onstraints iposed on the proess, desinin the appropriate equation with the introdution of two funtions Heaviside (Heaviside) [9,0] He = UnitStep V NV, He = UnitStep V + NV UnitStep V (4) With this in ind, as shown in [4], the expression for N will be as follows N = N He + N He (5) 0 where N should in disuss ase be deterined by (). The expression (5) results also in the lare 0 N to physially eaninful representations (Fi. 5, urve ). It is seen that the inrease in the nuber of ative entres in the opressible volue akes possible the withdrawal of oleules fro the as phase and by lare volue of the syste, that orresponds to the physial onepts. Of ourse, usin the Copyriht 07 Isaa Sientifi Publishin

9 00 Modern Orani Cheistry Researh, Vol., No., Auust 07 dependenies (5) we an obtain urves and in Fi. 5. In this ase, it is interestin that at sall N usin the prora "Matheatia 0" and without the use of funtions Heaviside ets the sae results. N V, м Fiure 5. The dependene of the nuber of oleules in the as phase fro V, the inequality to zero their own volue V. Calulated by (0) with: N = 0, N = 0, k = 0, k = 0, V = N 5 k k 0 V 0 5 N 8 k k 0 V 4 0 N =.5 0, = 0, =, =, = N = 0, = 0, =, =, = With equation (), siilar to the way it was done for the ase of ideal as, one an o over to the equation of state that takes into aount the size of its oleules. For this purpose, in aordane with van der Waals, we introdue in the equation of state () a orretion for volue of oleules and, in addition, aordin to the above, onsider the differene between their total nuber in the syste (N) and in the as phase ( N ), i.e. ( ) P V NV = N k T (6) where N at low N an be deterined by (), and at lare it is neessary to use the dependene (5). Four urves alulated usin these dependenies are shown in Fi. 6, the first and seond of the differ only in the paraeter k, and the third and fourth for reater larity, they are shifted relatively alon the axis V owin to the hane of paraeter N and differ fro eah other by the value N. It is seen that at sall k (urve ) orrespondin raph is a hyperbola with a vertial asyptote equal to NV in the reion of positive V. Inreasin k transfors the raphial representation in suh a way that it appears three phase (urve ). The first of the, lyin in the reion of lare volues, onsistent with the opression of as, the seond, the pressure of whih is onstantly hanin in volue - the proess of its ondensation, and the third with a sharp inrease in pressure with little hane in volue (aordin to the interpretations adopted in the analysis of the isothers of van der Waals) an be assoiated with opression of the liquid. An iportant, althouh aybe subtle is the fat that the last plot on the urve with a hih deree of auray oinides with the position of the border separatin the pure liquid reion fro the reion in whih is present in the syste and the as phase, i.e. with a line V= NV. As noted in [7] in the analysis of the van der Waals isother, the whole of it, inludin the reion losest to the P axis, lies to the riht of the line V= Nb, i.e. the orrespondin dependene an not desribe the opression of the liquid. Copyriht 07 Isaa Sientifi Publishin

10 Modern Orani Cheistry Researh, Vol., No., Auust 07 0 P 0 4, Pa V, Fiure 6. Isothers of a real as state in the presene of the entres of ondensation in the opressible volue. Calulated by (6) with (5) when: N, 4 0, N = 0, k = 0, k = 0, V = 0 5 N 5 k k, 0 V 0 5 N 0 k 0 k 6 7, 6 0 V N 9 k 0 k 6 7, 6 0 V 6 0 N =, 4 0, = 0, =, =, = N = 0, = 0, = 0, =, = 4 N = 0, = 0, = 0, =, = The inrease of k, as in the ase of Fi., akes lower saturation pressure (urve ), whih obviously oinides with the result expeted fro a real proess. If N is relatively sall, then the urve preserves all the above entioned eleents and it is possible to talk about the onept of saturation pressure (urves and of fiure 6). Curve 4 in Fiure 6 was obtained for the sae values of all onstants as urve with the exeption of N, whih is inreased by four orders of anitude. It is seen that for lare N a proess of foration of fluid oes in the whole area of the investiated volue and, as a onsequene, the orrespondin isother (urve 4 of fiure 6) has no areas of onstant pressure with hanin V. The output plots for the opression of fluid for urves and 4 oinide and do not depend on the nuber of ondensation entres ( N ) in the opressible volue. 6 Isother of Polyoleular Adsorbtion It is lear that the loseness to reality of the obtained results depends on the deree of orretness of the fundaental equation (9). To hek its effiieny will analyse the hanes in the interpretation of polyoleular adsorption, whih leads to the proposed approah to the study of the proess of as opression in the presene of the entres of ondensation in the opressible volue. Given that the nuber of adsorbed oleules N a is deterined by the followin expression N = N N (7) a soure dependene (9) an be transfored to ind k NN k ( N + N a a ) N + = N (8) a V V Fro the last equation N, we et a Copyriht 07 Isaa Sientifi Publishin

11 0 Modern Orani Cheistry Researh, Vol., No., Auust 07 k NN N = a V + k N k N If in this equation substitute V fro (), it is onverted to an interestin siht: k NP N = a k P k P+ kt Equation (0) represents the isother of polyoleular adsorption. We see iediately that in the absene of adsorption in the seond and subsequent layers, i.e. k equal to zero, it takes the for of the Lanuir isothers (Fi. 7, urve ). N a 0 0 (9) (0) P 0, Pa Fiure 7. The adsorption isother was alulated aordin to the equation (0) for different values of onstituent onstants: T = 50, k =,8 * 0 ; 0 4 N = 5 0, k = 0, k = 0 ; N = 5 0, k = 0, k = 7 0 ; 0 6,, 0 5 N = 0 k = 0? k =, If k is not zero, it is, of ourse, an be either less than k or ore. When k < k the alulation by the equation (0) ives the results, a raphial representation of whih oinides in for with the Lanuir isother (Fi. 7, urve ). More fillin is justified, as the urve in Fi. 7 is isother of polyoleular adsorption, while the urve of this fiure represents the onooleular adsorption isother. Apparently this is a new approah to the proble, beause in the literature it is ipossible to find stateents that the isothers of onooleular and poly oleular adsorption an be idential in for. It is lear that in addition to purely onitive interest of this assertion requires a ore areful approah to the use of the adsorption ethods for the deterination of solid surfaes. When k k inrease in pressure will result in a redution of the denoinator of expression (0), > and thus to inrease the fill. Thus, fro physial onsiderations, it is lear that the pressure in suh a syste ay not exeed a ertain value deterined by the requireent of equality to zero of the denoinator of expression (0), and represents the saturation pressure P. For its definition we have 0 kt P = () 0 k k If k in the latter equation is so sall that its value an be neleted, it oinides in for with the dependene (8), obtained for the syste not ontainin in the opressible volue of the entres of ondensation. Copyriht 07 Isaa Sientifi Publishin

12 Modern Orani Cheistry Researh, Vol., No., Auust 07 0 The results of the alulation in (0) for the ase when k > k is urve of fiure 7, as well as the urve, is the isother of polyoleular adsorption of a as on a solid surfae. This onsiderable differene in the shapes of isothers ours as a onsequene of different relationships between the onstants k and k for these ases. If we aept that k is onstant, everythin beins to depend on the harateristis of the surfae k. If the surfae is fored by the entres with k ore than k, the isother of poly oleular adsorption will be in the for of urve of fiure 7. If it is fored by entres with k less than k, the orrespondin isother will be in the for of a urve of this fiure.? An interestin situation is when the surfae is presented by the entres and the first and seond type. It is lear that total fillin will be equal to the su of fill entres of eah type. For the ases of urves and fiure 7 the suation leads to urve 4 of the sae fiure. We iediately see that this urve is well refleted by nuerous experiental data. Traditionally, the initial portion it is interpreted as the fillin of the onolayer and a subsequent inrease overae at hiher pressures due to the foration of a seond and next adsorbed layer on the surfae, i.e. the developent of polyoleular adsorption. ased on the foreoin it is lear that suh an understandin ay not always orrespond to objetive reality. The shape of the urve 4, for exaple, in every setion depends on polyoleular adsorption. Two ehaniss to inrease the fill are explained in the analysis of fiure 7 and they are onneted with the presene on the surfae of two types of entres with different values of the paraeter k. The proposed understandin enables hanes in a wide rane of urve 4 in fiure 7, and, as a onsequene, allows to obtain a satisfatory areeent of alulation results with experiental data. 7 Conlusion In onlusion of this work, we an say that the proposed approah to the analysis of the as opression proess in the presene of ondensation entres in the opressible volue allowed us to obtain a physially onsistent piture of it, explainin, apparently, all experientally known features of its flow.. It is shown for the first tie that the investiated proess is desribed by funtions havin hidden speial points, whih beoes apparent only under ertain onditions of its aintenane.. The presene of these points ade it possible to solve the old proble of desribin transitions fro the hyperboli as opression reion to its ondensation setion with onstant pressure under a varyin volue.. It is shown that an inrease in the nuber of ondensation entres akes the speial points hidden, the orrespondin funtional dependenies beoe onotonous. 4. Takin into aount these results, the equation of state of real ases is proposed, whih naturally desribes the hyperboli setion of their opression, the reion of their ondensation with onstant pressure at a varyin volue, and the opression setion of the fored liquid. 5. This analysis ade it possible to deterine the fillin of the ondensation entres by adsorbed as, i.e. obtain isothers of polyoleular adsorption for the studied syste. The latter differs fro those known by a sipler for and opens up the possibility of a physially ore realisti siulation of the proess desribed by the. The resultin equation of state and isother of polyoleular adsorption for the studied syste were obtained, leadin to a new understandin of the flowin in it proesses. Referenes. Diu, Guthann C, Lederer D, Roulet. Grundlaen der Statistishen Physik. erlin-n.y.: Walter de Gruyter, S.. Reid R, Prausnitz J., Sherwood T. Properties of ases and liquids. L.: Cheistry, p. 4. M. enedit, G.. Webb, L. C. Rubin J.Che.Phys., 940, No. 8. S Zakharov A. G. //Solid Fuel Cheistry No.. S Zakharov A. G. //Solid Fuel Cheistry No.5. S Zakharov A. G. //Journal of Physial Cheistry. 0. Vol. 85, No. 5. S. 85. Copyriht 07 Isaa Sientifi Publishin

13 04 Modern Orani Cheistry Researh, Vol., No., Auust Matthews, J., Walker, R. Matheatial ethods of physis. M.: Atoizdat, p Copyriht 07 Isaa Sientifi Publishin