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calculation of a heat of absorption

Quantity of a heat of absorption contains the direct information on a binding energy and the nature of the adsorption interaction of each pair an adsorbate/adsorbent, and dependence of heats of absorption on filling in monolayer field is the performance of energy homogeneity or heterogeneity of a surface of an adsorbent.

As a heat of absorption understand
Warmth which is oozed at sedimentation adsorbtiva on an adsorbent surface. There are different views of heats of absorption, but they can be parted on two basic groups [13]:

1. The warmth gained from equilibrium thermodynamic data (as a rule, from the isotherms of adsorption measured at different temperatures);

2. The warmth measured in a calorimeter; and measuring process corresponds to some nonequilibrium process and consequently the effect basically should depend on a construction of a calorimeter and from an expedient of carrying out of experience.

For kaloricheskoj performances of the adsorption processes it is necessary to enter four warmth - two, corresponding to a constancy davlenijai

Two, corresponding to a volume constancy the integrated heat of absorption corresponding to differential warmth

- The isosteric heat of absorption spotted sootnosheniemkotoroj corresponds integrated warmth

Adsorptions:

Coefficients gи sу labels of enthalpy Hотвечают to a gas phase and surplus accordingly. Znachkisootvetstvujut srednemolnomu and

To partial value of corresponding quantity. This circumstance essentially distinguishes adsorption from the volume processes which description can be spent in terms of two warmth - one at constant pressure and one at constant volume.

Thus ooze following basic methods of definition of heats of absorption:

1. From isotherms of adsorption by build-up of isosteres and calculation of isosteric heats of absorption qst;

2. From effects gazohromatograficheskih measurings of volumes of keeping V at the underload concentrations adsorbtiva and different temperatures in the form of so-called differential heats of absorption at zero filling of a surface by dependence build-up ln Vот 1/T;

3. From direct calorimeter measurings in the form of differential heats of absorption at different fillings of a surface.

In experiment can be calculated qintи qst, but especially important quantity in the adsorption theory is the isosteric warmth, spotted as a difference between molnoj an enthalpy gas fazyi partial

molnoj an adsorbate enthalpy.

It is possible to show, that an isosteric heat of absorption qst - it molnaja warmth which would be oozed at isothermal transference of very small quantity of substance from a gas phase in the adsorbed.

In the given operation on the basis of gained within the limits of classical MFP lateral views of local density calculation of isotherms of adsorption is yielded for flat and spherical adsorbents. Considering, that in ideality approach adsorbtiva and nedeformiruemosti an adsorbent it is possible to present an isosteric heat of absorption in a view

The basic idea of calculation of an isosteric heat of absorption is reduced to the following sequence of activities. At first, carrying out calculations at various values of temperature and pressure, we gain family of isotherms of adsorption (fig. 23). Fixing constant value of adsorption T, and using the gained family of isotherms of adsorption, we gain a gang of points (Ti, Pi), necessary for build-up of continuous function P = P (). We will especially score a problem

Approximations of a gang of points (Ti, P) (fig. 24). For carrying out of calculations the developed earlier and registered computer program developed earlier has been essentially finished taking into account that for calculations of heats of absorption 67

It is necessary to handle more number of isotherms of adsorption. On the one hand, with a view of reduction of an estimated time of the program it makes sense to break a temperature interval into rather small amount of points. On the other hand, the there are less than starting points we have, the it is more probability of occurrence of major errors of approximation. By us it has been erected, that for an approximation problem the optimal select is the modified method of approximation by the cubic splines, offered in operation [145]. Curves P gained thus = P (T), adsorptions corresponding to several fixed values, are given on fig. 25.

Fig. 23. Family of isotherms of redundant adsorption of methane on graphite at various values of temperature in system.

Also it is necessary to score that fact, that at the formula (2.18) there is a derivative of function P (T), and it imposes very high requirements of accuracy and monotony of gained dependence P (T), as any
"Unevennesses" of approach will be considerably strengthened by the subsequent differentiation.

Fig. 24. A finding of points (Ti, P.) on family of isotherms of redundant adsorption of methane on graphite at the fixed value of adsorption G = 0.35.

Our analysis of major number of methods of interpolation and approximation has allowed to come to conclusion, that it is the most expedient to use the linear interpolation between tochkamidlja a numerical finding of derivatives in

The intermediate points (fig. 26, 27). It is obvious, that derivative of the straight line equation will be a stationary value on gaps. In summary, appropriating value of a derivative on an interval it is easy to value of argument of the middle of an interval to gain a gang tochekkotoryj at

The help of the modified method of interpolation [145] with a split-hair accuracy features a functional connection of derivative function of pressure P (T) on temperature.

Fig. 25. Dependences P (T) methane on graphite at various fixed values of adsorption G (g/g).

In the observational operations, usually, it is accepted to give isosteric heats of absorption in dependence not from temperature, and from the content (adsorption), for example in a mole Our design procedure allows to gain functional connections of heats of absorption both from temperature, and from the content. Moreover, calculation is possible both for dokriticheskoj, and for zakriticheskoj fields of temperatures in a wide gamut of pressures and contents of the adsorbed substance in system.

On fig. 28 temperature dependences of isosteric heats of absorption of hydrogen on a graphite surface are presented. First of all it is necessary to explain not average dynamics of change of explored quantity, and peaks on dependence qst (T). On the basis of the effects presented on fig. 27, it is possible to draw a deduction, that peaks on a drawing of an isosteric heat of absorption are not "artefact" of applied mathematical approach. In other words, we would not observe such good consent between three essentially various methods

Fig. 26. Comparison of three computational methods of an isosteric heat of absorption in cases when differentiable function P (T) is interpolated between known points: 1 - a cubic spline, 2 - the modified cubic spline [145], 3 - pieces of straight lines. The substance content in system G = 0.21 g/g.

Fig. 27. Comparison of three computational methods of an isosteric heat of absorption in cases when differentiable function P (T) is interpolated between known points: 1 - a cubic spline, 2 - the modified cubic spline [1 45], 3 - pieces of straight lines. The substance content in system G = 0.27 g/g.

Fig. 28. Family of dependences of isosteric heats of absorption of hydrogen on graphite from temperature at various values of adsorption.

Approaches if viewed peaks were "artefact". Their physical sense becomes clear if to be converted to fig. 29, 30 and 31 on which heats of absorption and quantities of filling of stratums in the adsorption phase on a graphite surface are presented.

The Fig. 29 corresponds to isosteric warmth at the constant adsorption equal 0.1 g/g. It is simple to note, what exactly on peaks is observed the peak change of filling of an exterior (third) monolayer. And, for first three peaks (21, 22.5, 24) this change carries plus, and for the third (26) subzero character. The conventional fifth peak (28.4) also corresponds to a case of sharp growth of an exterior monolayer. As one would expect, between peaks we observe essentially less considerable change of filling of an outside monolayer. The similar effect can be observed and at constant value of adsorption 0.2 g/g (fig. 30).

Fig. 29. Temperature dependences of an isosteric heat of absorption of hydrogen on graphite from temperature. Numerals with arrows specify degrees of filling of the adsorption stratum (quantity of monolayers). Calculation was spent for constant value of adsorption 0.1 g/g.

Fig. 30. Temperature dependence of an isosteric heat of absorption of hydrogen on graphite from temperature. Arrows specify degrees of filling of the adsorption stratum (quantity of monolayers). Calculation was spent for constant value of adsorption 0.2 g/g.

Fig. 31. Temperature dependence of an isosteric heat of absorption of hydrogen on graphite from temperature. The given case corresponds to six filled monolayers. Calculation was spent for constant value of adsorption 0.25 g/g.

As well as in case of more thin adsorption stratum, the peak change of filling of an exterior (fifth) monolayer is necessary on peaks of an isosteric heat of absorption. It is interesting, that at practically limiting (for dokriticheskogo a case) adsorptions peaks of isosteric warmth, except the first, are not observed (fig. 31). It is obvious, that it is related to the underload change of filling of an exterior (sixth) monolayer in all gamut of temperatures. Concerning blanket (average) dynamics of change of explored quantity it is possible to tell the following: at temperatures, close to critical, the isosteric heat of absorption for rather thick a word (it is more than 4 monolayers) decreases with adsorption magnification (fig. 28). It is obvious, that it is related with bolshej mobility of molecules of the adsorbate which is on bolshem removal from a substrate and, as consequence, with essential reduction of influence podozhki. It, in turn, facilitates desorption process. Thus, we find out correlation between peaks on the temperature
Dependences of an isosteric heat of absorption and a degree of filling of monolayers. Nevertheless, the origin of the specified peaks remains to the full not clear.

On fig. 32 dependences of isosteric heats of absorption on quantity of adsorption are given at various fixed values of temperature. It is interesting, that some peaks are observed and on these dependences, and, these peaks have approximately on the same content (with some detrusion) for the curves corresponding to various constant temperatures. Obviously it also illustrates heterogeneity of filling of monolayers and is a valuable radiant of the information for the analysis of structural and thermodynamic properties of the adsorption stratums.

Fig. 32. Isosteric heats of absorption of hydrogen on graphite at various values of temperature.

Effects for methane on graphite are presented on fig. 33, 34. As we deal with more massive molecule, on dependence of an isosteric heat of absorption (fig. 33) for small values of adsorption ( 0.24 g/g) any peaks it is not observed, i.e. filling of an exterior stratum changes smoothly. Moreover, as one would expect, for high values of constant adsorption (0.25 and 0.45 g/g) distinctions between quantity of isosteric warmth are much less appreciable. Dependences qstот contents, are presented on fig. 34. It is visible, that besides the characteristic first maximum of isosteric warmth observed on all similar known observational curves, it is possible to observe and one more maximum characterising dependence of settlement quantity from structural changes in an adsorbent, namely from changes of quantity of filling of exterior monolayers in the adsorption stratum.

Fig. 33. Family of temperature dependences of isosteric heats of absorption of methane on graphite at various values of adsorption.

Comparison with experimental data and similar calculation E.A.

Ustinova within the limits of MFP it is presented on fig. 35.

Fig. 34. Isosteric heats of absorption of methane on graphite at various values of temperature.

Fig. 35. Dependence of an isosteric heat of absorption of nitrogen on graphite from quantity of adsorption (content) at temperature 77 To: 1 - our effects, 2 - experimental data [146], 3 - a settlement curve from operation [147].

As a whole the behaviour of the isosteric heat of absorption calculated within the limits of approach MFP used by us, for nitrogen on graphite will well be compounded with data from operations [146,147]. Moreover, on the central site of a curve, the odds between our calculation datas and the observational dependence [146] is underload.

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A source: Grinev Ilja Viktorovich. EXAMINATION of the ADSORPTION STRATUMS ON FLAT And CURVED SURFACES With USE of CLASSICAL METHOD FUNKTSIONALA of DENSITY. The DISSERTATION on competition of a scientific degree of the candidate of physical and mathematical sciences. Tver - 2014. 2014

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