Should be smaller than BET area). 9.6 The microporeanalysis method (MP method ) t-plot. Three point for 1 line. 9.5 Isotherm of N2 silica gel, Davidson 03 at 77.3. Standard Test Method for Specific Surface Area of Alumina or Quartz by Nitrogen Adsorption. D19) Standard Test Method for Precipitated Silica - Surface Area by Multipoint BET Nitrogen Adsorption. D56) Standard Test Methods for Precipitated Silica - Surface Area by Single Point B.E.T. Nitrogen Adsorption.
© A.W.Marczewski 2002
A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces
t-plot and t/F-plot
(see an example in a surface area / pore volume analysis report)
(see a short summary of t-plot, t/F-plot and αs methods)
(see a comparison of t-plot and αs-plot)
(see a short summary of t-plot, t/F-plot and αs methods)
(see a comparison of t-plot and αs-plot)
t-plot
This method of isotherm data analysis was introduced by de Boer. It assumes, that in a certain isotherm region, the micropres are already filled-up, whereas the adsorption in larger pores occurs according to some simple equation, characteristic for a large class of solids. This equation should approximate adsorption in mesopores, macropores and on a flat surface in a narrow pressure range just above complete filling of micropores, but below vapour condensation in mesopores. Such assumption may be valid, if micropores are small and super-micropores are not present. Then the adsorption within this pressure region may be described by a simple linear dependence :
a(x) = amicro,max + k Sext tlayer(x)
where:
x = p/ps
amicro,max - adsorption in saturated micropores,
Sext - 'external' surface area; here it is the surface area of pores larger than micropores,
tlayer(x) - estimated statistical thickness of adsorbed layer (in meso- and macropores) - usually by Harkins-Jura (HJ) and Halsey / Frenkel-Halsey-Hill (FHH) equations - often used are their experimental versions HJ and Halsey; in the t-plot calculated thickness values are within 0.354 nm (i.e. tmono) and 0.5 nm (it corresponds roughly to the range of relative pressures 0.1 - 0.2); for solids with small amounts of micropores, the Halsey isotherm may give negative value as an estimate of micropore volume/area, thus it sould be used with caution.
k - coefficient which depends on units used for the values of adsorption a(x), layer thickness t(x) and surface area Sext. E.g. for t [nm] , S [m2/g] and a [cm3/g STP] we obtain: k = 1 / (4.3532 tmono), where tmono = 0.354 nm is monolayer thickness for N2 adsorbed layers and 4.3532 is a coefficient for calculation of occupied area [m2/g] from adsorbed amount [cm3/g STP] (N2 adsorption at 78K).
a(x) = amicro,max + k Sext tlayer(x)
where:
x = p/ps
amicro,max - adsorption in saturated micropores,
Sext - 'external' surface area; here it is the surface area of pores larger than micropores,
tlayer(x) - estimated statistical thickness of adsorbed layer (in meso- and macropores) - usually by Harkins-Jura (HJ) and Halsey / Frenkel-Halsey-Hill (FHH) equations - often used are their experimental versions HJ and Halsey; in the t-plot calculated thickness values are within 0.354 nm (i.e. tmono) and 0.5 nm (it corresponds roughly to the range of relative pressures 0.1 - 0.2); for solids with small amounts of micropores, the Halsey isotherm may give negative value as an estimate of micropore volume/area, thus it sould be used with caution.
k - coefficient which depends on units used for the values of adsorption a(x), layer thickness t(x) and surface area Sext. E.g. for t [nm] , S [m2/g] and a [cm3/g STP] we obtain: k = 1 / (4.3532 tmono), where tmono = 0.354 nm is monolayer thickness for N2 adsorbed layers and 4.3532 is a coefficient for calculation of occupied area [m2/g] from adsorbed amount [cm3/g STP] (N2 adsorption at 78K).
t/F-plot
Introduced by Kadlec in order to increase precision of t-plot analysis. In this method the Dubinin-Radushkevich isotherm is fitted in the micropore range (low pressures / adsorption range). This method may help to compensate for t-plots non-linearity in the case of solids containing some super-micropores. However much wider data range is required and t-plot method looses its great simplicity.
A preferred method (see comparison) should be αs-method. This method requires some standard isotherm data for comparisons, however it gives information largely free of adsorption model influences and additionally provides description of mesopore and - to some extent - macropore system.
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Summary
Fifteen years ago, a party was thrown at a banker's home, where many intellectuals such a journalists and lawyers attended. During that party, the group in attendance had many lively discussions, ultimately turning to the topic of capital punishment.
As the group argued, the two sides of the debate coalesced into two representatives: the banker, who is for capital punishment and believes that it is more merciful, and a lawyer, who believes that life imprisonment is the better option, due to its preservation of life. The lawyer believes that any life is better than none, and that life cannot be taken away by the government, since life cannot be given back if the government realizes that it made a mistake.
The banker and the lawyer decide to enter into a bet, with the banker wagering that the lawyer could not withstand 5 years of imprisonment. The lawyer, young and idealistic, decides to up the ante and makes the bet longer: 15 years. If he could last to the end of his sentence, the lawyer would receive two million rubles for wining the bet.
The banker cannot fathom his good fortune, and even offers the young lawyer a way out, saying that he is being hasty and foolish. Nevertheless, the lawyer decides to stick to his word and the bet is carried out.
For fifteen years, the lawyer lives on the banker's property, in a small lodge, and has no human contact. He can have any item that he desires. At first, the lawyer does not comfort himself with any liquor or tobacco, confining himself to playing the piano. But as the years progress, he gives in and spends much of his time drunk or asleep.
Later, the main focus of his time becomes books, as he searches for adventures and comforts that he cannot possess physically. He takes great advantage of the banker's ability to provide any book, and asks that the banker test the result of his reading by firing two shots in the garden if his translations of several languages is indeed flawless. The banker acquiesces and confirms the lawyer's suspicion that he has mastered languages.
T Plot Method Between Two
As the years go by, the lawyer reads virtually every genre under the sun. He makes his way from the lighter reading of the early years, to the dense text of the Gospels and Shakespeare. The banker, by this time, has gone broke due to his own recklessness and gambling. He begins to worry that the lawyer's bet with him will ruin him financially.
The banker begins to hope against all hope that the lawyer will break his vow and lose the bet. He doesn't even feel remorse at his evil thoughts, excusing them on the basis that they are in his own best interest. In fact, the banker even manages to convince himself that the lawyer is getting the better end of the deal, since he will still be relatively young at 40, and, with the 2 million rubles, relatively rich.
With this in mind, the banker goes to investigate how the lawyer is doing. He finds that his prisoner is asleep at his desk, looking much older and careworn than he ever imagined him to be. After observing him for a few seconds, the banker notices a letter on the table.
In it, the lawyer proclaims his intention to renounce earthly goods in favor of the spiritual blessings. The prisoner has become entirely embittered during his captivity. He has developed an intense hatred for other humans and believes that there is nothing that he or they can do to ever reconcile this chasm. To prove his seriousness, the lawyer decides to leave his prison five hours before the appointed time, and renounces his claim to the two million, thereby freeing the banker from his debt and from financial ruin.
T-plot Method Bet
The banker cries and kisses the prisoner with relief. The next day, watchmen alert the banker of the lawyer's escape, and the banker is unsurprised. He walks over, takes the letter from the lodge, and locks it in a fireproof safe.
Analysis
In The Bet, Chekov decides to analyze which is worse: life imprisonment or capital punishment. In order to do this, he sets up a bet that would likely never take place in real life. This is typical of Chekov, who likes to examine philosophical questions (against the backdrop of a simple plot) as they might play out in real life, with real consequences, rather than simply examining them in the abstract.
Through this story, Chekov demonstrates the pitfalls of idealism and the foolishness of youth. Had the lawyer been older and wiser, he would never have decided so impulsively to go through with this bet. Had he had a family, a wife, children–any support structure that depended on him–he would not have agreed. So the bet also demonstrates the selfishness of man and youth. With nothing to lose, and two million to gain, the lawyer cannot think of a reason to reject the bet.
T Plot Method Bet
It is very interesting that Chekov does not show the readers the thoughts of the lawyer as he makes this bet. The only time that we see the thoughts of the lawyer clearly is later in the story, through a letter. We never see the lawyer's thought process wholly unvarnished and unfiltered, as we often see the thoughts of the banker. This allows the lawyer to remain a pure model of idealism, sacrificing years of his life to prove his moral principles, something that most would find hard to stomach in real life. It lends the lawyer a polished, holier aura.
The story also shows the toll that separation from human society can take on a person. Whereas at first the lawyer was full of virtue, eschewing wine and tobacco, he later gives himself in to his vices, drinking and smoking constantly. He has lost some of his idealism, even as he continues to seek to prove it, and himself, right.
The story is left rather open-ended, with the reader left with a sense that the story hasn't finished. Chekov may have done this on purpose, to prompt the reader into thinking about the consequences of the banker and the lawyer's actions. What ultimately is the fate of the lawyer? Does he live out his days happily? Is the banker able to live remorse-free, feeling no guilt over taking so many years away from a young, bright man? Maybe the old banker realized the vanity and emptiness of his life; we will never know.
![Plot Plot](https://www.researchgate.net/profile/Fernando_Vallejos-Burgos/post/How_can_I_interpret_the_external_surface_area_of_an_activated_carbon_which_is_greater_than_the_BET_surface/attachment/59d63fa2c49f478072ea9c67/AS:273780847382541%401442285849149/image/1.jpg)
The banker does feel some contempt for himself, but the story does not give the reader much more detail than that. It is possible that the banker struggles with his decisions for the rest of his life as he does choose to hold onto the lawyer's last letter, but it is equally possible that he simply forgets about the lawyer in a few years time, locking away all thought of him from his mind.