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La química en la historia, His life and work para la enseñanza Kurt Wohl
Jaime Wisniak*
Resumen After participating in the First World War he Kurt Wohl (1896-1962) fue un brillante científico went to study chemistry at the universities of the Free teórico que trabajó en áreas tan diferentes como la City of Danzig, Heildelberg, and Berlin. He did his fotosíntesis y las llamas. Amargos sucesos antes de Ph.D. studies at the University of Berlin under the la Segunda Guerra Mundial lo obligaron a cambiar direction of Walther Nernst (1864-1941; 1921 Nobel de ocupación varias veces, pero a pesar de ello, él Prize in Chemistry) and graduated in 1923, summa contribuyó en forma duradera al tema de ter- cum laude, with a thesis related to the specific heat modinámica de soluciones y propiedades de trans- and dissociation of diatomic gases. porte de los gases a altas temperaturas y presiones. After graduation he was appointed assistant in the Physikalisch-Chemischen Institut (Physical Chemis- Abstract try Institute) of Nernst, in Berlin, where he began Kurt Wohl (1896-1962) was a brilliant theoretician working with the group of Max Bodenstein (1871- who worked in areas as widely different as photosyn- 1943). At the Institute he continued his research on thesis and flames. Bitter events before World War II the specific heat of gases, their dissociation, explo- forced him to change activities several times, but in sion methods, and equilibrium states. In addition, he spite of this he made lasting contributions to the did work on the London theory of the van der Waals thermodynamics of solutions and transport proper- forces (Wohl, 1931). ties of gases at high temperatures and pressures. Bodenstein recognized the exceptional charac- teristics of Wohl and promptly incorporated him to the editorial board of the journal Zeitschrift Physical chemists and chemical engineers are famil- für Physikalische Chemie (Journal of Physical Chemistry). iar with Wohl through his model for describing the Between 1923 and 1933 Wohl had as doctoral stu- behavior and phase equilibrium of a real solution. dents Günther von Elbe, who would become one of They are generally unaware of his rich contributions the foremost authorities in the area of combustion in other scientific areas such as the application of and flames, and Michel Magat (Wohl and Magat, thermodynamics and reaction chemistry to photo- 1932). synthesis and plant respiration, determination of In January 1933 the National Socialist Party heat capacities and dissociation rates of gases at high assumed the government of Germany and in April temperatures, and the theory of combustion and 7 of the same year regulations were issued to purge flames. Here we describe his personal life and career, the Civil Service, including the universities, of social- his scientific achievements, and, in particular, his con- ists, democrats, and Jews. The first laws forbade the tributions to thermodynamics and plant physiology. employment of Jews in government establishments, except in some circumstances like having participat- Life and career (Jost, 1963; von Elbe, 1963) ing in WWI. Wohl’s life suffered a serious change and Kurt Wohl was born in Berlin, on December 3, 1896, he could continue working only because he had the son of Alfred Wohl, a professor of chemistry indeed participated in the First World War. He first at the University of Danzig, known for his contribu- accepted a temporary arrangement of semi-retire- tions to the theory of equations of state. ment from university and between 1936 and 1938 he could perform only work of a technical value, includ- ing a position at I. G. Farberindustrie. During this * Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel 84105. period he developed an interest in the theory of plant Correo electrónico: [email protected] assimilation and published several papers dealing with the kinetic and energetic aspects of the subject Recibido: 14 de junio de 2002; aceptado: 17 de agosto de 2002. (Gaffron and Wohl, 1936; Wohl, 1937).
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In September 1935 the Nuremberg laws were home in Zehlendorf (a district of Berlin) their musical enacted that prohibited the employment of Jews in evenings were well known. There they would play almost every activity. In the first supplementary de- clarinet, chamber music, and Jewish music, al- cree of Nov. 14, 1935, these laws were made more though Hitler had forbid playing the latter. Wohl specific by defining who was a Jew and by declaring remained interested all his life in art, literature, paint- explicitly that ‘‘a Jew cannot be a citizen of the Reich, ing, and music. After moving to Newark, Wohl’s he cannot exercise the right to vote; he cannot home continued to be a meeting place for musical occupy public office.’’ Other enactments were de- evenings. signed to complete the process of Jewish segregation. Kurt Wohl died on September 3, 1962, in In the beginning of 1939, as a result of these Newark, Delaware, of complications during a heart regulations, Wohl immigrated to England where he operation. stayed a little less than four years. He gave a very successful seminar about photosynthesis at the Bal- Scientific work liol College in Oxford; this seminar led to his ap- Wohl did research in very different fields that in a pointment to the Department of Botany of the uni- way reflect the influence of external events on his versity working with William Owen James. While at life. In Germany he dedicated himself mainly to Balliol he continued the work on the assimilation of thermodynamics (equations of state, osmotic pressu- CO2 that he had initiated in Germany with Gaffron re, heat effects), in England to heat balance of plant (Gaffron and Wohl, 1936). physiological processes, and in the US to thermody- By the end of 1942 Wohl immigrated to the US namics of solutions, and phenomena related to com- where his family had moved before. After some time, bustion. We will now discuss some of Wohl’s most and a stay at Princeton University, he accepted a po- important contributions. sition as professor of chemical engineering at the University of Delaware, Newark. His only son, 1. Energy processes in plants Hellmut, became a professor of art in the Fine Arts Wohl worked on the application of thermodynamics Department at Yale University. concepts to the analysis of energy processes in plants, At Delaware he dedicated himself to thermody- first in Germany and then in England. He analyzed namics and combustion processes. He was also in- in particular the processes of energy capture (photo- terested in gas turbines and jet engines. All these synthesis) and utilization (respiration) (Wohl, 1940; subjects were part of the more general area of trans- Wohl and James, 1942) and his results will be dis- port phenomena, flow processes, and chemical reac- cussed in what follows. tions that interested him. Wohl was also interested in the phenomena of diffusion in flames, laminar flames, and turbulent flames, and flame analysis 1.1 Photosynthesis (Wohl, 1937; 1940) spectroscopy in all wavelengths. His achievements Wohl work in this area is interesting because it re- were reflected in the successful conferences he gave flects what was known (and not known) about pho- during the seven symposia on Combustion and tosynthesis sixty years earlier. Photosynthesis in Flame and Explosion Phenomena that took place green plants means the transformation of carbon from 1948 on. The nineth (and last symposium) took dioxide and water into oxygen and carbohydrate by place in August 1962 at Cornell University, some the action of light. The overall equation for the time before he died. process is Among his achievements we can mention are nCO2 + nH2O → nO2 + (CH2O)n (1) his serving as a member of the Sub-committee on Combustion of the National Advisory Committee for Equation (1) indicates that photosynthesis is a Aeronautics, charter member of The Combustion light-energized oxidation-reduction process. Institute, and working as a consultant for the chemi- At that time it was not known how the synthesis cal and motor industry. In 1953 he was invited to was achieved, or the chemical intermediates and Göttingen University were he lectured on the ther- enzymes involved in the process. The product of modynamics and mechanisms of combustion. the reduction of CO2 had the empirical formula In 1924 he married the pianist Margarete (CH2O)n yet it was not known whether this group left Wocke. Music accompanied their life and at their the reaction site as formaldehyde (CH2O), or in some
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more condensed form (Wohl, 1940). The only infor- reducing centers were necessarily occupied by X4 mation available was the amount of incident light and that at each reducing center one oxygen mo- absorbed by chlorophyll, as well as the amount of the lecule would be formed every τ seconds. Wohl con- latter, and the rates of CO2 consumption and O2 clusions matched the experimental fact that at moderate production. temperatures and low to medium light intensities, the Wohl first analyzed the energy requirements of rate of photosynthesis increased as the intensity in- the process assuming, as was usual, that formaldehy- creased and was independent of the temperature. At de was the primary product of photosynthesis. Since higher light intensity levels the rate became increa-
eq (1) required 130 kcal/mol of CO2 or (O2) and for singly dependent on temperature and less dependent one photo-act the quantum energy available (red on intensity, a feature characteristic of chemical light) was 42.6 kcal, it meant that the reduction reactions.
of CO2 required at least three primary photo-acts. The next question was related to the structure of Thus it was of interest to learn how the rate of the the reducing center (photosynthetic unit). It had reaction increased with the increase in light intensity. already been assumed that each reducing center The available experimental evidence indicated that required about 2500 chlorophyll molecules. Gaffron the curve that described this process started linearly, and Wohl (Gaffron and Wohl, 1936) calculated that meaning that the energy of a quantum absorbed by a single chlorophyll molecule in a dimly illuminated
chlorophyll and transmitted to the CO2 molecule had plant would absorb a light quantum only once in to be stored until the next quantum arrived (Wohl, several minutes. At this rate the molecule would 1935). Now, according to the second law of ther- require nearly an hour to capture the quanta neces-
modynamics, the light energy necessary to imple- sary to produce one molecule of O2. When the plant ment the process should be larger that the heat of was fully illuminated the maximum rates of CO2 reaction, even if no photo-act was lost. It should be uptake and O2 evolution were fully established. So also true because the return to the initial state could Gaffron and Wohl postulated that the energy harvest only be inhibited by additional amounts of energy by a large set of chlorophyll molecules was conduc- (Gaffron and Wohl, 1936). ted at a single reaction center. Hence, four quanta of red light, and not three, This, and other concepts about the structure of as assumed by others, were the theoretical minimum the photosynthetic unit have been revised and by which photosynthesis could be achieved. Another improved many times during the last 50 years. important consequence was that the fourth photo The current explanation is that the photosynthetic product was produced in an irreversible way. unit is composed of three complexes, as explained
The next question was how does the CO2 mole- below. cule store up its four quanta? The only possibility was What do we know today about photosynthesis?
for the CO2 molecule to be firmly attached to a First, that the process takes place in the chloroplasts reducing center, which had to be energetically cou- of green plants and that all the light-harvesting chlo- pled to the source of quanta, that is, to chlorophyll. rophyll and the electron transport pathways are lo- The proposed mechanism was, then cated in the thylakoid membranes of the chloroplast. Second, the actual reactions occur in two stages: the O , (CH O) 2 2 light stage, consisting of photochemical reactions, and X0 → X1 → X3 → X4 H2O, CO2 (2) the dark stage, comprising chemical reactions control- led by enzymes. During the first stage the energy of
that is, the CO2 attached to the reducing center Xo, light is absorbed and used to drive a series of electron was transformed by four quanta taken from the transfers, resulting in the synthesis of ATP and
chlorophyll into three intermediate products, X1, X2, NADPH. During the dark stage NADPH and the
and X3, and finally into the photo-product X4. energy-loaded ATP are used to reduce CO2 to orga- Wohl proceeded then to analyze mathematically nic compounds. the consequences of the kinetics of reaction (2) by In the thylakoid membrane there are three com- comparing the mean time of chemical reaction (τ) plexes named photo system II (PSII), cytochrome bf with the mean time interval between the arrival of complex (Cyt bf ), and photo system I (PSI), respec- two light quanta at the reducing center (Wohl, 1937, tively. PSII and Cyt bf are connected by the electron p. 105-121). His conclusions were that nearly all the carrier plastoquinone, and Cyt bf and PSI are con-
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nected by plastocyanin. The whole arrangement is accompanied by respiration, it was of interest to find called Z arrangement. A photo system is then, a com- how did the plant used the decrease in G. plex of light-absorbing chlorophylls, a reaction-cen- To keep respiration going there must be a con- ter chlorophyll, and an electron transport system. tinuous supply of energy, which plants normally The chlorophyll molecules feeding excitation energy obtain from the sun. We have discussed before how to the centers are called antenna chlorophyll. In the plants transform the radiant energy of sunlight into case of PSII the reaction center chlorophyll is called chemical energy by converting carbon dioxide and P680 because it absorbs light up to that wavelength. water into certain activated intermediates that spon- PSI is called P700 for the equivalent reason. taneously continue their chemical way to oxygen The energy efficiency of photosynthesis is the and sugars. If we write now ratio of energy stored to the energy absorbed. The amount of energy stored can only be estimated ∆G = ∆H − T∆S (3) because of the myriad of products formed. If eq (1) we see that part of the available energy in sunlight is used to approximate the actual storage process utilized by the plant is dissipated as heat (∆H ) and then the production of one mole of oxygen and 1/6 part is absorbed as information (T∆S ) in building mole of glucose (formaldehyde) results in the storage complex molecules. The plant uses the heat released of about 117 kcal, at room temperature. Since plants to evaporate water (transpiration) from the cell sur- use that part of the solar spectrum that has an esti- faces of the mesophyll. The transpiration process is mated mean wave length of 570 nm, then the energy always accompanied by useful mechanical work absorbed is about 50 kcal/einstein. To calculate the (such as lifting the water from the soil to leaf level) amount of light energy involved in photosynthesis because the transpiration stream carries the water we need to know the number of einsteins absorbed and the nutrients needed by the leaves. Anyhow, this per mole of oxygen formed. This is called the quan- mechanical work is very small, for example, raising . The minimum quantum require- tum requirement one kilogram of water to the top of a 60-m high tree ment is about nine. Therefore, the maximum energy requires 0.588 kJ, which is 0.024% of the heat re- efficiency of photosynthesis is about 117/450, that is, quired to evaporate the same amount of water at about 26%. 25°C (2442 kJ/kg). An interesting result is that the actual percentage Plants utilize energy in two stages: (a) during of solar energy stored by plants is as much as 1% of growth, building up all the necessary organs and the total solar energy received. Also, that respiration structure and (b), on maintenance of the mature by living organisms and combustion of carbon fuels plant. During the active growth phase part of the consumes on the average about 10,000 tons of O2 sugar is oxidized to carbon dioxide and part is in- every second on the surface of the earth. At this rate, vested in building permanent plant constituents such all the O2 in the atmosphere would have been used up in about 3000 years (!) (Hall and Rao, 1994). as cellulose and proteins, or is stored as starch and fat. Fortunately, this loss is counterbalanced by the pro- When the plant has achieved his final structure, duction of carbohydrates and oxygen during photo- it may be considered as a thermodynamic system in synthesis. More than that, as stated by Wohl, long equilibrium with its environment and suffering inevi- ago, the rate of photosynthesis in the green parts of tably spontaneous changes with a changing environ- plants is about 30 times as much as the rate of respi- ment. The organism maintains itself in a steady state ration in the same tissues. by a continuous supply of energy by photosynthesis and the continuous breakdown of carbohydrate. The 1.2 Respiration (Wohl and James, 1942) important question is determining what fraction of In this second project on plant physiology, Wohl the energy is actually utilized by the organism and analyzed how plants utilized the energy that was what fraction is wasted. captured by photosynthesis, and particularly, the According to Wohl the experimental evidence respiration process. His main argument was that respi- indicated that all the energy decrease associated with ration in plants and animals was accompanied by respiration appeared as heat. The decrease in total transformations of energy. Respiration, like any other energy was given by the decrease in the heat com- process that went on spontaneously, involved a decrease bustion of the tissue during the period considered, in the Gibbs function (G), and since plant life was always and this was a measurable quantity.
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During growth, sugars were converted into sub- This reaction was not spontaneous, but since the stances of greater energy content, the additional oxygen released could be used to oxidize carbohy- energy was supplied by the oxidation to CO2 of part drate of the available sugar. If we consider the complete 1 oxidation of sugar or starch to CO2 as the only 2O + C H O → 2CO + 2H O 2 3 6 12 6 2 2 reaction taking place, then the decrease in the heat of combustion of the plant and its substrate, (A), ∆G = −230 kcal (5) would be equal to the heat of formation of the CO2 released, (B). In this case, the percentage of the then, the available energy released by eq (5) was more than enough to drive eq (4). Coupling of both respiratory energy used would be 100 (A − B) ⁄ B. Unfortunately, to determine the actual efficiency of equations allowed realizing the over all reaction respiration in maintaining the steady state required 1 NO− + 2H+ + C H O → NH+ + 2CO + H O knowing the details of many processes like the rate 3 3 6 12 6 4 2 2 of spontaneous breakdown of all substances other than respiratory substrates, how much sugar was ∆G = −165 kcal (6) consumed in making a unit replacement, etc. Present The energetic linkage of eqs (4) and (5) was knowledge of the metabolic processes was far from (100)(64/230) = 28% efficient. enough for making a reasonable estimation. It could then be concluded that by breaking the The most obvious energy wastage was found metabolic process into a series of driving and driven among primitive and especially among anaerobic chemical reactions the plant was able to utilize the saprophytes, For example, yeast under nitrogen available energy of respiration for all its needs. would continue to release energy from sugar by According to Wohl, this reasoning could be ex- vigorous fermentation even when growth was at a tended to many other vital processes such as organi- standstill and energy utilization was virtually nil. The zation, salt accumulation, cell division, and proto- action seemed to be the result of uncontrolled en- plasmic rotation. Only a suitable reaction sequence zyme activity on susceptible materials, and was not would allow them to use the decrease in available related to the metabolic needs of the organism. Start- energy caused by respiration. Wohl illustrated this ing from such a condition evolution could be envis- conclusion pointing out that although plants may aged as a progressive control of expenditure of ma- exhibit a large energy turnover of anaerobic respira- terial and energy. tion, the available energy released was essentially Metabolic reactions may be spontaneous or non lost because the lack of a suitable chemical sequence spontaneous. Realization of the latter will require to utilize it. assistance from another reaction or from another Wohl’s example of anaerobic respiration is an source of energy We may then speak of driving reac- excellent illustration of the difference between the tions (spontaneous) and driven reactions (non sponta- First law of thermodynamics (quantity of energy) and neous). the Second law (quality of the energy). The fact that Wohl stated that respiration processes were an a certain amount of energy is released is not a excellent example of this situation. They occurred requirement that it must be partly converted into through numerous chemical steps by which a variety work. In the example in question all the energy of intermediates and by products was formed. These released is converted into lost work. afforded as many opportunities for the coupling of At the time of Wohl’s review of driving and respiration to synthetic steps. The overall reaction driven equations it was not known that the normal could be considered as a complex package of driving strategy used by living organisms is to couple a and driven equations. In the particular publication process that releases available energy (downhill) (Wohl and James, 1942) Wohl and James gave several with another that requires energy (uphill). The strat- examples of this coupling. One of them was the egy is very general and includes not only chemical oxidation of nitrate into ammonia with the help of reactions (as illustrated by Wohl), but also simple water only mass transfer of compounds and ions. For example, − + + in the intestinal lumen, after a meal, there is a high NO3 + 2H + H2O → NH4 + 2O2 ∆G = 64 kcal (4) level of Na_ and a low level of glucose compared to
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the level of the same in the epithelial cells. The First, let us relate the value of the constants a, b, glucose-Na symport protein brings the glucose to the c, to the critical properties, by using eq (7) together lumen using the energy released by the downhill with the conditions for an inflection at the critical + 2 2 export of Na . Another example is the cell of the point, (∂P ⁄ ∂v) = 0 and (∂ P ⁄ ∂v )Tc. The result is Tc bacteria Escherichia coli; here an H_symport pro- v tein coupled to a proton pump uses the same princi- b = c ple to export H+ and import lactose. 4 (11) 2 a = 6vc Pc (12) 2. Thermodynamics 3 R 3 8 c = 0.075852 2 Tc = ab (13) 2.1 Wohl’s equation of state for gases Pc 3 In 1914 Wohl’s father proposed the following equa- From z = P v / T we get z = 0.267. (14) tion of state for gases (Wohl, 1916) that did not gain c c c c c much popularity In order to study the behavior of the second virial coefficient we transform eq (7) into its virial af (T) cf (T) 1 2 form, as follows p + − 3 v − b = RT v(v − b) v (7) Pv v α v β 1 In eq (7) a, b, and c are the parameters of the = − + 2 RT v − b RT v(v − b) RT v (15) equation, and f1(T) and f2 (T) functions of the tem- perature defined as where α = af1(T) and β = cf2(T). Equation (15) may be written T f (T) = c 1 T (8) Pv 1 α 1 β 1 = b − b + 2 2 RT 1 − ⁄v RT v (1 − ⁄v) RT v Tc f (T) = 2 T (9) Taking advantage that b/v is a very small number we have This equation presented a series of interesting Pv b b2 b3 α 1 b b2 features, first it transformed the van der Waals pa- = 1 + + + + … − 1 + + + … RT v v2 v3 RT v v v2 rameters a and b, from being constant, into functions of the temperature, and second, it predicted a con- β 1 + 2 stant value of 0.267 for the critical compressibility zc, RT v zc = Pcvc / RTc). The equation was not considered very and rearranging according to powers of the volume successful because it did not predict the second virial coefficient properly and hence, it was not very accu- Pv 1 α 1 2 αb β = 1 + b − + 2 b − + + … rate for high pressures. RT v RT v RT RT (16) The fact that zc = 0.267 is important for the From here we get that the second and third virial following reasons: (a) The experimental values of zc coefficients are vary between 0.200 and 0.300, with about 60% of the data in the region 0.26 to 0.28, (b) all modern equa- af1(T ) B = b − tions of state like Redlich-Kwong, Soave, and Peng- RT (17) Robinson predict a value of zc larger than 0.300, way abf1(T ) cf2(T ) out the experimental range (Walas, 1985). C = b2 − + Kurt Wohl analyzed the reasons for the failure RT RT (18) of eq (7) (Wohl, 1928) and concluded that its perform- Since the second virial coefficient becomes zero ance could be improved substantially by simple at Boyle’s temperature (TB), from eqs (11-13) and (17) changing the expression of f2(T) from than given by we get eq (9) to aTc 4 ⁄ 3 TB = √ Tc Rb (19) f (T) = 2 T (10) Wohl compared the behavior predicted by eq (17)
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with the experimental one for a large set of sub- problem consisted now in expressing the deviations stances that exhibited a quadrupole (such as CO2) caused by solution, it was necessary that in each and others having a high dipole moment (such as cluster at least one molecule be different from the
CHCl3, HCl, N2O, ethyl ether, and SO2). The results rest. The resulting expression was (Wohl, 1946) indicated that the equation held very well for normal ∆gE 1 substances in the entire gas domain, with an error = z z a + z z z a RT ∑ i j ij ∑ i j k ijk + … (20) of 1-2% for the pressure. The second virial coeffi- ∑qixi ij ijk cient calculated using eq (17) agreed very well with the calculated one in the range 0.8Tc to 2.5 Tc. where qi represents the effective molar volume of com- ponent i and zi its volumetric fraction , given by 2.2 The Wohl model for solution behavior qx z = i i The estimation of the thermodynamics properties of i n (21) a real fluid or fluid mixture without experimental qx data is a very complicated problem involving de- ∑ i i i = 1 tailed spectroscopic, structural and interaction po- tential data, and the use of mechanical statistics. The The effective volume was considered to be the behavior of liquid solutions cannot be adequately volume of influence of the molecule and hence its expressed in terms of an equation of state and for this actual value would vary from solution to solution, reason it is common practice to describe the devia- depending on the type of interactions present. tion from ideal solution behavior by means of excess Since actual molecules do not touch each other the functions, particularly that of the Gibbs function, GE. minimum value of the effective volume will clearly No general theory exists that adequately describes be somewhat larger than the actual molar volume of the composition dependence of liquid solution prop- the molecule. erties; many theoretical approaches have been tried Parameter aij represents the interaction between but they have met very little success. molecules i and j, parameter aijk the interaction be- For these reasons, the preferred approach is to tween molecules i, j, and k, and so on. Clearly then develop empirical correlations. In general GE/RT is the order of the indexes is not important. The first a function of T, P, and the composition, but for sum represents the total of interactions in clusters liquids at low to moderate pressures it is a very weak formed by two molecules, the second the total of function of P. Therefore the pressure dependence of interactions present in clusters constituted by three activity coefficients is usually neglected. molecules and so on (notice here the similarity with The many models available today are charac- the meaning of the virial coefficients). Each addi- terized by expressing the deviation of the liquid tional sum after the first one considers the interac- phase alone. They contain no information whatso- tions not taken into account by the previous sum. In ever about the vapor phase. It is desirable that the addition, each sum is preceded by an appropriate GE/RT model be of sufficient flexibility to represent expression of the volumetric fraction to assure that E E the various types of behavior of the function G (x1, the value of ∆G /RT be zero for x = 1 and x = 0. Since x2, xn, T). Preferable the model should have a sound we are describing the excess property, in every physicochemical basis so that the numerical val- expression at least one of the z must carry an index ues of the parameters in the expression be suscepti- different from the rest. ble to correlation and estimation. No known model To illustrate the meaning of the different con- for GE meets all the requirements; the choice of an tributions, let us consider the expansion of the term expression for E is usually made on an ad hoc basis. G ∑ zi zj aij for a binary and ternary solution. For a Wohl selected to express the excess value of the ij molar Gibbs function, ∆gE/RT, on similar arguments binary solution we have
as those used by Heike Kamerlingh Onnes (1853- 2 1926; 1913 Nobel Prize in Physics) to develop the z z a = z z a + z z a = 2z z a (22) virial equation of state for gases. The many-body ∑ i j ij 1 2 12 2 1 21 1 2 12 1 problem was assumed to be the sum of the interac- tions of clusters of increasing complexity: two body and for ternary one clusters, three-body clusters, and so on. Since the
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3
∑zi zj aij = zi z2 a12 + z1 z3 a13 + z2 z1 a21 + z2 z3 a23 + 1
z3 z1 z31 + z3 z2 a32
3
∑zi zj aij = 2zi z2 a12 + 2z1 z3 a13 + 2z2 z3 a23 1 (23) For practical reasons the Wohl expansion is usu- ally limited to the third or fourth order in z. Consider now the expansion of order three for a binary solution. We have ∆gE = q x + q x 2z z a + 3z2 z a + 3z z2 a RT 1 1 2 2 1 2 12 1 2 112 1 2 122 (24) Figure 1. Activity coefficients as a function of concentration.
Using the fact that for a binary solution z1 + z2 = 1 the definition of A and B, stated by eqs (25) and (26). and z1/z2 = q1/q2, and defining the two new constants In addition to interaction between molecules (mi-
A = q12a12 + 3a112 (25) croscopic interpretation) the constants represent the limiting values of the activity coefficients. B = q 2a 3a (26) 2 12 122 To illustrate this point consider the shape of the we can transform eq (24), after some algebra, to curves γi(x) for solutions that have a positive devia- tion form ideality given in Figure 1 (more than 90% E ∆g 2 2 of the known binary systems exhibit this kind of = Ax z + Bx z RT 1 2 2 1 (27) behavior). The shape of these curves will be re- stricted by the condition that the value of the activity We now multiply eq (27) by the total number of coefficient of a component will change from A (or B) moles n and then differentiate with respect to ni, to at infinite dilution, to unity as the component be- obtain the activity coefficient of component I comes more and more concentrated. In other words, (Prigogine and Defay, 1954) parameter q1 ⁄ q2 represents the ‘‘length’’ of the γi(x) ∂ ∆GE curves, that is, for the same values of parameters A = lnγ i and , we can define a whole family of curves that ∂ni RT B n,T ∞ j start from γi and end at γi = 1. Each of these curves
The final result, after some algebra, is will correspond to a different value of q1 ⁄ q2. Parame- ter q1/q2 represents then, the flexibility of the model. 2 q2 ln γi = z2 A + 2z1B − A q1 (28) Calculation of the Wohl parameters for binary systems q1 ln γ = z2 B + 2z A − B Let us assume now that vapor liquid equilibria data 2 1 2 q 2 (29) at constant temperature or constant pressure are Equations (28) and (29) represent the Wohl ex- available for a given system. How can these be used pressions of the third order for the activity coefficients to obtain the pertinent values of the activity coeffi- in a binary system and they show that these are cient? Unfortunately Wohl left this question unan- function of the three parameters A, B, and q1 ⁄ q2. swered for two very simple reasons: computers were For the case of infinite dilution (∞) we obtain not available at his time and two of the three parame- ters of his model appear multiplying each other (see ∞ A = ln γ1 (30) eqs 28 and 29). ∞ Today these two facts are not a problem and B = ln γ2 (31) several procedures are available for answering the which allow us to give a macroscopic interpretation to question.
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2 Table 1. Wohl parameters for typical systems (Apelblat and Wisniak, 1989). ∗ ∗ A x1 x1 q1 q1 A B q1 = gmax + 2gmax ∗ − gmax ∗ + gmax RT x2 x2 q2 q2 (32) q2 2 benzene + heptane 0.4881 0.6587 1.12 ∗ ∗ B x1 x2 q2 q2 = gmax + 2gmax − gmax + gmax butanol + chlorobenzene 0.6197 0.8260 2.47 RT ∗ ∗ q q x2 x1 1 1 (33) 1,2-dichloroethane + pentanol 0.6250 1.6003 0.25 E where gmax is the extreme value of g /RT (Wisniak and diethylamine + methanol --1.9507 --0.6104 0.89 Segura, 1995). ethyl acetate + propanol 0.7957 0.5705 0.40 2. Optimization hexane + benzene 0.6946 0.4724 1.00 The Wohl parameters may be determined using methyl acetate + 2-propanol 0.8328 0.7627 0.67 standard optimization techniques such as Simplex, methyl ethyl ketone + 2-propanol 0.4474 0.4500 0.29 maximum likelihood, etc., together with an objective function. In should be understood that the results 2,2,4-trimethylpentane + toluene 0.4120 0.0087 0.41 produced by this method (mathematical roots) do not necessarily have a valid physical meaning con- sistent with the theory behind the model. What we are doing is forcing the parameters to comply with a 1. Infinite dilution certain mathematical restriction and that is all. In Equations (28) and (29) can be used to determine other words, the model may not have ‘‘physical parameters A and B by extrapolation of the γi(x) roots’’ but it will be possible to find ‘‘mathemati- curves to zero concentration, as illustrated in Figure 1. cal roots’’ that will give a very good fit of the data. It Although this procedure is in principle correct, in is then up to the user to decide the kind of solution practice it does not work properly because the ana- he desires. lytical methods available today are not accurate Table 1 gives values of the Wohl parameters for enough to justify the results of the extrapolation. The some representative systems, which were obtained analytical methods become less and less accurate as using the method proposed by Wisniak and Apelblat dilution is increased so that the corresponding values (1989). of the activity coefficient carry a larger error than the We have indicated before that parameter repre- activity coefficients calculated at a higher concentra- sents the flexibility of the Wohl model. By making tion. Possible future improvement of the analytical certain assumptions regarding its value the Wohl techniques may make this procedure viable. model may be transformed into other known mod- els, as indicated in Table 2. With this assumption the 2. Maximum value of gE number of parameters is reduced to two, namely A An additional method for calculating the , , pa- A B and B. rameters is to take advantage of the fact that the L In table 2 vi represents the molar liquid volume. curve gE(x) always has an extreme internal value at
x* and that at that particular concentration γ1 = γ2. In E 2.3 Liquid-liquid equilibrium and the Wohl model other words, besides eq (27) we have d(g /RT)/dx1 How general is Wohl’s equation for representing = 0. These two conditions lead to the following phase equilibrium? Wisniak and Segura have shown relations (Wisniak and Segura, 1995) that eq (27) is also capa- ble of representing liquid-liquid equilibrium and Table 2. Special cases of the Wohl model. have determined the range of values of the parame- ters for this situation. Their results have indicated q1 / q2 z(qi) Model that systems that are highly immiscible correspond L L L v1 ⁄ v2 z(vi ) Scatchard-Hamer to small values of the ratio q1 ⁄ q2 . In other words, we see that the latter ratio should be considered a highly 1 Margules zi = xi flexible parameter that allows representation of many cases of vapor-liquid equilibrium and many A/B zi = zi Van Laar immiscible systems.
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2.4 Flames and combustion processes In a following work (Wohl and Kadow, 1925) (von Elbe, 1963) reported that improved methods of calculation The present state of knowledge about combustion yielded that the specific heats at constant volume, of processes makes it difficult to recall the rudimentary chlorine and HCl, were better represented by the state of combustion science in the early 1920’s when equations 4.963 + (ECl -- 93)/(T -- 291) for chlorine (βν Wohl entered this field of research as a Ph.D. candi- = 902), and 4.963 + EHCl / (T -- 291) for HCl (βν = date. Some basic phenomena such as detonation 3420). waves and combustion waves had already been de- Wohl’s results were a breakthrough in the knowl- scribed but little was known about its possible impact edge of explosions and removed existing doubts and on the growing technology of fuel utilization. Not misconceptions of the thermochemical approach to only that, there was a widespread disbelief that it the determination of flame gas parameters. His re- might be practically impossible to use thermody- search in the field gave impetus to the accumulation namics and thermochemical methods to determine of the large body of data and computational proce- combustion gas parameters. Nernst, then director of dures, which eventually would make engine and the Physics Institute of the University of Berlin, had rocket technology possible. initiated an experimental study of the subject of After moving to the University of Delaware determining explosion pressures under optimized Wohl renewed his interest in combustion phenom- conditions, i.e., a spherical vessel with central igni- ena and studied a wide range of flame stability tion. The basic equipment had been built by Mathias conditions; the combined flow and mixing problems Pier (1882-1965). Wohl choose hydrogen and chlo- in laminar and turbulent jets of burning fuel gas; and rine as his test mixture and after some brilliant the gradient structure of combustion waves. At Dela- theoretical and experimental work he was able to ware, he determined a vast amount of experimental demonstrate an excellent correlation between the- data on the transport properties of gases at tempera- ory and experiment, based on the calculation of tures in the range 1000 to 2000 K, which were dissociation equilibria (Wohl, 1924). The dissociation important for testing the rigorous kinetic theory of energy of chlorine at 0 K, then unknown, was found dilute gases, without the occurrence of disturbing to be 57 ± 2 kcal⋅mol--1, as compared to the now effects such as dissociation, excitation, or ionization. accepted value of 57.22 kcal⋅mol--1; for hydrogen the They also became important for the analysis of prob- dissociation energy was found to be about lems in gaseous reactions. 100 kcal⋅mol--1, as compared to the modern value of As stated by von Elbe, his former Ph.D. student, 103.2 kcal⋅mol--1. The heat of dissociation of chlorine Wohl’s contributions to the science of combustion as a function of the temperature, at constant pressure, are lasting and destined to enter into textbooks and --1 --3 was given by Q (kcal⋅mol ) = 57+ 2.978 × 10 T - ET handbooks for generations of scientists and engi- where ET is the energy of atomic vibration in the neers (von Elbe, 1963). chlorine molecule, calculated as an Einstein function with 2.2 degrees of freedom and βν = 902. The Conclusions degree of dissociation of chlorine varied from Wohl was a pioneer not only in combustion chemis- 0.051% at 800 K to 86% at 2300 K, and for hydrogen try but also in many other fields. His broad interest, from 0.16% at 1700 K to 3% at 2300 K. coupled to political events that took place in pre-war Wohl also measured the mean value of the specific Germany led him to other areas such as thermodynamic heat at constant volume of chlorine, hydrogen, and of solutions and applications of thermodynamics to HCl and found that for chlorine it was 6.83 cal⋅mol--1 processes taking place in green plants. Many of his (±1.3%) between 291 and 1335 K, and that up to 2000 K results continue to be actual and useful and have set −1 it was given by 4.963 + (ET -- E291)/(T -- 291) cal⋅mol , the way to the development of important areas fuel where again ET was the energy of atomic vibration for rockets and combustion engines. in the chlorine molecule, calculated as an Einstein function with 2.2 degrees of freedom and βν = 902. Acknowledgment For hydrogen the specific heat at constant volume The help of Mrs. Karine Barker, Librarian, Docu- --1 was 4.963 + (ET -- E291)/(T -- 291) cal⋅mol , calculated ment Supply Department, Radcliffe Science Library, as an Einstein function with 2 degrees of freedom Oxford, in providing copies of hard-to-get docu- and βν = 3420. ments is gratefully acknowledged.
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References Wohl, K., An Empirical detcrmination of the Potential of the van Apelblat, A., Wisniak, J., A Simple Method for Evaluating the der Waals Forces ir1 tlie Vicitiity of a Molccule, Z. Physik. Wilson Chnstants, Fluid Phase EquiL;SO, 1-13 (1993). Chem., B14, 31i-65 (1!)31). Galfron, H:; Wohl, K., The Theory of Assimilation, Nalurwissen, Wohl, K., The Ener IBEROAMERICANA LAKRDADNOS HARA LIBRES -- Es un grupo constituido por profesores del Departamento de Ing. y Ciencias Químicas, dedicados desde 1990 a promover en México el uso de técnicas de laboratorio en microescala. nNormalmente nuestros talleres duran treinta horas y se imparten a lo largo de 4 días, excepto donde se indique. Podemos ofrecer talleres especiales desde 4 hasta 50 nhoras, tanto en el CMQM como en otras instituciones. Favor de contactar al Coordinador del Centro. En 2003 cada taller cuesta $ 1,800 pesos por persona (3o r% personas de una misma institución, 20% de descuento; 5 o más, 25 % dcto.).Descuentos especiales a Asistentes Frecuentes (1 0% por taller).Los dctos. Son aditivos hasta un 50%.Cada participante tiene derecho a adquirir un equipo básico de microescaia a precio de descuento (sujeto a disponibilidad). Habrá algunas becas para los talleres de Q. Verde, Q. Orgánica y Q. Nivel Medio. patrocinadas por Monsanto, Poiioies y Provitec. Favor de solicitarlascon anticipación.Favor de inscribirse por lo menos con dos sernanasde anticipación. I 1 2 profesores de la UIAe invitados especiales. Algunas Maravillas de la Química Orgánica Junio 10-13 VI Química Verde* Mtro. Arturo Fregoco o Dr. Jorge ibáñez Cornejo Fotoquímica** Julio+Julio 8-1114-16 Universidad Iberoamericana, A.C. Centro Mexicano de Química en Microescala Química Ambiental +Julio 16-18 Depto. de ing. y Ciencias Químicas Química para el Nivel Medio Agosto 12-15 Proi. Paseo de la Reforma # 880, 01 21 0 México, D.F. Química Analítica*** Octubre 14-17 Tels (55)5950-4074.5950 71 31,5950 41 76 'ImpartidoporelDr Kenneth Ooxsee. U.ofOregon(eningles) y MtraCarmenDoria, UIA '7impartido por el Dr. Michaei Tausch, Gerhard Mercator Un~v.,Alemania (en ingles). Fax (55)5950 4279.5950 4063 t Son dos talleres contiguos Fotaquimica, del 14 al 16 de Julio, y Quimica Ambiental (MC E-mail [email protected] [email protected] Margarita Hernandez, UIA) del 16 al 18 (20 h cada uno). Se pueden tomar sólos o juntos "'Impartido por el Dr. Alejandro Baeza de la UNAM y otro profesor por designar. 46 Educación @dmica 14[1]