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Reply to comments on “the thermodynamics of cluster formation in nucleation theory”
Surface Science 104 (1981) L217-L221 North-Holland Publishing Company
SURFACE SCIENCE LETTERS REPLY TO COMMENTS ON “THE THERMODYNAMICS FORMATION
IN NUCLEATION
OF CLUSTER
THEORY”
M. BLANDER
and
Received 6 October 1980
It is grat~fy~g that our 1972 paper [I ] will get additions exposure as a result of this exchange of comments since we think it is an important paper which disproves the so called Lothe-Pound (L-P) hypothesis [2]. It is unfortunate that the editor of this journal has allowed us only one month to write a reply which can be published simultaneously with the comments of Nishioka and Russell (N-R) [3] ; this time limitation makes it difficult to present our major points as clearly as possible. In brief, we believe that N-R (as well as L-P) have misunderstood Gibbs and misunderstood the concept of a thermodynamic state. In our 1972 paper [l] we derive the following equation thermodynamically Ac=-(n-l)RTlnS+[~~-~1’:--(n-I)~,],
(1)
where n is the number of molecules in an n-mer, S is a supersaturation ratio given by ~i/pr~ where pi is the partial pressure of monomer, pie is the partial pressure of monomer at equilibrium with liquid, /J,” and py are the chemical potentials of n-mer and monomer in their standard states, pe is the chemical potential of bulk liquid and where the quantity Ac is defined by the equation &&J I ) = (N,INI I= ev-4’R
0,
(2)
where pn and Iv;2 are the pressure and number density of n-mer, respectively, and pi and Ni are the pressure and number density of monomer, respectively. This equation is valid for an ideal gas and can be readily generalized for real gases. Nevertheless, we shall restrict ourselves to ideal gases since the language is then simpler and the important ideas can be more clearly presented. Since the ratios p,/p, and Nn/
L218
hf. Blander, J.L. Katz / Thermodynamics
of cluster formation
Nr are generally chosen as unitless, AC must be independent of the choice of units or of whether we choose pressure or density in this equation. The only requirement is that the units be the same for n-mer and monomer. In nucleation theory, the rate of formation of nuclei is proportional to N,. In thermodynamic calculations, the units of measure of pressure or density are fixed by the choice of a standard state and can be chosen arbitrarily if one wishes. Thus, the chemical potential of the ith vapor species is given by pi = /.I: + R T ln(&/p’)
(or /_Li= pp + RT ln(Ni/N”)),
(3)
where p” (or Ne) is the pressure (or density) of the standard state and is usually chosen as the unit for expressing pi (or Nj). This is normally not specifically stated in common texts. Thus, if p” is 1 atm then up is seen to be the chemical potential of the species i at 1 atm and the units of pi are atmospheres; if p” is taken to be 1 Torr (i.e., l/760 atm) the standard chemical potential is lowered by RTln 760. In our eq. (1) (i.e., eq. (9) in our 1972 paper and eq. (3) in N-R) if one changes units (and changes the standard states), pi and py change equally, and, as is required thermodynamically, this produces no change in AG. N-R’s subsequent eq. (3b) (and L-P’s equation) does not have this property. It is valid for only one choice of unit, i.e., pi E 1 unit of pressure, because N-R have manipulated our equation by defining a standard state asp, =p” so that one can write for the chemical potential of the liquid ,LL~-p~=RTln(p,,/p,)=-RTlnS.
(4)
They have thus set pe - py to be numerically equal to -RT In S. This numerical equality can only be valid for their peculiar standard state which is a function of temperature and of kinetic factors. Using eq. (4) to substitute for pz - py in eq. (1) they obtain AG = -nRT In S + [$, - n)_t,],
(5)
where the term in brackets in their equation (our eq. (5)) is larger than our bracketed term in eq. (1) by RT In S. They then claim that this keeps viable the L-P hypothesis [2] that corrections to pz for translational, rotational, and vibrational degrees of freedom must be included in the bracketed term. The L-P procedure leads to very large corrections to classical nucleation theory (e.g., of the order of 10” in the prefactor for water which arises from a term equal to -RT In 10” in the bracketed term). We have shown that these degrees of freedom for the term & ~ yy in our bracketed term lead to a far smaller correction of the order of 105. Thus, if Lothe and Pound are corect, by transferring a small term RT In S (where S is, e.g., 2-100 in nucleation measurements on organics and water) into our bracketed term, the bracketed term decreases by an extremely large quantity of the order of e.g., RTln 1012 and incorrectly increases the prefactor in nucleation theory by a factor of about lOi rather than by a factor of S-’ as it should. This argument can be reinforced in a way which emphasizes our point. Instead
M. Blander, J.L. Katz / Thermodynamics
of cluster formation
L219
of transferring one RT In S into our term in brackets let us transfer J’ such terms out of this term (v-here j < n). By proper substitution for -jRT In S in the brackets from eq. (4) one obtains AC=-nRTlnS+(jt
l)RTlnS+
[pi-(1
+j)&(n-j-
l)p,].
(6)
If one applies the Lothe-Pound hypothesis, then one must subtract, not add corrections for translational, rotational, and vibrational degrees of freedom for j molecules to the bracketed term so that instead of our correction of e.g., -RT In lo5 the “Lothe-Pound correction” will be -(lZj - 5)ln 10 which for j = 2 leads to a change in the prefactor of -lo-” rather than our value of perhaps 10’ (for j= 0) or the original Lothe-Pound value of -10” (for j = -1). Of course, these prefactors would also have to include the usually small term S-(j+‘) which derives from the (j + 1)RT In S term in eq. (6). Obviously, the Lothe-Pound hypothesis, if carried through in this manner, can be used to increase or decrease the prefactor in nucleation theory by very large factors depending on the sign of j. This illogical result proves the incorrectness of the Lothe-Pound procedure by reductio ad absurdum. N-R’s and L-P’s conclusions stem from their misunderstanding of the concept of standard states and of the fact that the choice of standard states merely dictates the units of pressure (or density). If pn and pr are both measured in the same units,p,/ p1 is unitless and the standard state pressure for the n-mer and monomer must be the same. The assumption of the same standard states for n-mer and monomer is a standard thermodynamic procedure which leads to a simpler representation of the thermodynamics. If the standard states for n-mer and monomer are different, pn/pI is no longer unitless and the ratio of the numerical values of p,/pI on the lhs of eq. (2) to the unitless value is correspondingly reflected by a factor on the rhs equal to the ratio of the two values of exp((-pz + py)/RT) for the two sets of standard states which is also equal to the ratio of the two values of exp(-AGIRT). Their odd choice of standard states necessarily leads to the same conclusion as ours but adds a needless complication which impedes understanding. The other and perhaps more important question concerns the relationship between our eq. (1) and the Gibbs equation [4] for the work to make a new (spherical) phase W = - VAp + 4w20 - -nRT
In S + 4&o,
(7)
where the symbols have been defined by N-R (except that Ap is the pressure difference between the interior and exterior of the droplet and not what they state). As pointed out by N-R, the terms nRT In S is merely an approximation which depends on the factors they mention as well as on the incompressibility of the liquid which they ignore. Our term (n - 1)RT In S (usually) differs insignificantly from this approximation for VAp and is at least as valid as nRT In S. From compressibility data on water or oganics it can be readily shown from equations we have discussed [5] that the error generalIy made by ignoring the compressibility to calculate the
L220
M. Blander,
J.L,. Katz / Thermodynamics
of cluster formation
term nR T In S (several percent of the value of nRT In S) is more negative than the difference between our term and nRT In S. In fact, for such fluids our term is closer to Gibbs VAp than is nRT In S. As a consequence, our bracketed term is more closely comparable to the Gibbs term 4&o than is the N-R bracketed term. We must be cautious in equating the continuum droplet of Gibbs (and 4nr20) with the molecular n-mer of nucleation theory. One clue to a problem is most evident in a footnote by Gibbs [6]. His new phase is made by extrusion through an orifice and remains fixed to a membrane. There is no translational degree of freedom. If there were, Gibbs would have had to define the gaseous volume, u, available to a gaseous droplet and the value of W would have to have a term RT In e where R (=1/u) is the number density of the standard state. Since u andfl is not specified and since there is no logarithmic term, Gibbs continuum droplet can not have a translational entropy nor can its properties be related in any way to the number density of a standard state *. In addition, Gibbs continuum droplet is also a sphere. From quantum statistical mechanical considerations one can deduce that a sphere must have a zero rotational entropy since all configurations of a spherical continuum which depend only on rotating it are exactly equivalent and indistinguishable. Thus, any term set equal to Gibbs’ term 4nr2a cannot contain translational or rotational contributions or any dependence on mass, density or a chosen standard state. Since our term in brackets in eq. (1) comes closest to these constraints it is certainly a far more reasonable approximation to Gibb’s term than that suggested by L-P or N-R. There are a number of other criticisms of N-R’s comments. Of these, the only ones which are significant for the points at hand are: (A) Their equation (our eq. (5)) destroys the clean separation we made (in our eq. (1)) between a kinetically controlled term (n - 1)RT In S and our purely thermodynamic bracketed factor. Their bracketed term contains S (in another guise) so that they must equate a term containing a kinetic factor with Gibbs’ purely thermodynamic term 4nr20. This dilemma can be solved if one could choose the dividing surface so that the first term in Gibbs’ equation is exactly equal to -(rr - 1)RT In S and if our approach is used. (B) N-R make a big point of an irrelevant typographical error (of which there are more) we made in one of our equations in ref. [l] (their eq. (6)), where P,, should be PI. Our calculations later in the paper (e.g., eq. (11) in ref. [l]) and all our deductions were made correctly using p ’ for this. Since our calculations are independent of standard state, our conclusions are valid at PI which was 0.075 atm in L-P’s paper. Contrary to N-R’s assertion in their conclusion, our misquote of L-P turns out to be a misprint. * To follow our method of reductio ad absurdurn, if Gibbs’ exterior (vapor) phase is infinite in extent and the droplet were free to move throughout the vapor, W would be minus infinity and not given by eq. (7).
M. Blander, J.L. Katz / Thermodynamics
of cluster formation
L221
(C) Many specific words and statements in N-R’s conclusions (and elsewhere) are incorrect or misleading. Some of these are (1) “. . . but not original . . .“, (2) “ . . . inconsistent with Gibbs . . .“, (3) “. . . is erroneous . .“, (4j “. . . B-K misquote . . .“, (5) “. . disproof or confirmation of their (the L-P) treatment must be based on a statistical mechanical model . . .“. It is unfortunate that such statements were retained since they are based on assertions or illogical arguments. We will not attempt to refute such statements directly since this would serve no useful purpose. Instead, we suggest replacing their conclusions with the following (1) We have again disproven the Lothe-Pound hypothesis, this time by reductio ad absurdurn. (2) One must be extremely careful in equating Gibbs’ W with AC. Our treatment is consistent with Gibbs’, N-R’s and L-P’s are not. In no treatment can Gibbs’ W be proven to be exactly AC but ours is closest. (3) L-P and N-R have obfuscated the issues. If they admit that our equation is exact then they must criticize our equations directly or by conventional transformations, not transform it by an unorthodox and confusing thermodynamic calculation. (4) Most of the discussion by N-R concerning Gibbs is irrevelant in the light of our discussion above. This work wasperformed under the auspices of the Division of Material Sciences, Office of Basic Energy Sciences of the US Department of Energy.
References [l] [2] [3] [4]
M. Blander and J.L. Katz, J. Statist. Phys. 4 (1972) 55. J. Lothe and G.M. Pound, J. Chem. Phys. 36 (1962) 2080. K. Nishioka and K.C. Russel, Surface Sci. 104 (1981) L213. J. Gibbs, The Collected Works of J. Willard Gibbs, Vol. I (Yale University CT, 1948) pp. 252-258. [5] J.L. Katz and M. Blander, J. ColIoid Interface Sci. 42 (1973) 496. [6] Ref. [4], pp. 257-258.
Press, New Haven,
SURFACE SCIENCE LETTERS REPLY TO COMMENTS ON “THE THERMODYNAMICS FORMATION
IN NUCLEATION
OF CLUSTER
THEORY”
M. BLANDER
and
Received 6 October 1980
It is grat~fy~g that our 1972 paper [I ] will get additions exposure as a result of this exchange of comments since we think it is an important paper which disproves the so called Lothe-Pound (L-P) hypothesis [2]. It is unfortunate that the editor of this journal has allowed us only one month to write a reply which can be published simultaneously with the comments of Nishioka and Russell (N-R) [3] ; this time limitation makes it difficult to present our major points as clearly as possible. In brief, we believe that N-R (as well as L-P) have misunderstood Gibbs and misunderstood the concept of a thermodynamic state. In our 1972 paper [l] we derive the following equation thermodynamically Ac=-(n-l)RTlnS+[~~-~1’:--(n-I)~,],
(1)
where n is the number of molecules in an n-mer, S is a supersaturation ratio given by ~i/pr~ where pi is the partial pressure of monomer, pie is the partial pressure of monomer at equilibrium with liquid, /J,” and py are the chemical potentials of n-mer and monomer in their standard states, pe is the chemical potential of bulk liquid and where the quantity Ac is defined by the equation &&J I ) = (N,INI I= ev-4’R
0,
(2)
where pn and Iv;2 are the pressure and number density of n-mer, respectively, and pi and Ni are the pressure and number density of monomer, respectively. This equation is valid for an ideal gas and can be readily generalized for real gases. Nevertheless, we shall restrict ourselves to ideal gases since the language is then simpler and the important ideas can be more clearly presented. Since the ratios p,/p, and Nn/
L218
hf. Blander, J.L. Katz / Thermodynamics
of cluster formation
Nr are generally chosen as unitless, AC must be independent of the choice of units or of whether we choose pressure or density in this equation. The only requirement is that the units be the same for n-mer and monomer. In nucleation theory, the rate of formation of nuclei is proportional to N,. In thermodynamic calculations, the units of measure of pressure or density are fixed by the choice of a standard state and can be chosen arbitrarily if one wishes. Thus, the chemical potential of the ith vapor species is given by pi = /.I: + R T ln(&/p’)
(or /_Li= pp + RT ln(Ni/N”)),
(3)
where p” (or Ne) is the pressure (or density) of the standard state and is usually chosen as the unit for expressing pi (or Nj). This is normally not specifically stated in common texts. Thus, if p” is 1 atm then up is seen to be the chemical potential of the species i at 1 atm and the units of pi are atmospheres; if p” is taken to be 1 Torr (i.e., l/760 atm) the standard chemical potential is lowered by RTln 760. In our eq. (1) (i.e., eq. (9) in our 1972 paper and eq. (3) in N-R) if one changes units (and changes the standard states), pi and py change equally, and, as is required thermodynamically, this produces no change in AG. N-R’s subsequent eq. (3b) (and L-P’s equation) does not have this property. It is valid for only one choice of unit, i.e., pi E 1 unit of pressure, because N-R have manipulated our equation by defining a standard state asp, =p” so that one can write for the chemical potential of the liquid ,LL~-p~=RTln(p,,/p,)=-RTlnS.
(4)
They have thus set pe - py to be numerically equal to -RT In S. This numerical equality can only be valid for their peculiar standard state which is a function of temperature and of kinetic factors. Using eq. (4) to substitute for pz - py in eq. (1) they obtain AG = -nRT In S + [$, - n)_t,],
(5)
where the term in brackets in their equation (our eq. (5)) is larger than our bracketed term in eq. (1) by RT In S. They then claim that this keeps viable the L-P hypothesis [2] that corrections to pz for translational, rotational, and vibrational degrees of freedom must be included in the bracketed term. The L-P procedure leads to very large corrections to classical nucleation theory (e.g., of the order of 10” in the prefactor for water which arises from a term equal to -RT In 10” in the bracketed term). We have shown that these degrees of freedom for the term & ~ yy in our bracketed term lead to a far smaller correction of the order of 105. Thus, if Lothe and Pound are corect, by transferring a small term RT In S (where S is, e.g., 2-100 in nucleation measurements on organics and water) into our bracketed term, the bracketed term decreases by an extremely large quantity of the order of e.g., RTln 1012 and incorrectly increases the prefactor in nucleation theory by a factor of about lOi rather than by a factor of S-’ as it should. This argument can be reinforced in a way which emphasizes our point. Instead
M. Blander, J.L. Katz / Thermodynamics
of cluster formation
L219
of transferring one RT In S into our term in brackets let us transfer J’ such terms out of this term (v-here j < n). By proper substitution for -jRT In S in the brackets from eq. (4) one obtains AC=-nRTlnS+(jt
l)RTlnS+
[pi-(1
+j)&(n-j-
l)p,].
(6)
If one applies the Lothe-Pound hypothesis, then one must subtract, not add corrections for translational, rotational, and vibrational degrees of freedom for j molecules to the bracketed term so that instead of our correction of e.g., -RT In lo5 the “Lothe-Pound correction” will be -(lZj - 5)ln 10 which for j = 2 leads to a change in the prefactor of -lo-” rather than our value of perhaps 10’ (for j= 0) or the original Lothe-Pound value of -10” (for j = -1). Of course, these prefactors would also have to include the usually small term S-(j+‘) which derives from the (j + 1)RT In S term in eq. (6). Obviously, the Lothe-Pound hypothesis, if carried through in this manner, can be used to increase or decrease the prefactor in nucleation theory by very large factors depending on the sign of j. This illogical result proves the incorrectness of the Lothe-Pound procedure by reductio ad absurdum. N-R’s and L-P’s conclusions stem from their misunderstanding of the concept of standard states and of the fact that the choice of standard states merely dictates the units of pressure (or density). If pn and pr are both measured in the same units,p,/ p1 is unitless and the standard state pressure for the n-mer and monomer must be the same. The assumption of the same standard states for n-mer and monomer is a standard thermodynamic procedure which leads to a simpler representation of the thermodynamics. If the standard states for n-mer and monomer are different, pn/pI is no longer unitless and the ratio of the numerical values of p,/pI on the lhs of eq. (2) to the unitless value is correspondingly reflected by a factor on the rhs equal to the ratio of the two values of exp((-pz + py)/RT) for the two sets of standard states which is also equal to the ratio of the two values of exp(-AGIRT). Their odd choice of standard states necessarily leads to the same conclusion as ours but adds a needless complication which impedes understanding. The other and perhaps more important question concerns the relationship between our eq. (1) and the Gibbs equation [4] for the work to make a new (spherical) phase W = - VAp + 4w20 - -nRT
In S + 4&o,
(7)
where the symbols have been defined by N-R (except that Ap is the pressure difference between the interior and exterior of the droplet and not what they state). As pointed out by N-R, the terms nRT In S is merely an approximation which depends on the factors they mention as well as on the incompressibility of the liquid which they ignore. Our term (n - 1)RT In S (usually) differs insignificantly from this approximation for VAp and is at least as valid as nRT In S. From compressibility data on water or oganics it can be readily shown from equations we have discussed [5] that the error generalIy made by ignoring the compressibility to calculate the
L220
M. Blander,
J.L,. Katz / Thermodynamics
of cluster formation
term nR T In S (several percent of the value of nRT In S) is more negative than the difference between our term and nRT In S. In fact, for such fluids our term is closer to Gibbs VAp than is nRT In S. As a consequence, our bracketed term is more closely comparable to the Gibbs term 4&o than is the N-R bracketed term. We must be cautious in equating the continuum droplet of Gibbs (and 4nr20) with the molecular n-mer of nucleation theory. One clue to a problem is most evident in a footnote by Gibbs [6]. His new phase is made by extrusion through an orifice and remains fixed to a membrane. There is no translational degree of freedom. If there were, Gibbs would have had to define the gaseous volume, u, available to a gaseous droplet and the value of W would have to have a term RT In e where R (=1/u) is the number density of the standard state. Since u andfl is not specified and since there is no logarithmic term, Gibbs continuum droplet can not have a translational entropy nor can its properties be related in any way to the number density of a standard state *. In addition, Gibbs continuum droplet is also a sphere. From quantum statistical mechanical considerations one can deduce that a sphere must have a zero rotational entropy since all configurations of a spherical continuum which depend only on rotating it are exactly equivalent and indistinguishable. Thus, any term set equal to Gibbs’ term 4nr2a cannot contain translational or rotational contributions or any dependence on mass, density or a chosen standard state. Since our term in brackets in eq. (1) comes closest to these constraints it is certainly a far more reasonable approximation to Gibb’s term than that suggested by L-P or N-R. There are a number of other criticisms of N-R’s comments. Of these, the only ones which are significant for the points at hand are: (A) Their equation (our eq. (5)) destroys the clean separation we made (in our eq. (1)) between a kinetically controlled term (n - 1)RT In S and our purely thermodynamic bracketed factor. Their bracketed term contains S (in another guise) so that they must equate a term containing a kinetic factor with Gibbs’ purely thermodynamic term 4nr20. This dilemma can be solved if one could choose the dividing surface so that the first term in Gibbs’ equation is exactly equal to -(rr - 1)RT In S and if our approach is used. (B) N-R make a big point of an irrelevant typographical error (of which there are more) we made in one of our equations in ref. [l] (their eq. (6)), where P,, should be PI. Our calculations later in the paper (e.g., eq. (11) in ref. [l]) and all our deductions were made correctly using p ’ for this. Since our calculations are independent of standard state, our conclusions are valid at PI which was 0.075 atm in L-P’s paper. Contrary to N-R’s assertion in their conclusion, our misquote of L-P turns out to be a misprint. * To follow our method of reductio ad absurdurn, if Gibbs’ exterior (vapor) phase is infinite in extent and the droplet were free to move throughout the vapor, W would be minus infinity and not given by eq. (7).
M. Blander, J.L. Katz / Thermodynamics
of cluster formation
L221
(C) Many specific words and statements in N-R’s conclusions (and elsewhere) are incorrect or misleading. Some of these are (1) “. . . but not original . . .“, (2) “ . . . inconsistent with Gibbs . . .“, (3) “. . . is erroneous . .“, (4j “. . . B-K misquote . . .“, (5) “. . disproof or confirmation of their (the L-P) treatment must be based on a statistical mechanical model . . .“. It is unfortunate that such statements were retained since they are based on assertions or illogical arguments. We will not attempt to refute such statements directly since this would serve no useful purpose. Instead, we suggest replacing their conclusions with the following (1) We have again disproven the Lothe-Pound hypothesis, this time by reductio ad absurdurn. (2) One must be extremely careful in equating Gibbs’ W with AC. Our treatment is consistent with Gibbs’, N-R’s and L-P’s are not. In no treatment can Gibbs’ W be proven to be exactly AC but ours is closest. (3) L-P and N-R have obfuscated the issues. If they admit that our equation is exact then they must criticize our equations directly or by conventional transformations, not transform it by an unorthodox and confusing thermodynamic calculation. (4) Most of the discussion by N-R concerning Gibbs is irrevelant in the light of our discussion above. This work wasperformed under the auspices of the Division of Material Sciences, Office of Basic Energy Sciences of the US Department of Energy.
References [l] [2] [3] [4]
M. Blander and J.L. Katz, J. Statist. Phys. 4 (1972) 55. J. Lothe and G.M. Pound, J. Chem. Phys. 36 (1962) 2080. K. Nishioka and K.C. Russel, Surface Sci. 104 (1981) L213. J. Gibbs, The Collected Works of J. Willard Gibbs, Vol. I (Yale University CT, 1948) pp. 252-258. [5] J.L. Katz and M. Blander, J. ColIoid Interface Sci. 42 (1973) 496. [6] Ref. [4], pp. 257-258.
Press, New Haven,