In the Classroom
A Note on Dalton’s Law: Myths, Facts, and Implementation Ronald W. Missen Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 3E5 William R. Smith* Faculty of Science, University of Ontario Institute of Technology, Oshawa, Ontario, Canada L1H 7K4; *
[email protected] Dalton’s law for gas mixtures provides one method for predicting the pressure–volume–temperature (PVT ) behavior of a gas mixture from the PVT behavior of the individual pure gases that comprise it. Many treatments of the law unfortunately promulgate what we term “myths” about it. The genesis of these myths apparently occurs if a treatment originates from, and is restricted to, an ideal gas mixture (IGM), for which the law is exact but unnecessary. Then the prior adoption of the myths makes the transition from an IGM to a nonideal gas mixture (NIGM) misleading and conceptually puzzling. In firstyear general chemistry, the law is typically only discussed in the context of an IGM.
Our purpose here is to separate fact from myth, to enlarge on a treatment (such as that of Dodge) of possible cases for application, and to provide contemporary means of implementation, for example, by use of metacomputing software. In achieving this, we do not attempt to develop any new concept or explanation concerning Dalton’s law.
Myth 1
Statement A In his 1808 book, A New System of Chemical Philosophy, Part I, Dalton wrote (9, pp 191–192):
The name is “Dalton’s law of partial pressures”.
This name is misleading, since partial pressure (pi) has nothing intrinsically to do with the law, as shown below. Over many years to the present time, even such prominent authors as Planck (1), Partington (2), Hatsopoulos and Keenan (3), Guggenheim (4), Laidler and Meiser (5), and Atkins (6), among many others, use this name, although Guggenheim states that “a better name [is] the law of additivity of pressures” (4). Myth 2 The partial pressure of an individual gas in a gas mixture at (T, V ) is the pressure that the gas exerts alone at the same (T, V ).
Most of the authors listed above, and many others, use this definition of partial pressure in the context of Dalton’s law, even though in most cases they introduce the generally accepted definition of pi (see below) at this, or another, point. In contrast to these “myths”, the “facts” about Dalton’s law emerge from an appropriate statement, followed by a clear distinction between partial pressure (pi) and pure component pressure (Pi). Pi is the quantity defined in Myth 2 and pi is defined differently (see below). These lead to operational forms of the law. Implementation of the law includes identification of the various types of problems that it can address and procedures for their solution. This is the plan of treatment in the sections to follow. At the end, recommendations are made about which parts of the material presented here are suitable for inclusion in a general chemistry course, and which are suited to courses in physical chemistry and thermodynamics. It is particularly unfortunate that such apparent “myths” persist in the literature, since treatments taking them properly into account have existed for some time. Dodge (7) provides the earliest such detailed account that we are aware of, although Gibbs expressed the law correctly much earlier (8). www.JCE.DivCHED.org
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Statements of Dalton’s Law
Dalton (1808) John Dalton first conceived of “elastic fluids” acting independently in 1801, and this gave rise to various experiments, the results of which were published in 1802 (9, pp. 153–154).
When any two or more mixed gases acquire an equilibrium, the elastic energy of each against the surface of the vessel or of any liquid, is precisely the same as if it were the only gas present occupying the whole space, and all the rest were withdrawn.
We note, by way of interpretation, that the concept of gases acting independently does not imply that the gases are necessarily ideal.
Gibbs (1876) In 1876, Gibbs (8, p 155) wrote: Statement B The pressure in a mixture of different gases is equal to the sum of the pressures of the different gases as existing each by itself at the same temperature and with the same value of its potential.
This statement has been called the Gibbs–Dalton law by Gillespie (10). It was used by Gibbs to obtain consequences for various properties of gas mixtures at (T, V ), including P, as expressed by the statement below.
A Contemporary Statement As a contemporary statement, we express Dalton’s law as follows: Statement C The pressure (P ) of a mixture of different gases at temperature T and volume V is equal to the sum of the pressures of the individual gases such that the same amount of each gas exists alone at (T, V ) of the mixture.
Gibbs (8, p 156) made essentially the same statement (of a “familiar principle”) in his consideration of gas mixtures.
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It is statement C that we are concerned with here, and not the wider implications of statement B. Statement C assumes that the gases are either chemically inert to each other or are at chemical equilibrium. Thus, the idea of equilibrium is included implicitly. Partial Pressure (pi) and Pure Component Pressure (Pi) As recommended by IUPAC (11), and as widely observed, the partial pressure of gaseous species i in a gas mixture of N species, whether ideal or nonideal, is defined by (1)
pi ≡ x i P ; i = 1, 2, … , N
where xi is the mole fraction of species i in the mixture:
(
N
∑ ni
∑ pi
=
i
∑ xi P i
(3) v r′ ≡
= P ∑ x i = P
zm ≡
)
Pi = Pi T , V , ni ; i = 1, 2, … , N
(5)
(
)
(6)
Pc v RTc
(10)
PV nt RT
(11)
zi zm
(12)
then, from eq 1, 2, 9, and 11, Pi =
pi
Thus, Pi and pi are identically equal at all values of (T, V, n) for a given mixture only when zi ⬅ zm, which occurs only for a mixture of ideal gases, when both are equal to unity. Operational Forms of Dalton’s law When the given problem variables are (T, V, n), where n is the vector of species mole numbers, explicit operational forms of Dalton’s law follow from statement C together with one of eq 5, 6, 7, or 9 with 10: N
= Pi T , vi
P =
∑ Pi (T, V, ni )
i =1
N
=
∑ Pi (T, vi )
i =1
N
=
(
= Pi T , v, x i
)
(7)
where vi(= V/ni) is the molar volume of pure i, and v (= V/nt) is the molar volume of the mixture. Equations 6 and 7 arise because P is an intensive variable.
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(9)
where v is the molar volume. Alternatively, z(Tr , Pr) and z(Tr, vr) are available numerically (12, 13) as noted above. The quantities Pi and pi are related as follows. If the compressibility factor for the gas mixture is defined as
(4)
i
since Σxi ≡ 1. Identity 4 does not represent a law of nature, but only the consequence of a definition. Dalton’s law is thus not a law of additivity of partial pressures. In contrast to partial pressure, the individual gas pressure referred to in any statement of Dalton’s law, usually called the pure component (or pure substance or pure species) pressure, Pi , is defined as the pressure exerted by ni moles of pure i at (T, V ). Pi may be obtained from a pressure-explicit or volume-explicit algebraic equation of state (EOS) for the pure substance, or from a generalized compressibility factor (z) correlation stemming from the principle of corresponding states (PCS), usually in tabular or graphical form (12), but also by means of an essentially exact numerical calculation (13). If an EOS is explicit in P, the usual case, then
(
zi ni R T ; i = 1, 2, … , N V
where R is the universal gas constant. In graphical and tabular correlations, zi is available as z(Tr, Pr) and may be available as z(Tr , vr), where subscript r refers to a reduced quantity, for example, T/Tc, where Tc is the critical temperature, and vr is defined (12) as
i =1
The partial pressure is not the quantity referred to in any of the statements above. With pi defined by eq 1, the sum of the partial pressures is identically equal to the total pressure:
(8)
where Vi denotes the functional form of the EOS for substance i. For given (T, V, ni ), Pi can be calculated directly from an EOS of the form of eq 5, but use of eq 8 requires solution of a nonlinear equation. This latter can be facilitated greatly by the use of metacomputing software such as Maple1 or Mathematica.2 In terms of the compressibility factor for the pure substance, zi , Pi is given as
(2)
where ni is the number of moles of species i and nt is the total number of moles of the mixture:
)
V = Vi T , P , ni ; i = 1, 2, … , N
Pi =
n xi ≡ i ; i = 1, 2, … , N nt
nt ≡
If an EOS is explicit in V, which is a relatively rare case, then eq 5 can be written in inverted form as
∑ Pi (T, v, x i )
from (5), (6), (7)
(13)
i =1
=
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RT V
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N
∑ zi(Tr i , v ′ r i ) ni
from (9), (10)
i =1
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In the Classroom
When the given problem variables are (T, P, n), implicit forms (nonlinear equations) arise from use of eq 8 or eq 9 involving zi(Tri , Pri): N
∑ Pi T , Vi (T , Pi ) , ni
P =
(15)
i =1
RT V
=
N
∑ zi (Tri , Pri) ni
(16)
i =1
It is these four equations, 13–16, that form the basis for implementation of Dalton’s law for various cases of calculating P or V described in the next section. Alternative useful operational forms of Dalton’s law can be constructed involving zm , defined by eq 11, particularly in the context of the PCS. Thus, if we are given (T, V, ni), or equivalently (T, vi), eq 11, in combination with 6, 9, and 2, yields PV V zm (T, V , n) = = nt RT nt RT N
=
N
∑ Pi (T , vi )
i =1 N
∑ zi (T , vi ) xi
=
i =1
∑ zi (Tri , v ri′ ) xi
(17)
i =1
According to eq 17, use of Dalton’s law corresponds to evaluation of zm as a mole-fraction average of the pure substance zi ’s, each zi being evaluated at (T, vi) or equivalently at (Tri, vri). Alternatively, if we are given (T, P, n), a corresponding treatment yields N
zm (T , P , n) =
N
∑ zi (T, Pi ) xi
i =1
=
∑ zi (Tri , Pri ) xi
(18)
i =1
In this case, zm is a mole-fraction average with each zi evaluated at (T, Pi) or equivalently at (Tri , Pri) . Statement C in the form of eq 13 emphasizes that the law should be referred to as “Dalton’s law of additive (pure component) pressures” (7). It is thus analogous to the Amagat–Leduc law of additive (pure component) volumes (14, 15, 16). The Amagat–Leduc law is more comprehensive than Dalton’s law, since it can also be usefully applied to mixtures of liquids. Dalton’s law is exact for a mixture of ideal gases, as can be seen by setting zi = 1 (for all i ) in eq 14 or 16. This can also be realized more subtly by determining the significance for zm if each zi is evaluated at (T, pi) (analogous to the result in eq 17 or 18). Use of Dalton’s law for an IGM, however, adds nothing to its PVTn description, since it merely recaptures the EOS for an IGM, which is known a priori. By contrast with this and eq 4, it follows from eq 12 that the sum of the pure component pressures is not generally equal to the total pressure: N
∑ Pi
i =1
=
1 N ∑ zi pi ≠ P zm i = 1
(19)
In other words, Dalton’s law is an approximation for a mixture of nonideal gases. Comparisons with experimental data and other methods, together with conclusions about its validity are given by Gillespie (10), Dodge (7), and Smith and
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Van Ness (17). These last conclude that Dalton’s law is a useful approximation for relatively low P (or relatively low density), and should probably not be used for P > 5000 kPa. Beattie (18) states that “Amagat’s law gives results superior to Dalton’s law...at temperatures well above the critical temperature of each constituent, while Dalton’s law seems to give the better results if one of the constituent gases is below or only slightly above the critical temperature.” Implementation of Dalton’s Law of Additive Pressures As noted following eq 19, Dalton’s law as articulated by statement C is one way of predicting the approximate PVTn behavior of a mixture of nonideal gases from the PVTni behavior of the individual pure gases (the Amagat–Leduc law is another way). We may construct eight types of problems depending on the following factors: • Whether the calculation is (1) for P, given (T, V, n); or (2) for V, given (T, P, n) • Whether the PVTni description is given in pressureexplicit form—that is, the independent variables are (T, V, ni )—or in volume-explicit form, the relatively rare situation for which the independent variables are (T, P, ni). There are 4 cases, depending on the specific form of the PVTni description available: (a) Pi(T, V, ni), (b) zi(Tri, vri), (c) Vi(T, P, ni), (d) zi(Tri, Pri).
Problem Types for Solving Additive Pressures The eight types of problem arise from combinations of the above two items. Furthermore, in cases b and d, implemented by numerical solution of eq 14 and 16, respectively, the solutions can be carried out by means of exact calculation of each function zi, or tabular or graphical representation of each function zi. The various types of problems or cases are shown in Figure 1, denoted by la,...,2d, where 1 and 2 refer to calculation of P and V, respectively. We describe solution procedures for each of these briefly in turn. Case 1: P (T, V, n) 1a. P can be calculated directly from eq 13. 1b. P can be calculated directly from eq 14 (whether zi is available in exact or in tabular or graphical form). 1c. P can be calculated by numerical solution (e.g., with metacomputing software) of nonlinear eq 15 for each Pi and summation of the N Pi’s. 1d. P can be calculated by solution of nonlinear eq 16, the implementation of which depends upon the available data. If zi is available exactly, eq 16 is solved numerically. If only tabular or graphical data are available, an iterative procedure to solve the equation may be used as follows: Set Pi(0) niRT/V Determine zi(1) Calculate Pi(1) zi(1)niRT/V Repeat until zi converges Calculate zm Σxizi Then calculate P zmntRT/ V
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In the Classroom Dalton's law to calculate
1: P(T,V,n) from/via (Eq.)
1a: Pi (T,V,ni)
2: V(T,P,n) from/via (Eq.)
1c: Vi (T,P,ni) 1b: zi (Tri,vri′ )
(13) †
num
tab/graph
(14) †
(14) †
2a: Pi (T,V,ni) 2b: zi (Tri,vri′ )
1d: zi (Tri,Pri)
(15) ‡
2c: Vi (T,P,ni)
num
tab/graph
(16) ‡
(16) §
(13) ‡
num
tab/graph
(14) ‡
(14) §
2d: zi (Tri,Pri)
(15) ‡
num
tab/graph
(16) ‡
(16) §
Figure 1. Types of problems addressed by Dalton’s law: The symbol † indicates a direct calculation; The symbol ‡ indicates a numerical solution of nonlinear equation(s); The symbol § indicates an iterative procedure for nonlinear equation(s).
Case 2: V (T, P, n)
use of the Amagat–Leduc law or of pseudocritical constants (e.g., Kay’s rule, ref 19).
2a. V can be calculated by solution of the single nonlinear eq 13. 2b. V can be calculated by solution of nonlinear eq 14. If zi is available exactly, eq 14 is solved numerically. If only tabular or graphical data are available, an iterative procedure to solve the equation may be used: Set Pi(0) pi Calculate vi(1) RT/Pi(0), then vri(1) Obtain zi(1) Calculate zm(1), V (1), vi(2) V (1)/ni Repeat until V converges 2c. V can be calculated by numerical solution (e.g., with metacomputing software) of (N 1) nonlinear equations of the type of eq 15: 1 for V V(T, P, nt ) and N for V Vi(T, Pi, ni); in which the unknowns are V and N Pi’s. 2d. V can be calculated by solution of nonlinear eq 16 by means of procedures analogous to those for case ld. If zi is available exactly, eq 16 is solved numerically. If only tabular or graphical data are available, an iterative solution procedure may be used as follows: Set Pi(0) = pi Determine zi(1)(T, Pi(0)) Calculate zm(1) = Σzi(1)xi Calculate Pi(1) from eq 12 Determine zi(2)(T, Pi(1)) Calculate zm(2) Repeat until zm converges Then V zmntRT/P
Corresponding cases can be constructed for other methods of predicting approximate PVTn behavior, such as the
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Conclusions 1. Dalton’s law for gas mixtures should properly be called “Dalton’s law of additive (pure component) pressures” and not “Dalton’s law of partial pressures” as is commonly done. 2. The pure component pressure of a species in a gas mixture should be distinguished from its partial pressure; the two are identical only for a mixture of ideal gases. 3. Dalton’s law can be used to solve several types of problems at relatively low density for the approximate calculation of P or V of a nonideal gas mixture from information on pure component PVT behavior. It is, however, exact for an ideal gas mixture. 4. The problems vary because of the way in which the pure component information is available, and their solution may involve direct calculations or solution of one or more than one nonlinear equation. 5. Calculations involving the solution of nonlinear equations in conjunction with an equation of state can be facilitated by metacomputing software.1, 2
Recommendations The following are recommendations to treat Dalton’s law at two levels of undergraduate instruction.
General Chemistry Courses and Dalton’s Law In general chemistry, Dalton’s law should be introduced to students with conclusions 1 and 2 in mind. This includes Myth 1 and Myth 2 above and parts of the fol-
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In the Classroom
lowing three sections, making use of eq 1–4, and the essence of eq 5 and 13. It could be demonstrated that the law is exact for mixtures of ideal gases, and the significance of this examined. It should also be noted (without demonstration) that Dalton’s law is an approximation for mixtures of nonideal gases, most suitably at relatively low pressures. The simplest problems involving ideal gases could be used in conjunction with eq 13. This recommendation assumes that the PVTn behavior of nonideal gas mixtures is not discussed.
Physical Chemistry Courses and Dalton’s Law In physical chemistry (including thermodynamics), at a more advanced level in which the behavior of nonideal gas mixtures is introduced, the full treatment described here, encompassing conclusions 1–5, could be used. The extent would depend on the time available and the desires of the instructor. Students could be involved in formulating and solving various problems addressed by Dalton’s law. The solution of certain of these problems could provide valuable experience in using metacomputing software packages (e.g., for solving nonlinear equations). Acknowledgment Financial assistance has been received from the Natural Sciences and Engineering Research Council of Canada. Notes 1. Maple software is available from Maplesoft: http:// www.maplesoft.com/index.aspx (accessed Apr 2005). 2. Mathematica software is available from Wolfram Research, Inc.: http://www.wolfram.com/products/mathematica/ index.html (accessed Apr 2005).
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Literature Cited 1. Planck, M. Treatise on Thermodynamics, 3rd English ed.; Dover: New York, 1945; pp 11, 21 (translated by Alexander Ogg). 2. Partington, J. R. An Advanced Treatise on Physical Chemistry, Vol. 1; Longmans, Green and Co.: London, 1949; pp 200, 609–612. 3. Hatsopoulos, G. N.; Keenan, J. H. Principles of General Thermodynamics; Wiley: New York, 1965; p 283. 4. Guggenheim, E. A. Thermodynamics: An Advanced Treatment for Chemists and Physicists, 5th ed.; North-Holland Publishing Co.: Amsterdam, 1967; pp 175, 177. 5. Laidler, K. J.; Meiser, J. H. Physical Chemistry, 2nd ed.; Houghton Mifflin: Boston, 1995; p 21. 6. Atkins, P. W. Physical Chemistry, 6th ed.; Oxford University Press, 1998; pp 21–22. 7. Dodge, B. F. Chemical Engineering Thermodynamics; McGrawHill: New York, 1944; pp 188–196. 8. Gibbs, J. W. The Scientific Papers of J. Willard Gibbs, Vol. I: Thermodynamics; Dover: New York, 1961. 9. Dalton, J. A New System of Chemical Philosophy, Part I; Bickerstaff: Manchester and London, 1808. 10. Gillespie, L. J. Phys. Rev. 1930, 36, 121. 11. Mills, I. M., et al. Quantities, Units, and Symbols in Physical Chemistry, 2nd ed.; Blackwell: Cambridge, MA, 1993; p 42. 12. Lee, B. I.; Kesler, M. G. AIChE J. 1975, 21, 510. 13. Smith, W. R.; Missen, R. W. Chem. Eng. Ed. 2001, 35, 68. 14. Amagat, E-H. Ann. Chim. 1880, Ser. 5, 19, 345. 15. Amagat, E-H. Compt. Rend. de l’Académie des Sciences, Paris 1898, 127, 88. 16. Leduc, A. Compt. Rend. de l’Académie des Sciences, Paris 1898, 126, 218. 17. Smith, J. M.; Van Ness, H. C. Introduction to Chemical Engineering Thermodynamics, 2nd ed.; McGraw-Hill: New York, 1959; pp 104–105. 18. Beattie, J. A. Chem. Reviews 1949, 44, 141. 19. Kay, W. B. Ind. Eng. Chem. 1936, 28, 1014.
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