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The Physical Meaning of the Mathematical Formalism Present in Limiting Chemical Equations; Or, How Dilute Is Dilute? C. Contreras-Ortega,* N. Bustamante, J. L. Guevara, C. Portillo, and V. Kesternich Facultad de Ciencias, Universidad Católica del Norte, Casilla 1280‚ Antofagasta, Chile; *
[email protected] In teaching and research, we frequently talk about solutions being concentrated or dilute, or gases being at low or high pressures or temperatures. But, just how “dilute” is dilute? And just how “low” is a low pressure or temperature? Consider a 0.001 M solution: While a physical chemist teaching a general chemistry laboratory might consider this to be dilute, an analytical chemist working with trace levels might consider this extremely concentrated. Similar complications exist when discussing the usefulness or limitations of a given algorithm. To one chemist an equation that is functional to within 1.0% of accepted values might be considered a good equation, but the same equation may not be acceptable to another chemist. Key to understanding the important dialectic illustrated in the above examples is the realization that physical conditions that control or limit the behavior of the phenomenon or system also control the mathematical representations generated to describe it. While professional chemists understand the important link between the conditions and limitations of the phenomenon and its algorithm, students typically do not understand that link. Let’s take, for example, some commonly used algorithms such as the ideal gas law, Nernst’s equation, and Raoult’s and Henry’s laws. The mathematical formalisms present in these algorithms are commonly expressed as mathematical tendencies and limits. Unfortunately, the general nature of these formalisms is a source of ambiguity and confusion for students because the exact physical meaning of the mathematical tendencies and limits is not always denoted. This problem is compounded by the fact that general chemistry textbooks tend to include only equations that are strictly valid only for systems behaving under limiting physical conditions (1–6). In this article, we propose a general mathematical treatment for chemical systems to help overcome the problems outlined above, while providing students a better understanding of the real scope of the mathematical equations they will encounter. Tendencies and Limits Let us start by considering some important terms needed in the larger discussion. Tendency is the behavior of an independent mathematical variable to approach a limiting value. For example, the physical phenomenon of pressure can tend toward a low, high, or constant value. Thus, pressure is commonly expressed as a value tending to zero, infinity, or a constant, respectively. The limiting value of the tendency (zero, infinity, or a constant) fixes the boundary conditions of the algorithm in use. A limit is the particular limiting value or the particular limiting expression of an equation, a numerical limit and an equational limit, respectively. Consider the following examples for clarification. The unity value for the activity coefficient 788
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of a solute as its concentration tends to a low value is an example of a numerical limit. An example of an equational limit is the concentration-based expression for the chemical equilibrium constant as the concentrations of the chemical species (solutes) tend to low values.
Limit Validity A fundamental question that should now be raised is: When does the numerical limit or the equational limit become valid or apply? Limits become valid at those values of their respective tendency where the difference between (concordance) an accepted result (an experimentally or theoretically reliable result) and the obtained experimental result is acceptably small. This concordance can be expressed as a mathematical tolerance, such as, for example, a linear regression index, a deviation (an arithmetical quotient or difference) or percent variation, and so forth. We will name any one of these criteria a concordance criterion. Values of Numerical Limits and Tendencies When the limit of an equation (a function) is a numerical value (a numerical limit), the explicit mathematical relationship between the function, its limit, and the tendency of its independent variable is frequently expressed as: lim f ( x ) = k
x → l
(1)
In the above expression, k is the theoretical numerical limit for the expression f ( x) (a mathematical function) as x (its independent variable) tends to its value l. Values l and k can be any two constants from zero to infinity. While f ( x) approaches the limit k in the mathematical sense (it is infinitely close to the limit, but it is never at that value), x is not obligated to reach its limit l. The fact is that in most of the physical cases, f ( x) moves towards its limit much more rapidly than x moves toward l. Therefore, while f ( x) approaches k as a limit, x appears only as a tendency moving toward the limiting value l (as an example, see eq 5 below). The differences between the accepted value and the experimental result can be examined using the concordance criteria mentioned above and the concordances expressed as general equations, which we will name as the concordance equations. An analysis of these concordance equations starts with the equation for the arithmetical difference: lim
x → l
f (x ) − kexp
= k − kexp = 0cc
(2)
In the above equation, kexp is the experimental value for f ( x) at x ⫽ lexp, and lexp is an experimental value of x at which it is expected that the concordance equation result is close
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enough to zero to be acceptable for the experimenter. The zero in eq 2, an absolute value, is not a real zero value; rather it is an acceptably small value of the chosen concordance criterion. This is the reason for the subscript cc. We will name that symbol as the concordance criterion zero. Alternatively, if kexp ⫽ 0, the expression for the arithmetical quotient can be used: f (x ) kexp
lim
x → l
=
k kexp
= 1cc
Real gases obey this law at normal laboratory conditions quite well. Therefore, in real gas systems the quotient PVnT, computed with pressure values of about 1 atm and temperature values of about 298 K, for any given volume and number of gas moles, differs from the R value to within a few percent (see Table 1). In general, the ideality of the gas and its agreement with the ideal gas law improve as the pressure goes down and temperature goes up. The agreement should be absolute in the limiting values of both pressure and temperature. Analogous to eq 1, eq 5 describes the limiting situations corresponding to an ideal gas:
(3)
Here, the symbol 1cc is similar to the symbol 0cc in eq 2. As such, we will name it as the concordance criterion unity. The concordance equation for the deviation percent from the accepted value, as derived from eq 2, is as follows: deviation% ⫽ (0cc k) ⫻ 100.
lim
P → 0 T → ∞
= R
(5)
Equation 5 raises a fundamentally important point: the mathematical notation used for a tendency demands that a parameter must move in the direction indicated until it reaches the magnitude that makes possible the value of the limit. The tendency must fit with an appropriate limit. Tendency values that lead to limits of no physical sense or that are mathematically contradictory must be ruled out even if they are mathematically allowed. For example, one can obtain an absurd result (undetermined value) ⫽ R (a fixed number) from eq 5 if the limiting values of the tendencies are taken exactly as written. The only meaningful values for pressure and tem-
Ideal-Gas Limiting Behavior Let us consider a common limiting case: the relationship between the volume, pressure, and temperature of a gas when it behaves as an ideal gas. By definition, ideal gases are those that at any pressure and temperature obey the equation known as the ideal gas law. This law is expressed as: PV = R nT
PV nT
(4)
Table 1. Concordance Values for the Gas Constant for Helium and Water at a Fixed Temperature, by Pressurea P, atm
0.1
0.2
R e xp
0.08206
0.08207
0 cc
0
0.3
0.4
0.5
1
2
5
10
50
0.08208
0.08209
0.08209
0.08213
0.08221
0.08243
0.08281
0.08582
0.00001
0.00002
0.00003
0.00003
0.00007
0.00015
0.00037
0.00075
0.00376
1.000
1.000
1.000
0.9996
0.9996
0.9991
0.9981
0.9955
0.9909
0.9561
0
0.01
0.02
0.04
0.04
0.09
0.18
0.45
0.91
4.58
He
1 cc Deviation %
b
Degree of acceptabilityc
Excellent
Very good
Good
Poor
Bad
Qualitative values d
Extremely low
Very low
Low
Moderately low
Very high
H 2O R e xp Deviation %b
0.08199
0.08193
0.08186
0.08180
0.08173
0.08140
0.08074
0.07868
0.07499
0.09
0.16
0.24
0.32
0.40
0.80
1.61
4.12
8.62
0.006095 92.57
Degree of acceptabilityc
Good
Poor
Bad
Bad
Unacceptable
Qualitative values d
Low
Moderately low
High
Very high
Extremely high
aCalculated from eqs 6 and 7, where R ⫽ 0.082056 (atm L)/(mol K) and (PV/nT ) exp are the values named as Rexp in the table. Rexp values were calculated at any given pressure with n ⫽ 1 mole, T ⫽ 298 K, and corresponding V values obtained from the van der Waals equation and data from reference 5. bDeviation % ⫽ (0 /R) ⫻ 100; a deviation value of 0.1% was considered acceptable: this was arbitrarily selected for the purposes of this article. cc cDegree of acceptability of results conveys how well the equation and the result correspond for the experimenter, ranging from unacceptable to excellent. dQualitative pressure values range from low to extremely high.
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perature are those found at that point where pressure reaches a value small enough (P → 0) to make product PV decrease, and simultaneously temperature reaches a value high enough (T → ∞) to make the product nT increase. The concurrent decrease of pressure and increase of temperature occur until the quotient of both products becomes equal to R. Concordance eq 2 or 3 can be applied to eq 5. They are, respectively, lim
P → 0 T → ∞
PV nT
lim
P → 0 T → ∞
PV nT
−
PV nT
= R − exp
PV nT = PV nT exp
R PV nT
= 0cc exp
= 1 cc
(6)
(7)
exp
These equations show that the pressure must be so low (its value so small) and the temperature must be so high (its value so high) that the value for the expression (PVnT)exp becomes so close to the R value that the difference between (PVnT)exp and R is nonsignificant to the experimenter. Therefore, “low pressure” and “high temperature” must be understood as pressures and temperatures fulfilling the above conditions. Table 1 shows the results for two gases when eq 5 is analyzed with eqs 6 and 7 for pressure variations. Observe that at 298 K, pressures for He can be considered as “zero” (P0) from 1 atm and lower, while pressures for H2O can be considered “zero” (P0) from 0.1 atm and lower. The same analysis made for temperature variation shows that at 1 atm pressure, temperatures for He can be considered as “infinite” (T∞) from 298 K and higher, while temperatures for H2O can be considered as “infinite” (T∞) from 750 K and higher.
Thus, T∞ and P0 are not infinite temperature and zero pressure without physical meaning, nor are they necessarily an extremely high temperature (many millions of degrees) and an extremely low pressure (a few millionths of an atmosphere). Note also that the same temperature and pressure values can be considered high for one system and low for another. Therefore, the limiting values of the tendencies are real experimental values that define equations, to the extent that equations define real chemical systems. Concordance values provide magnitudes for values of tendencies that are low or high and values for limits, as well as equations that are useful, acceptable, good, or bad. In that regard, qualitative values and acceptability degree scales are proposed as shown in Tables 1 and 2. For another application of eq 1, we now discuss the relationship between the activity, a, and the concentration of a species in a solution as it becomes “infinitely dilute.” For a solute i and a solvent j, that relationship is expressed in the mole fraction composition scale as: lim (ai /xi) ⫽ lim γi ⫽ 1 and lim (aj /xj) ⫽ lim γj ⫽ 1 as xj → 1 This is the infinitely dilute boundary condition for solute and solvent in their standard states, respectively. The γ’s are the activity coefficients. Let us take a brief look at the mercury– thallium amalgam system. We will assume arbitrarily that an acceptable concordance criterion for the γ values in that system is a deviation less than 3%. According to reported values, γHg ⫽ 0.986 and γTl ⫽ 1.80 at xHg ⫽ 0.95 and 20 °C (7). This implies that deviation percents are 1.4% and 80% for mercury and thallium, respectively. It is interesting to note that this same system may be considered as dilute or very dilute relative to the solvent behavior (mercury), while it may simultaneously be considered as very or highly concentrated relative to the solute behavior (thallium).
Table 2. Concordance Values for the Dissociation Constant of Dinitrophenol in Pure Watera Ci,exp ⫻ 104, moles L᎑1
7.018
5.505
3.991
2.395
1.5364
0.9245
K(Ci,0) ⫻ 105
8.152
8.152
8.152
8.152
8.152
8.152
8.152
K exp ⫻ 1 0 5
8.449
8.435
8.377
8.325
8.310
8.289
(8.15…)e
0 cc ⫻ 1 0 5
0.297
0.283
0.225
0.173
0.158
0.137
(0.00…)e
0.9648
0.9664
0.9731
0.9792
0.9810
0.9835
(1.00…)e
3.66
3.47
2.76
2.12
1.94
1.68
(0.00 ...)e
1 cc Deviation % b Degree of acceptability c Qualitative values d
(……...)e
Acceptable
Good
Very good
Excellent
Moderately dilute
Dilute
Very dilute
Highly dilute
aCalculated from eqs 16 and 17. Values for K(C ) and K i,0 exp for any of the Ci,exp values shown in the table were obtained with data from reference 9; Ci,exp ⫻ 104 are undissociated acid concentrations. b Deviation % ⫽ (0cc /Ci,0) ⫻ 100; a deviation value of 3% was considered acceptable: this was arbitrarily selected for the purposes of this article. cDegree of acceptability of results conveys how well the equation and the result correspond for the experimenter, ranging from acceptable to excellent. dQualitative concentration values range from moderately dilute to highly dilute. e All values in parenthesis are hypothetical limiting values.
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Values of Equational Limits and Tendencies
the expression for the equilibrium constant is:
When the limit of an equation (a function) is a particular expression (an equational limit), the explicit mathematical relationship of the function, its limit, and the tendencies of their respective independent variables can be expressed as: lim
y → x → l
f ( y ) = g (x l )
(8)
In the above equation, g ( x l ) is the equational limit of function f (y) when its variable y becomes equal to a new variable x, as x tends to value l, namely, xl. The value l can be any constant from zero to infinity. As in eq 1, while f (y) must reach the limiting expression g( x l ), x is not obligated to reach its limiting value l. As before—in most of the physical cases— f (y) moves toward the limiting expression g ( x l ) much more rapidly than x does to its limiting value l. Therefore, while f (y) appears as an equational limit, x appears only as a tendency. (As an example, see eq 14 below for the chemical equilibrium constant.) Having to write both tendencies in eq 8 is redundant because the tendency of x toward its limiting value l obligatorily implies the tendency of y towards the new variable x. Therefore, this equation can be rewritten as follows: lim f ( y ) = g ( x l )
(9)
x → l
Normally, we work with the expression for g ( x l ) in the form g ( x), where the implicit presence of tendencies and limits in the results are not obvious. (See eq 13 below.) The concordance or difference between the theoretical prediction and the experimental result can be examined with the following concordance equation:
(
lim f ( y ) − g x l exp
x → l
(
)
= g ( x l ) − g x l exp
)
= 0cc (10)
In the above equation, g(xl,exp) is the value for g ( x l ) at the experimental value of xl ⫽ xl,exp. As before, xl,exp is an experimental value of x at which the concordance equation should show a value acceptable for the experimenter. Alternatively, if g ( xl,exp) ⫽ 0, the following expression can be used:
K (Ci ) =
c
d
a
b
CC CD
(13)
CA CB
In the above expression, Ci represents the concentration of each of the species in the reaction in a given concentration scale and the exponents are the respective reaction stoichiometric coefficients. In eq 13, the infinitely dilute solution boundary condition was taken for all the chemical reactants in its deduction. This condition is not explicitly shown in the expression. We say that the relationship between the equational limit and the concentration limiting values is intrinsically contained in the result. We can explicitly show this relationship as follows:
a Cc aDd
lim
ai → Ci → 0 ∀ i ≠ j C j → constant
aAa aBb
=
c
d
a
b
CC CD CA CB
The quotient in the brackets on the left side of eq 14 is the expression for the equilibrium constant in terms of the activities of the reaction species, which we will name as K(ai). In that equation, i is any of the solutes, j is the solvent if it participates as reactant; the molar concentration scale has been used. With the boundary condition of an infinitely dilute solution, the concentration of the solvent—as one of the reaction species—normally does not explicitly appear on the right side of the above expression, because the solvent concentration is almost a constant value and, as such, it is taken as part of the equilibrium constant value. The same thing happens with the solvent activity on the left side of that expression.1 Replacing the quotients in eq 14 for K(ai) and K(Ci), but making explicit in the concentration equilibrium constant the limiting values for the concentrations of the solutes and omitting solvent terms as well, eq 14 can be rewritten in the general form of eq 9 as: lim K (ai ) = K (C i ,0 )
Ci → 0
lim
x → l
f (y)
(
g x l exp
)
=
g ( xl )
(
g x l exp
)
= 1 cc
(11)
Concentration-Based Equilibrium Constants Let us consider the concentration-based expression for the chemical equilibrium constant that is valid, as it is usually said, at “dilute” to “highly dilute” concentrations of the chemical reaction participants. Thus, for the general chemical reaction equilibrium
aA + bB
(12)
cC + dD
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(14)
(15)
Equation 13 is the expression for the equilibrium constant usually introduced in general chemistry courses and simply indicated as K(Ci). The activity concept is needed to work with the canonic expression given at the left side of eq 14, a concept that is not generally introduced in first-year chemistry courses. Artificially, all the equilibria studied in first-year courses involve equilibria and concentrations where activities can be replaced by concentration values and acceptable values for K are obtainable. It is experimentally found that the more dilute the concentration of species in solution, the more acceptable are the values for K. These particular concentrations are normally referred to as “very dilute concentrations”. For other situations, eq 13 is valid just as an
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estimate, since it is only an approximation of the general expression. In the above case, equations of the form 10 and 11 apply to analyze the concordance criterion. Hence:
lim K (ai ) − K exp
Ci → 0
= K (C i ,0 ) − K exp = 0cc
(16)
or lim
Ci → 0
K ( ai ) K exp
=
K (C i ,0 ) K exp
= 1 cc
(17)
As shown in eqs 10 and 11, Kexp in the above equations is the value for K(Ci,0) calculated with eq 13 and the equilibrium concentration values of each one of the chemical species participating in the reaction: Ci,0 ⫽ Ci,exp. The Ci,exp terms are the experimental values of the species concentrations at which concordance equations are expected to show acceptable concordance values. In eqs 15–17, K(Ci,0) is numerically equal to K(ai) obtained at any concentration value because the equilibrium constant value, as such, must be the same in the whole concentration range of the reaction participants. This value can be obtained by means of many experimental procedures depending on the chemical nature of the system. Once K(ai) has been determined, it can be used with either eq 16 or 17 to determine when Kexp becomes acceptable. Concentration values in these equations need to be small enough (approaching zero) to make both values for K closely equal, such that their differences cannot be experimentally detectable or their differences are insignificant. Therefore, a “dilute solution” must be understood as providing an acceptable low concordance value, while a “non-dilute” or “concentrated solution” must be understood as having the opposite meaning. Depending on the nature and the concentration of the participants, it is possible for a concentration value to be considered low (dilute) for one system and high (concentrated) for another. For example, a concordance of 5–8% is observed for the dissociation constant of 3-nitrobenzoic acid at a concentration range of 2 ⫻ 10᎑3 to 4 ⫻ 10᎑3 mol L᎑1 (8), while a concordance value of 2% and 4% is observed for the dissociation constant of dinitrophenol in the range of 1 ⫻ 10᎑4 to 7 ⫻ 10᎑4 mol L᎑1 (9). Both equilibria were in pure water at 25 ⬚C. Details for the latter system are given in Table 2. Conclusions While expressed in formal mathematical terms as limits of tendencies, the limiting values of practical chemical variables are not abstract, undetermined values. Limiting values are as understandable as concrete values: measurable, determined, and bound by experiment. These practical limits of variables can be used to fully understand the limits of equations that define chemical phenomena. A zero-value tendency (e.g., P → 0) must be understood as the need of a parameter to diminish its magnitude so that the expression it is involved in reaches its limiting form. In such a regard, a zero-tendency value (P0) is understood as the largest experimental value of a given parameter that produces acceptable concordance criterion values for the experimental result and the predicted result. Smaller experimental values 792
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produce smaller concordance criterion values and smaller percent deviations. An infinite-value tendency (e.g., T → ⬁) must be understood as the need of a parameter to increase its magnitude with the same purpose as above. Thus, an infinite-tendency value (T⬁) is understood as the smallest experimental value of a given variable that produces acceptable concordance criterion values for the experimental result and the predicted result. Larger experimental values produce smaller concordance criterion values and smaller percent deviations. An understanding of these limits helps experimenters correctly consider variables such as pressures and temperatures that are “low” or “high” or solution concentrations that are “dilute” or “concentrated”. The concordance criterion values between theoretical and experimental results are determined from limiting values of tendencies. The extent to which an equation acceptably describes a chemical system is determined from the limiting values of practical chemical variables and the concordance value defined by given experimenters. An understanding of the role of these limits and their relationship to experimental values allow experimenters to label equations as applicable or non-applicable. While sounding theoretical, an understanding of tendencies, limiting values of tendencies, and the limits of equations is absolutely practical. Acknowledgments The authors wish to thank Mickey Sarquis, Director of the Center for Chemistry Education at Miami University (OH, USA), and Susan Hershberger, Amy Hudepohl, and Amy Stander, also with the Center for Chemistry Education, for their helpful suggestions to this article. The authors also wish to thank Ana Tejeda, from the Faculty of Humanities at the Universidad Católica del Norte, for helpful assistance in editing this article. Financial support from DGIP/UCN is gratefully acknowledged. Note 1. If the solvent is water, the value for Cj is 55.5 mol L᎑1 at 25 ⬚C.
Literature Cited 1. Mortimer, C. E. Chemistry, 5th ed.; Wadsworth: Belmont, CA, 1983. 2. Petrucci, R. H; Harwood, W. S. General Chemistry: Principle and Modern Applications, 7th ed.; Prentice Hall: Upper Saddle River, NJ, 1997. 3. Brown, T. L.; LeMay, H. E., Jr.; Bursten, B. E. Chemistry: The Central Science, 7th ed.; Prentice Hall: Upper Saddle River, NJ, 1997. 4. Chang, R. Chemistry, 6th ed.; McGraw-Hill: Boston, MA, 1998. 5. Umland, J. B.; Bellama, J. M. General Chemistry, 3rd ed.; Brooks/Cole: Pacific Grove, CA, 1999. 6. Hein, M.; Arena, S. Foundations of College Chemistry, 10th ed.; Brooks/Cole: Pacific Grove, CA, 2000. 7. Lewis, G. N.; Randall, M. Thermodynamics, 2nd ed. (revised by K. S. Pitzer and L. Brewer); McGraw-Hill: New York, 1961. 8. Contreras, M.; Silva, S. Bol. Soc. Chil. Quím. 1998, 43, 147– 153. 9. Ramette, R. W. Chemical Equilibrium and Analysis, AddisonWesley: Boston, MA, 1981.
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