Integrated Approach to a Basic Understanding of Chemical Reactivity and Reaction Order with Illustrated Applications to Selected General Chemistry Problems
	Kenneth M. Maloney Chemistry and Physics Division, Baton Rouge Community College, 201 Community College Drive, LA 70806
	The complete formulations of three(3) new problems, in general chemistry, are illustrated with the selective applications of the significant and basic equation or learning tool{i.e., WF = t^p•exp[p(1 - t)]} to three distinctly different experimental systems. For each one of the reaction systems, formulation of the respective reduced dependent variable WF(proportion reacted) and the corresponding reduced independent variable t(i.e., reduced time, etc.) are instructive. Importantly, for a given reaction system, the geometry factor, p, offers an expanded perspective from which a system, or integrated comprehension of reaction order and the conservation laws, can be visualized and appreciated by the student. Significantly, a student’s three - part question, catalyzed a substantial enrichment of an important basic general chemistry course. Additionally, presented, herein, are enlightening aspects of what also transpired, as a result of questions which were posed by students, in a series of various learning scenarios, where listening vs hearing and critical thinking were integral components of the learning environment.
Introduction Two of the best methods to foster and enhance critical thinking skills, in general chemistry, are open inquiry experiments and the practice of solving challenging problems. Occasionally, the process of solving a well formulated chemistry problem can clear up a great deal of misconceptions. This is especially true when simple pictures, images, and analogies, familiar to the students, are constructively used in the explanations. It is also particularly effective when the learning occasion allows a return to basic considerations which enable seemingly unrelated topics to be seamlessly interwoven into clearly defined patterns (1 - 4). Stoichiometry and the law of the conservation of mass present such an opportunity. Generally, the law of the conservation of mass is broadly familiar to most individuals. Of course, it is only one of the basic conservation laws considered to be the most fundamental laws in nature. In general chemistry, the area of study in which the mass quantities of reactants and products are calculated, conserved, and balanced is broadly called stoichiometry. An essential part of a basic knowledge of general chemistry begins with a fundamental understanding of stoichiometry. Previously, the results of an open inquiry experiment were reported (1). This article is an example of the synergistic effects of solving a general chemistry problem where the questions of a student required the thoughtful integration of stoichiometry and some of the conventional perceptions with regards to the law of the conservation of mass. The Answers to a Curious Student’s Questions Part one. During the process of administering an examination, in a first course of basic general chemistry, a student asked a question about some apparent difficulties encountered when attempting to balance the chemical reactions presented as integral parts of a particular problem on the examination. Because the stoichiometry, in the problem, was particularly uncomplicated, it was even more important to carefully listen to the essence of the question being asked rather than, at the moment, simply hearing the particular vocabulary employed by the student. The problem, in question, involved stoichiometry as follows; Problem Statement: A sample containing only CaCO3 and MgCO3 is ignited and the following reaction occurs:

CaCO₃(s) ® CaO(s) + CO₂(g)

MgCO₃(s) ® MgO(s) + CO₂(g)

The ignited sample has half the mass of the original. What percentage of the original sample is CaCO₃?

As it happened, a typographical error, in one of the equations, caused the difficulty. After elimination of the error in question, the examination proceeded uneventfully. Subsequently, in a post-exam review, the solution to the problem was presented in the following typical manner:

Solution A: Let x = original mass of CaCO₃ and 100-x = the original mass of MgCO₃, then 100 grams = mass of the original mixture(CaCO₃ + MgCO₃) and 50 grams = mass of the product mixture(CaO + MgO). Thus,

(1)

where M_m(CaO) is the molar mass of CaO, M_m(CaCO₃) is the molar mass of CaCO₃, M_m(MgO) is the molar mass of MgO, and M_m(MgCO₃) is the molar mass of MgCO₃. Substituting the appropriate values gives

[0.560×x] + [0.478×(100-x)] = 50 grams

which yields

x = grams of CaCO₃ in a sample size of 100 grams = 26.8 grams

Thus, CaCO3 is, simply, 26.8% of the original sample. Part two. On a separate occasion, this same student inquired about a generalized approach to similar problems. Interestingly, a generalized approach surfaced as the course transitioned into discussions of formal aspects of the quantitative treatment of chemical transport, that is, the conversion of reactants into new chemical species. Suppose one has the general reaction

a×A + b×B + ... ® Products

(2)

where the coefficients a, b,.. are the numbers of moles of the respective reactants A, B,.. and A is the limiting reactant. If m represents the total mass of the original reactants and x is the mass of A remaining unreacted after a reaction time of t > 0, then

WF = 1 - W

(3)

where WF is the proportion reacted and the unreacted fraction is W = x/m. From algebraic division, we introduce the following expression

1 ―—— = 1 – x + x^2 – x^3 +…. @ 1 – x 1 + x

for x << 1

(4)

For x = W, eq 4 becomes

1 WF = 1 - W @ ――― 1 + W

for W << 1

(5)

Since the number of moles of A = x/molar weight(A), eq 5 takes the form

A w1×t WF = a× ――― = ――― A0 + A 1 + w2×t

for 0 £ W £ 1

(6)

where the coefficient of proportionality, a, restores equality over the complete range of W(= x/m = A/A0)[since A0 = m/molar weight(A)]. If an effective reaction rate that also incorporates the reaction rate at initiation (i.e., at t = 0) is defined as R*, then the initial concentration A0 = R*×tL where tL denotes the time required for the reaction to go to completion and t = t/tL. An overall reaction rate of R results in A = R×t which defines the coefficients as w1 = a(R/R*) and w2 = (R/R*). Notably, the Peleg equation(3, 4) for osmotic hydration, is a specialized form of eq 6. It is defined as

t w1×t WF = a× ———― = ——— btL + kwt 1 + w2×t

for 0 £ W £ 1

(7)

where t_L corresponds to the terminal value of the time. Thus, the interrelationship, between the regression correlation coefficients, b and k_w is simply

a

b = ——— - k_w

WF_L

(8)

where WF_L is the terminal value of the proportion reacted, w₁ = a/b, and w₂ = k_w/b. The limits of eq 7 are readily apparent when the data, for a given reaction, are normalized to the terminal values of WF_L and t_L (see Problem 1 below). Clearly, from eq 6, the coefficients defined by the specialized application of the Peleg equation are k₁ = t_L/w₁ and k₂ = w₂×(k₁/t_L).

Reaction Geometry Factor, Order, and Consistency Requirements

In addition to eq 5, it has been previously shown (1,2 ) that the conservation of mass (i.e., stiochiometric considerations) and the well defined consistency requirements, for chemical reactions, seamlessly integrate into a single robust function. For any single chemical reaction sequence, the function can be expressed as

WF = ¡×t^p× exp(- k*×t )

(9)

where ¡ is the scaling factor, p is the geometry factor, k*(= p/t_L) is the reaction sequence time constant, t_L (= b×t_f) is the value of the characteristic time length for the reaction sequence. Hence, the correlation coefficient, b, is defined by the differences in the terminal values that result from the nonlinear regressions of eqs 7 and 9 with identical data sets (i.e., b = t_f /t_L).

WF = 0, t = 0 at initiation

and

WF = 1, t = 1 at termination

which gives

¡ = exp(p)

Thus, for a fully normalized single reaction sequence, the proportion reacted is

WF = exp(p)×t^p×exp(- k×t )

or

WF = t^p×exp{p(1 - k×t/p)}

In essence, the relative magnitudes of k∙t versus p result in the following limiting conditions for chemical reactivity

WF @ t^p× exp(p) = ¡∙t^p for k∙t << p

(10a)

WF = t^p for k∙t = p

(10b)

WF @ t^p× exp(- k×t ) for k∙t >> p

(10c)

Since k* = p/t_L, eq 9 becomes

WF = t^p× exp[p(1 - t)]

(11)

where eqs 7 and 9 combine to yield

w₁×t

WF = t^p× exp[p(1 - t)] = ———

1 + w₂×t

(12)

Based on the conventional definition of reaction order(represented as n), the proportion reacted (1) is

WF = 1 – {1 - [(1 - n)×t/t_C] }^1/(1-n)

(13)

where t_C = k^-1∙[A]₀ ^{(1 - n)} and W = {1 - [(1 - n)×t/t_C] }^1/(1-n). Of course, the first order case of n = 1 defines the singularity and eq 13 becomes

WF = 1 – exp(-k×t)

(14)

where W = exp(-k×t). Significantly, eqs 11 and 13 are distinctly different analytical relationships which equivalently assess the progress of chemical reaction sequences. Nonlinear regression analysis of the continuous functional relationship between the conventional reaction order, n, and the geometry factor, p, yields the following quantitative expression

p = 0.487 × tanh(4.316 – [3.550 ×ïnï^0.493]) + 0.510

(15)

A graphical representation of eq 15 is illustrated by Figure 1.

Figure 1. A graphical representation of the p-geometry factor vs n, the conventional order parameter, quantitatively expressed by eq 15.

Thus, the singularity, defined by the first order case for the conventional rate equation, is algebraically reformulated and, simply, restated as a point in a continuum; namely, the boundary condition where t_L = finite for p = 1(see Appendix).

New Problems with Solutions

Significantly, the relationships for the proportion reacted, WF, which followed directly from conservation of mass considerations, provide an additional analytical approach to the analysis of a given set of reaction rate data. To illustrate these properties of WF, two additional general chemistry problems which necessarily involve sets of reaction rate data that exhibit zero, first, and second order behavior are systematically examined. The analyses that follow are presented in the form of problem statements that may be easily incorporated into texts, homework assignments, or tests in general chemistry.

Problem 1

During an examination of the sample repository of a space probe recently returned from the unmanned exploration of a distant asteroid, an unknown compound, A_xB_y, composed of the two unknown elements A and B, was recovered. After a series of tests, it was found that the compound reacted with hydrochloric acid as follows

A_xB_y(s) + z HCl(aq) ® A_xB_yCl_Z(aq) + (z/2) H₂(g)

(I)

In a statistically representative test, a 327 mg sample of A_xB_y was added to a solution of 0.0622 M HCl. The reaction was investigated at atmospheric pressure (i.e., under constant pressure conditions) and 50.4 °C. The data were collected and are reported in the following table as

Table 1. Symbolic Kinetic Data for the Hypothetical Chemical Reaction(II)
Time (s)	[HCl] (mol/L)	H2-Volume (cc)
0	0.0622	0.00
50	0.0514	2.66
100	0.0448	5.33
150	0.0402	8.00
200	0.0368	10.7
250	0.0341	13.3
300	0.0319	16.0
350	0.0301	18.6
400	0.0286	21.3
450	0.0273	23.9
500	0.0261	26.6
550	0.0251	29.2
600	0.0242	31.9
650	0.0234	34.5
700	0.0227	37.2
750	0.0220	40.0

Based on additional tests and analyses, it was found that z = 3. From the data collected in the table, A_xB_y is the limiting reactant. Use the data table to obtain the following: (a) determine the molar mass of the unknown compound A_xB_y and the initial volume of the HCl reactant; (b) determine the order of the reaction with respect to the individual reactants as well as the overall order of the reaction; (c) determine the specific rate constant; (d) plot graphs of the WF vs t relationships for HCl consumption and H₂ production respectively; (e) plot a graph which illustrates the summary relationships between the geometry factors k_w(eq 7) and p(eq 11) versus n(eq 13). [Density of H₂ (15 °C) = 0.085 kg/m³ = 0.04217 mol/L]

Solution

(a) The combined gas law, at constant pressure(i.e., Charles’ law), is

V₁/T₁ = V₂/T₂

(16)

where V₁ is the gas volume at temperature(in degrees Kelvin) T₁ and V₂ is the volume at temperature T₂. Since density = r = mass/volume = m/V, eq 16 can also be expressed as

(r₁×T₁) = (r₂×T₂)

(16a)

(b) Thus, from eq 16a, the density of H₂ gas produced, in the reaction for constant pressure conditions, is

r₂ = r₁×(T₁/T₂)

where substitution of the appropriate values yields,

r₂ = (0.04217 mol/L)×(15.0 °C + 273.15 K)/(50.4 °C + 273.15 K) = 0.03756 mol/L

The molar quantity of H₂ is

(0.03756 mol/L)×(40 cc)×(10^-3 L/cc) = 0.00150 mol

Therefore, the molar mass of A_xB_y is

(327 mg)×(10^-3 g/mg)/(2/3)×(0.00150 mol) = 327 g/mol

The initial volume of the HCl reactant is determined as follows:

(initial concentration)×(initial volume) = moles reacted + moles unreacted

The data table and the reaction stoichiometry yield

(0.0622 M)×V = 2(0.00150 mol) + (0.0220)×V

Therefore,

V = 0.00300 mol/(0.0622 – 0.0220)M = 0.0746 L = 74.6 mL

(c) The reciprocal of the square of the HCl concentration (i.e. [HCl]^-2) plotted vs time yields a linear relationship (Figure 2- Part A). The volume of H₂ plotted vs time yields a linear relationship (Figure 2-Part B).

A. B.

Figure 2. Part A. Plot of [HCl]^-2 against time. Part B. Plot of the volume of H₂ against time.

The simple third – order rate law, for the disappearance of HCl, has the form,

d[HCl/dt = - k[HCl]³

which, according to calculus, gives the integrated form

½ × {(1/[HCl]²) – (1/[HCl]₀ ²)} = kt

Therefore, the reaction is pseudo - third order in HCl. Since the reactant A_xB_y(s) is a solid, the reaction order is zero with respect to A_xB_y(s). Hence, overall, the order of the reaction is third order.

(d) The specific rate constant k = 1.21 L² mol^-2 sec^-1.

(e) The Figure 3 (Part A) is a graphical illustration of the proportion reacted(WF) vs reduced time(t) for p = 0.0495 (i.e.,

n = 3) which represents the progress of the reaction system in terms of the consumption of the acid reactant HCl. Part B, which represents the parallel production of H₂ gas{i.e., corresponds to the consumption …. of A_xB_y(s)}, is an illustration of p = 1 (i.e., n = 0). Hence,

A. B.

Figure 3. Part A. represents the proportion reacted(WF) vs. the reduced time(t) for the consumption of the acid reactant, HCl. Part B. represents the corresponding relationship for the simultaneous production of H₂ as indicated by reaction eq(II).

(f) Illustrated in figure 4 are the summary results of a comparative evaluation of the geometry factors k_w(eq 7) and p(eq 11)   vs n(eq 13).

Figure 4. A graphical illustration of the quantitative relationship between the geometry factors, k_w() and p(o), vs n (conventional order parameter) where the proportion reacted(WF) vs the reduced time(t) relationship is constrained to the normalized form(i.e., WF_L = 1, t_L = 1)[see eq 7].

Clearly, the point of intersection at k_w = p = n @ 0.9 is unique.

Problem 2

A student decided to investigate the characteristics of the solid-liquid magnesium hydrochloric acid single replacement reaction

Mg(s) + 2 HCl(aq) ® MgCl₂(aq) + H₂(g)

(II)

The student performed the experiments in a simple reaction-controlled flow unstirred reactor (RCFUR) (1) (i.e., constant pressure). The system was maintained at a constant temperature of 32.7 °C and pressure of 1.00 atm. The data are tabulated below.

Table 2. Data for the Mg(s) + 2 HCl(aq) ® MgCl2(aq) + H2(g) … Reaction
Time (min)	[HCl] (mol/L)	H2-Volume (cm^3)
0	0.0368	0.0
1	0.0106	1.54
2	0.00618	3.09
3	0.00436	4.63
4	0.00337	6.17
5	0.00275	7.72
6	0.00232	9.26
7	0.00200	10.8
8	0.00178	12.3
9	0.00158	13.8
10	0.00143	15.4
11	0.00130	16.9
12	0.00120	18.5
13	0.00111	20.0

Use the data table to obtain the following: (a) determine the initial volume of the HCl reactant; (b) calculate the grams of Mg metal initially present; (c) express the rate law and determine the specific rate constant for this reaction; (d) calculate the p – geometry factor and construct a graph of the WF vs t relationship for the HCl reactant consumption sequence.

Solution

(a) The density of H₂ gas produced, in the reaction for constant pressure conditions(see eq 16 and problem 1), is

r₂ = r₁×(T₁/T₂)

where substitution of the appropriate values yields,

r₂ = (0.04217 mol/L)×(15.0 °C + 273.15 K)/(32.7 °C + 273.15 K) = 0.03929 mol/L

The molar quantity of H₂ is

(0.03929 mol/L)×(20 cc)×(10^-3 L/cc) = 0.000786 mol

The initial volume of the HCl reactant is determined as follows:

(initial concentration)×(initial volume) = moles reacted + moles unreacted

The data table and the reaction stoichiometry yield

(0.0368 M)×V = 2(0.000786 mol) + (0.00111)×V

Therefore,

V = 0.00157 mol/(0.0368 – 0.00111)M = 0.0440 L = 44.0 mL

(b) The quantity of Mg metal initially present is

(0.000786 mol)(24.3 g/mol) = 0.0191 g = 19.1 mg

(c) As illustrated in Figure 5, a plot of the reciprocal of [HCl] vs time yields a straight line relationship

Figure 5. Plot of [HCl]^-1 against time.

The second – order rate law, for the disappearance of HCl, has the form,

R = -d[HCl]/dt = k[HCl]²

Inasmuch as Mg(s) is a solid(i.e., zero order-see Problem 1) and the reaction is second order with respect to [HCl] consumption, the overall order for the reaction is second order. Therefore, the integrated form of the rate law, gives

1/[HCl] = k×t + 1/[HCl]_o

Regression of the data (Table 2, Figure 5) yields the specific rate constant k = 67.4 L mol^-1 min^-1.

(d) Based on eq 15, p = 0.221 for a second order reaction. Figure 6 is a graph of the WF vs t relationship for the HCl

reactant consumption sequence.

Figure 6. A graphical representation of the proportion reacted(WF) vs. the reduced time(t) for the consumption of the acid reactant, HCl(see

Figure 5).

Problem 3

The copious amounts of acid oxides (i.e., SO₂, NO₂) released into the atmosphere, by industrialized countries, results in acid rain. Subsequently, this acid rain does serious environmental and property damage by reacting with the calcium carbonate content of Earth, limestone, marble, buildings, monuments, etc. A method for the determination of calcium carbonate in eggshells was developed by means of a systematic investigation of the corrosive reaction of CaCO₃ with HCl(5):

CaCO₃(s) + 2 H⁺(aq) ® Ca²⁺(aq) + CO₂(g) + H₂O(l)

(III)

The mass of the sample sizes were 0.01 g for sample 1, 0.03 g for sample 2, and 0.05 g for sample 2. All experiments were performed with the acid in large excess. The reaction vessel had a gas containment volume of 43.2 mL and the temperature was maintained at 298.15 K. The data for three representative experiments are tabulated below.

Table 3. Data for the … CaCO3(s) + 2 H^+(aq) ® Ca^2+(aq) + CO2(g) + H2O(l) … Reaction
Time (min)	1.) 0.01 g CaCO3(s)	2.) 0.03 g CaCO3(s)	3.) 0.05 g CaCO3(s)
	CO2 (kPa)	CO2 (kPa)	CO2 (kPa)
0	0.00	0.00	0.00
1	1.68	5.05	8.42
2	2.87	8.60	14.3
3	3.70	11.1	18.5
4	4.29	12.9	21.4
5		14.1	23.5
6		15.0	25.0
7		15.6	26.0
8			26.7

Use the data table to obtain the following: (a) for each reaction sequence(i.e., 1, 2, and 3), calculate the expected final pressure from the mass of CaCO₃ initially present; (b) determine the reaction order and express the rate law for this reaction; (c) determine the specific rate constant for this reaction; (d) calculate the p – geometry factor of the reaction and construct a graph of the WF vs t relationship for the reaction; (e) compare the first order case as a singularity with a single quadrant of a circle and the golden ratio.

Solution

(a) Based on the stoichiometry of reaction(III) and assuming ideal gas behavior and the initial mass CaCO₃sample, of each CaCO₃ sample, the expected final CO₂ pressure, from the decomposition of each original sample, is

P = [(Grams of CaCO₃)/M_m(CaCO₃)]×(RT/V)

where M_m(CaCO₃) is the molar mass of CaCO₃, P is the final pressure, V is the volume, R is the gas constant and T is

the absolute temperature. Hence, the expected final CO₂ pressures, for the reaction sequences, are respectively:

1) Pressure of CO₂ = [0.01 g/(100.01g/mol)]×[(0.08206 L atm K^-1mol^-1)×(298.15 K)/(0.0432 L)]×(101.3 kPa/atm) =

5.74 kPa

2) Grams of CaCO₃ = [0.03 g/(100.01g/mol)]×[(0.08206 L atm K^-1mol^-1)×(298.15 K)/(0.0432 L)]×(101.3 kPa/atm) =

17.2 kPa

3) Grams of CaCO₃ = [0.05 g/(100.01g/mol)]×[(0.08206 L atm K^-1mol^-1)× (298.15 K)/(0.0432 L)]×(101.3 kPa/atm) =

28.7 kPa

(b) Since the progress and stoichiometry of reaction(III) requires

d[HCl] dP(CO₂)

- ¾¾¾¾ = + ¾¾¾¾

dt dt

Thus, for a given sequence, the difference between the final CO₂ pressure (P_f) and the CO₂ pressure, at any other time (i.e., t ¹ t_f) during the progress of the reaction (P_t), defines the quantity P(DCO₂) = (P_f - P_t). In essence, each change in P(DCO₂) directly reflects the corresponding change in [HCl]. Plots of the logarithm of the CO₂ pressure changes, in terms of P(DCO₂) against time (Table 3), are shown in Figure 7.

Figure 7. The natural logarithm of CO₂ pressure changes [i.e., Ln{P(DCO₂)}] plotted against time. The CaCO₃ masses are represented as (¨) = 0.01 g, (·) = 0.03 g, and (�) = 0.05 g.

The straight line relationships confirm first order dependence. Thus, reaction(III) is pseudo-first order with respect to the concentration of [HCl]. Hence, the rate law

R = -d[HCl]/dt = k[HCl]

which, from calculus, integrates to

kt = ln[HCl]₀ - ln[HCl]_t = lnP_f – ln{P(DCO₂)}

[HCl]₀ represents the initial concentration of HCl and [HCl]_t represents the HCl concentration at time t.

(c) From (b) and figure 7, the specific rate constant is found to be k = 0.0059 sec^-1.

(d) From eq 15, p = 0.824 for the first order case(i.e., n = 1) and the following figure(i.e., fig 8) is a graphical illustration of the WF vs t relationship.

Figure 8. A graphical representation of the proportion reacted(WF) vs. the reduced time(t) for the first order case [see Reaction(III)].

(e) The first order case defines a singularity. For a complete discussion, see Appendix.

Conclusions

In the present article, it has been demonstrated that conservation of mass considerations, fundamental to balancing chemical reactions, can be seamlessly extended to directly incorporate reaction rate and order. Expressed in terms of the proportion reacted versus an appropriate independent variable, such as time, results in an analytical relationship defined by a geometry/curvature factor, p, and the length of the reaction sequence, t_L. Application of this relationship to three different problems in general chemistry, demonstrate the utility of this relationship as a diagnostic as well as a learning tool. In fact, it is suggested that important aspects of basic stoichiometric and reaction rate considerations can be naturally integrated much earlier in the general chemistry curriculum.

Importantly, it is seen that the removal of the singularity, inherent to the first order case for the conventional rate equation, is an algebraic reformulation in which the singular point is simply redefined as the boundary condition t_L = finite for p = 1. In effect, the first order singularity is restated as a point in a continuum. The fact that the progress of many other chemical reactions can be expressed in terms of a single dimensionless variable enhances the utility of this approach as an additional tool in the analysis of reaction rate data.

Finally, it is shown how a student’s three - part question resulted in the significant enrichment of a general chemistry course. It is yet another example of what can happen when enlightening questions are strategically interjected at appropriate moments in the learning process. Occasionally, it is important to be reminded of the significant difference between listening and hearing; because, questions are the windows to understanding. To listen well, becomes even more critical when exchanging information with students during the formative stages of learning and understanding basic science.

Appendix

Reference Standard. The singularity or first order case, defined by n = 1 for conventional reaction order (eq 13), is redefined, by the conservation of mass approach (eq 11), to the non-zero equivalent of p = 0.824 (eq 15). Inasmuch as the curvature of a given circle is always defined by the reciprocal of the radius of the circle in question, the circle is designated as the appropriate internal reference standard for the p-geometry factor. A single quadrant of a circle (Figure 9) yields a characteristic value of p = 0.628(i.e., n = 1.32).

Figure 9. An illustration of p = 0.628(■), for one quadrant of a circle, versus the conventional first order case of n = 1[p = 0.824](o) where red. dep. var. = reduced dependent variable(WF) and red. indep. var. = reduced independent variable(t).

In essence, since 2p radians @ 6.28 for one revolution, the value of p = 0.628(@ 2p/10) is consistence with that of a circle (Figure 10).

Figure 10. An illustration of p = 0.628(■), for one quadrant of a circle, versus the conventional first order case of n = 1[p = 0.824](o) where red. dep. var. represents the reduced dependent variable (WF) and red. indep. var. represents the reduced independent variable (t).

It is somewhat instructive to note that the value of p = 0.628 is positioned as follows

p = 0.824 < 0.628 < p = 1/j = 0.618

where j = golden ratio. Furthermore, the absolute value of the ratio of the differences between values of p and n for the singularity relates to j as follows

[p(singularity) – p(j)]/[n(singularity) – (j)] = (0.824 – 0.618)/(1 – 0.333) = 0.618

Additionally, the value of p = 0.628 is only 1.6% greater than p = 0.618(i.e., n = 1.333) @ 1 - j @ 1/j which is exactly the value obtained for one quadrant of the logarithmic spiral (Figure 11).

Figure 11. Right-handed logarithmic spiral(, i.e., r = e^-bq ) where the radial distance(r) = red. dep. var., the angle of rotation(q) = red. indep. var., and b is a constant.

The ratio of the respective values of p for the singularity (i.e., n = 1) to that of the reference standard (i.e., a quadrant of a circle) is 1.312(= 0.824/0.628). The ratio of p for the singularity to p = 0.618 is 1.333(= 0.824/0.618). In essence, based on eq 10, the ratio of the value of the p - geometry factor for the singularity (i.e., n = 1) to the p - geometry factor value for the reference standard (i.e., a quadrant of a circle) is 1.31( = 0.825/0.628). The value of the corresponding ratio is 1.33 (= 0.825/0.618) for the golden ratio.

Finally, based on eq 13, the singularity as uniquely defined in terms of conventional order (i.e., n = 1), yields

t_C = k^-1∙[A]₀ ^{(1 - n)} = k^-1

Since k = k* = p/t_L, one has

t_C = t_L/p

(17)

which is a disarmingly simple and significant result that follows directly from the boundary constraints of the consistency requirements (1,2).

Also, presented elsewhere (3), application of eqs 11 and 12 to the analysis of first order osmosis kinetics of hydration data result in additional considerations and implications relative to the Peleg equation (4).

References

1. Maloney, K. M.; Frye, A. C. http://kennethmaloney.com/Publications/ArticleBasic2Eqn.htm.

2. Maloney, K. M. http://kennethmaloney.com/Publications/Article1BasicEqn2002July16.htm, 2003.

3. Maloney, K. M. http://kennethmaloney.com/Publications/AspectsOsmosis1.htm.

4. Pinto, G.; Esin, A. J. Chem. Educ. 2004, 81, 532-536.

5. Choi, M. M. F.; Wong, P. S. J. Chem. Educ. 2004, 81, 859-861.

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