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An Interdisciplinary Approach: A General
Perspective of Characteristic Aspects of the Kinetics of Osmotic Hydration
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Kenneth M.
Maloney
Chemistry and Physics Division,
Baton Rouge Community College, 201 Community College Drive, LA 70806
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The Peleg
equation, which is an empirical relationship, has been shown to adequately
predict, characterize, and model the hydration process for a variety of
important food products. Additionally, the hydration,
dehydration-rehydration and adsorption relationships are established over a
range of temperatures. Thus, it is, particularly, significant that the
empirically based Peleg equation is also a special limiting form of a more
basic relationship which follows directly from fundamental transport, conservation,
and consistency requirements/considerations.
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Introduction
The balance
between water loss and water absorption in geological (1), biological (2),
engineering systems(3), and etc (4), is one of the oldest scientific problems.
The regulation of the transport of water across biological membranes is
fundamental to life, the maintenance of homeostasis between body
fluid compartments, and the preservation of the organism under
adverse conditions. In fact, the fundamental discovery and characterization
of the first water channel protein highlights the crucial role of
specialized ion channel functions in the maintenance of water balance in
all organisms (5). On the other hand, reverse
osmosis is often used in commercial and residential water filtration. It is
also one of the methods used to desalinate seawater. Sometimes reverse
osmosis is used to purify liquids in which water is an undesirable
impurity. Therefore, fundamentally, osmosis naturally lends itself
to an interdisciplinary approach to understanding the interdependence of a
number of basic principles and laws in several important scientific
disciplines routinely offered at the basic levels of college instruction
(6).
In addition
to being a colligative property, osmosis is formally defined as the
diffusion of a solvent, such as water, through a semi-permeable membrane
(7). The solvent usually travels from the side with the greater
concentration of the solution to the side with the lesser concentration.
The movement of solvent (i.e., rate) occurs in opposition to the
countervailing potential barriers established by the applicable
intermolecular interactions (e.q., Van der Waals, London dispersion,
hydrogen bonding, etc.) and components within the solution as well as the
selective permeability of the membrane(i.e., resistance). As is the case
with other basic phenomena, such as heat and electrical conduction (8), the
primary relationship of rate = potential/resistance is also embodied in the
characteristic transport processes fundamental to osmosis. Specifically, in
terms of mass transfer under isothermal conditions, the relationship, rate
= potential/resistance, can be expressed as
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DC
Rate = D∙A∙
DL
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(1)
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where D is the diffusion constant for mass transfer
through a channel of length L(resistance) and cross-sectional area A. The
driving force is the difference in concentration[DC](potential) between
the ends of the channel(DL). Eq (1) is referred to as
Ficks law of diffusion (8). From eq 1, the mass transfer kinetics, limited
to the process of osmosis, can take the form
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dM(t)
- = ktr∙A∙C(t)
dt
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(2)
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where ktr = D/DL is the transfer
coefficient and C(t) is the concentration from which the driving force for
osmotic hydration originates. In effect, the potential difference arises as
a result of the concentration difference between the immersion liquid[CIL(t)]
and the sample[CS(t)], respectively (i.e., C(t) = [CIL(t)]
- [CS(t)]). Restatement of Eq (2) in an equivalent form yields
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dC(t)
- = k∙C(t)
dt
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(3)
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where k(time-1) is the specific rate constant
for osmotic hydration. Integration of eq (3) yields
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C(t) = Cmax∙exp(-kt)
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(4)
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where Cmax is the maximum concentration of solvent
in the immersion liquid at t = 0. In terms of the proportion hydrated(WF),
one has
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WF = 1 - C(t)/Cmax
= 1 - exp(-kt)
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(5)
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Ostensibly, the
mass transfer kinetics, inherent to the processes of diffusion and osmosis,
take the general form applicable to first order processes. Therefore, eq
(5) also applies to first order chemical reactions. In effect, based on the
fundamental principles of the conservation of mass and energy, in addition
to a well defined set of consistency requirements - a quantitative
relationship, which expresses the proportion reacted of a chemical reaction
as a function of an independent variable, was formulated (9,10) as
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WF = tpΧ
exp[p(1 - t)]
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(6)
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The dependent variable, WF, represents the proportion reacted,
p is the geometry factor, and the independent variable, t(=
t/tL), represents the reduced time where tL equals
the terminal value of t. Importantly, however, when WF is redefined to be
the proportion hydrated, the same general relationship also applies to
osmosis. Consider the fact that an expression, which is analytically
equivalent to eq (6), is simply
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w1Χt
WF =
1 + w2Χt
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(7)
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where w1 and w2
are scaling coefficients and the geometry factor p = w2
/w1.
However, in the special limiting cases, where the final values of t(i.e., t
= tf) are less than the actual terminal values which results
from the constraints of the consistency requirements (9, 10)(i.e., tL
> tf = k1), the scaling coefficients will then
assume the values of w1 = tL/k1
and w2
= tL∙k2/k1. Given that the Peleg
equation (11), for osmotic hydration, is
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t
WF =
k1
+ k2Χt
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(8)
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the proportion hydrated (WF) becomes WF = {M(t) M0}/M(k1)
where the moisture contents M(t), M0, and M(k1) are
the respective values of the solid(w/w) for the times 0 < t < k1,
t = 0, and t = k1. Clearly, one has the fact that the
equivalence of eqs (6), (7), and (8) establishes a fundamental basis for
the Peleg equation which extends far beyond its empirical origins.
Typically, the prevailing experimental requirements/limitations are such
that k1 < tL. In essence, eq 7 is a useful
generalization of eq 8 where k2 = kw∙p.
The factor kw
is a scaling accommodation coefficient of the osmotic hydration and, thus,
ideal behavior pertains whenever kw
= 1.
Application
Recently, a
systematic experimental investigation of the osmotic hydration kinetics of
chickpeas was presented (6). Figure 1 is a representative plot of the data
presented as the proportion hydrated(WF) versus reduced time.
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Figure 1.
Representative plot of the proportion hydrated(WF) vs reduced time(t).
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Table 1. Hydration Parameters, [NaCl] = 0
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T
°C
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k1
min
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tL
min
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w1
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k2
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w2
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p
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kw
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5
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194
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435
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2.24
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1.02
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2.29
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0.88
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1.16
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20
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132
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365
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2.76
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1.00
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2.75
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0.83
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1.19
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30
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94.5
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345
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3.65
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0.91
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3.34
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0.77
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1.19
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40
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75.5
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314
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4.16
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0.82
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3.43
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0.82
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1.07
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50
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59.6
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284
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4.77
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0.78
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3.71
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0.76
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1.03
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Table 2.
Hydration Parameters at T = 20 °C
vs [NaCl]
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[NaCl]
Mol/L
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k1
min
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tL
min
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w1
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k2
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w2
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p
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kw
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0.0
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132
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365
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2.76
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1.00
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2.75
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0.83
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1.19
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0.5
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139
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287
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2.07
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1.62
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3.34
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0.80
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2.03
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1.0
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157
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289
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1.84
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1.92
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3.52
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0.77
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2.48
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3.0
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180
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287
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1.60
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2.04
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3.26
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0.80
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2.55
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On the basis of eq (6), the osmotic hydration, R(t) =
dWF/dt, is
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dWF dWF
R(t) = = =
p∙{(1/t) - )}∙WF/tL
dt tL∙ dt
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(9)
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Interestingly, a simple arithmetical average of the geometry
factors, as presented in Tables 1 and 2, yields the average value of
p = 0.81. Substitution of p = 0.81 and t
= 0.5 into eq (9) yields the unique result of R(t) = 0.693 where the
initial and terminal values of t and WF are both 0 and 1
respectively. Thus, one has the expected agreement with the well known
relationship, k = 0.693/ t1/2, for first order
reactions/processes where k is the specific rate constant and t1/2
is the corresponding half life.
The values of
k, determined for the first order case [i.e., eq (5)], results in an
Arrhenius activation energy(Ea) = 11.3 kJ/mol (Figure 2).
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Figure
2. Logarithm[rate(min-1)
= k, IHR] vs reciprocal temperature (k1-1 = initial
hydration rate[IHR]).
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Figure 2 presents a comparative illustration of the
logarithm of the k and k1-1(= initial hydration rate[IHR])
values vs reciprocal temperature. Note that, although IHR decreases as the
concentration of NaCl increases, the value of the geometry factor (p)
remains essentially invariant. The role of non-Fickian fluxes, associated
with active ion (Na+ and Cl ) transport
of water across biological membranes, is an important component in a number
of different mechanisms and pathways which necessarily involve different
energetics (12-18). As was indicated elsewhere (6), the activation energy,
derived from the initial hydration rate[IHR], gives a value of Ea
= 19.5 kJ/mol. Significantly, both results for the hydration of the chickpeas
are either, approximately, equal to (i.e., 19.5 kcal/mol) or less than
(11.3 kJ/mol) the activation energy of 19.2 kJ/mol for the self-diffusion
of water (19).
Notably,
molecular interactions are typically between 0.1 10 kJ/mol for Van Der Waals
forces and 10 40 kJ/mol for hydrogen bonding; whereas, chemical bonds are
routinely in the range of 100 1000 kJ/mol. Processes driven by osmotic
energy or salinity gradient energy have been reported in the range between
0.1 100 kJ/mol (12, 20) which approach that of weak chemical bonds at the
upper end of the range. Thus, the development of methods and experiments
that demonstrate the relative importance of different mechanisms/energetics
of water transport across well characterized membranes are
excellent enrichment opportunities for the undergraduate science program
across the disciplines.
Summary
It is shown
that the same basic equation which provides an integrated approach to the fundamental
considerations of stoichiometry and reaction kinetics also applies to the
kinetics of osmotic hydration. The moisture uptake is expressed in terms of
the proportion hydrated which is analogous to the proportion reacted
approach previously formulated and applied to chemical reactions.
Significantly, it is shown that the Peleg equation is a special limiting
form of this basic equation. As an important illustration, the utility of
this approach is demonstrated by the application of this basic equation to
the hydration of chickpeas. It is suggested that this complimentary
approach is a learning tool which explicitly illustrates the
interdependence of the conservation of mass in multifaceted processes such
as osmosis as well as chemical reactions systems routinely presented at the
basic levels of instruction. Such interdisciplinary approaches are needed
in the education of undergraduates in order to minimize the mistaken ideas
currently held by many that there are well defined barriers of separation between
the various scientific and engineering disciplines.
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Copyright © Kenneth M. Maloney, Ph.D.
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