Chem. Senses 26: 1015-1022,
2001
© Oxford University Press 2001
Determination of Saltiness from the Laws of Thermodynamics—Estimating the Gas Constant from Psychophysical Experiments
Department of Physiology and Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Canada
Correspondence to be sent to: K.H. Norwich, Institute of Biomaterials and Biomedical Engineering, University of Toronto, 4 Taddle Creek Road, Toronto, Ontario, Canada M5S 3G9. e-mail: k.norwich{at}utoronto.ca
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Abstract |
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 Top Abstract IntroductionMethodsResultsDiscussionConclusionsAppendix: the molar entropy...Glossary of symbols used...References |
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One can relate the saltiness of a solution of a given substance to the concentration of the solution by means of one of the well-known psychophysical laws. One can also compare the saltiness of solutions of different solutes which have the same concentration, since different substances are intrinsically more salty or less salty. We develop here an equation that relates saltiness both to the concentration of the substance (psychophysical) and to a distinguishing physical property of the salt (intrinsic). For a fixed standard molar entropy of the salt being tasted, the equation simplifies to Fechner's law. When one allows for the intrinsic `noise' in the chemoreceptor, the equation generalizes to include Stevens's law, with corresponding decrease in the threshold for taste. This threshold reduction exemplifies the principle of stochastic resonance. The theory is validated with reference to experimental data.
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Introduction |
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TopAbstractIntroductionMethodsResultsDiscussionConclusionsAppendix: the molar entropy...Glossary of symbols used...References |
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Sensory organs, the organelles of gustation not excepted, are adept at detecting changes in the sensory environment. Sustained sensory signals, particularly those regarding taste, tend to become muted due to processes of adaptation. Since `change' implies an element of uncertainty, the process of sensing can be represented by an uncertainty or entropy function with some success (Norwich, 1977
,
1993). However, the entropy
function employed in these earlier works was a generic function, whose
parameters had to be evaluated from the data. We attempt here to represent the
taste of salinity in terms of the standard thermodynamic entropy of the
substance being tasted. The chemical senses, especially the sense of taste of
saltiness, seem particularly suitable for this trial, because of the relative
simplicity of the chemical stimulus compared, for example, with visual or
auditory stimuli. The endeavor to represent the intensity of taste solely in
terms of physical and chemical quantities, which we undertake here, is
compelling from a number of points of view. It can potentially cast some light
on the fundamental concept of sensation—what it means to sense the
environment—and it permits, in principle, `modeling' the sensory
activity with minimal use of adjustable parameters: most parameters should
emerge from the associated physics and chemistry. A mathematical function
developed in this way, based largely on quantities that are exterior to
biology, either succeeds in its nascent state or it does not succeed; there
can be very little adjustment of parameters. We shall develop such a model and
confirm its accuracy to a degree using existing experimental data. We are concerned here with saltiness. The particular physical quantity that we suggest as a measure of the sensation of saltiness is the partial molar entropy of the tastant solution. It should be stressed that, although physical entropy is regarded as a measure of uncertainty or molecular disorder, it is, none the less, measured in the laboratory by very tangible means: calorimeters, thermometers, etc. The units of entropy are joule.degree-1.mole-1. The reason for selecting saltiness for a first effort is because we shall need some standard values of entropy to evaluate our theoretical equation, and some of these values have been measured and are readily obtained for strong electrolytes such as sodium and potassium chloride, which convey a salty taste.
The equations we require for our `law of sensation' are well known in physical chemistry, and we shall just state them without derivation in the body of the paper. However, in order to give the reader a feel for these equations, we provide a brief derivation, in the Appendix, beginning from the first two laws of thermodynamics. The Appendix leads us from rudiments to equation (1).
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Methods |
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TopAbstractIntroductionMethodsResultsDiscussionConclusionsAppendix: the molar entropy...Glossary of symbols used...References |
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Development of theory
We consider a solution of a strong electrolyte, such as sodium chloride,
which dissociates into two monovalent ions (1:1 electrolyte):
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 | (2) |
 | (3) |
 | (4) |
In order to complete the derivation, we must allow for the absolute
threshold for the taste of saltiness. Modern methods for evaluating thresholds
employ quite sophisticated statistical techniques. However, for the
approximate model presented here, we shall introduce the threshold by adding
the constant 2S0 to both sides of equation (4). We thus
obtain
 | (5) |
 | (6) |
is the activity coefficient, which governs the deviation of the
solution from ideality. For our current approximation, we shall take =
1, its value for an ideal solution. Thus our approximate equation for 1:1
salts can be written
 | (7) |
Comparison with Fechner's law
Equation (7) can now be compared with Fechner's law, which is usually
written in the form
 | (8) |
is a measure of saltiness, C is the concentration of the
tasted substance, and 
and 
are constants that are >0. For
concentrations that are small, we can approximate equation (8) by
 | (9) |
, with = S0
and ß = 2R. In other words,
 | (10) |
We can now proceed to validate the theoretical equation (7) using data that
have already been published by von Skramlik
(von Skramlik, 1926) and cited
by Pfaffmann et al. (Pfaffmann
et al., 1971). These results are now summarized by the
equation
 | (11) |
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Results |
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TopAbstractIntroductionMethodsResultsDiscussionConclusionsAppendix: the molar entropy...Glossary of symbols used...References |
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Validation of the theory based on the data of von Skramlik
Von Skramlik (von Skramlik,
1926) compared the saltiness of five salts to the saltiness of
sodium chloride. The five salts were NH4Cl, KCl, CaCl2,
LiCl and MgCl2. In Tables
1 and
2 are given the values of the
ratio (M NaCl/M salt), the molar concentrations of sodium chloride required to
match the saltiness of a comparison salt. Implicit in these data is the
assumption that altering the molar concentration of a salt will not alter the
value of the above comparison ratio.
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Von Skramlik made similar comparisons of sodium chloride to iodides, bromides, sulfates, nitrates and bicarbonates of NH4, K, Ca, Li and Mg. However, we shall restrict our discussion to chlorides.
It is now a simple matter to adapt our generalized form of Fechner's law for analysis of the von Skramlik experiments.
Salts with monovalent cations (1:1 salts)
We now evaluate von Skramlik's ratio, (M NaCl/M salt), using equation (7).
Since the two substances, NaCl and the comparison (1:1) salt, tasted equally
salty, we can equate the right-hand side of equation (7), written for NaCl,
with the corresponding quantity written for a comparison salt. That is
 | (12) |
 | (13) |
For concentrations that are not too great, we can take the ratio of mole
fractions on the left-hand side of this equation as a good approximation of
the von Skramlik ratio introduced in the previous section. The quantities,
S0, can be evaluated with reference to equation (4). When
a = X = 1, then S = S0. That
is, when the mole fraction of the salt is equal to unity, the molar entropy is
equal to the standard molar entropy at the reference temperature of 25°C.
But when the mole fraction is equal to unity, the molar entropy might be taken
to be that of the pure (liquid) form of the salt [hypothetically, as discussed
by Smith (Smith, 1967)].
This `fused liquid salt' may or may not exist in reality; however, we can
approximate S0 by the molar entropy of the pure
crystalline solid at 25°C. S0 for the crystalline
solid will be less than S0 for the hypothetical fused salt
because of the extra degrees of freedom possessed by the liquid form. However,
because of the identical crystal structures of LiCl, NaCl, KCl and
NH4Cl, the differences between S0 for the fused
salt and S0 for the crystalline solid can be expected to
remain approximately constant. Because of this constant difference, we shall
be able to use values of S0 for the crystalline form, to
estimate the value of R from biological data. Therefore, we take
S0 equal to the molar entropy of the pure crystalline salt
at 25°C. For chloride salts, these entropies have been measured in
joule.mole-1.deg-1 or
cal.mole-1deg-1 and can be obtained from standard books
of tables (Weast and Astle,
1981).
We can now plot the values of the von Skramlik ratio for each salt with
monovalent cation (LiCl, KCl and NH4Cl) against the standard molar
entropy for that salt (Table
1). The result is shown in
Figure 1. The corresponding
points for the two salts with divalent cations (calcium and magnesium) have
also been plotted for comparison. It may be seen that the four salts with
monovalent cations are arrayed linearly along the regression line given by
 | (14) |
 | (15) |
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Some of this material has been introduced recently
(Norwich, 2000).
Salts with divalent cations (2:1 salts)
The corresponding case for CaCl2 and MgCl2 is not
quite as simple as that of the monovalent ions. We now must write one equation
of the form of equation (1) for each of three ions, e.g. for Ca++,
Cl-. We now obtain the equation
 | (16) |
Validation of the theory based on Stevens's data
If equation (7) were valid for very low concentrations of salt, the
sensation of taste would equal zero (i.e. threshold) when
 | (17) |
The mole fraction, X, is given approximately by
 | (18) |
 | (19) |
The value of M is termed a quasi- or seeming-threshold because
Fechner's law is not valid for very small concentrations. Rather we must
employ Stevens's power law in this region, as shown for sodium chloride in the
data in Figure 2. Inverting the
latter equation,
 | (20) |
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The gas constant, R, is equal to the product of Boltzmann's
constant, k, with Avogadro's number, N0, so we
can express equation (20) in an interesting alternative format. Dividing both
sides of this equation by N0, we obtain
 | (21) |
Since the quasi-threshold, M, may be read off a graph of magnitude
estimate against ln (molarity) and the standard molal entropy,
S0, is found in tables, one can, quite unexpectedly,
obtain a psychophysical measure of the gas constant, R. With
reference to Stevens's data (Stevens,
1969), the right-most six data points fall on a straight line on a
semi-log plot (Figure 2), in
accordance with Fechner's law and equation (7). The least-squares regression
line for these points is
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However, calculation of R from equation (20) may contain sizeable errors. First, the calculation depends on the value of M, which issues from measurements of subjective magnitude that are quite variable. Second, M may be too small due to dilution by saliva etc., which will increase the calculated value of R. Third, S0 for the crystalline solid is less than it would be for the hypothetical fused salt, which will, again, reduce the calculated value of R.
The same kind of calculation can be made from the magnitude estimation data
of Meiselman et al. (Meiselman
et al., 1972) for sodium chloride. There are fewer data
points available, and one can use only the upper three points, obtaining
M = 0.181 molar and R = 6.32
joule.mole-1.deg-1. It must be appreciated that these
estimates are not offered as a substitute for proper physicochemical
evaluations of the gas constant; they are, rather, offered as a means of
partial validation of the principle of physical entropy as a measure of taste
sensation.
Validation of the theory: stochastic resonance
Stochastic resonance, in its broadest context—see, for example,
Wiesenfeld and Moss (Wiesenfeld and Moss,
1995), stochastic resonance type 3—can refer to the
reduction in threshold that follows the introduction of noise into a system.
This phenomenon can be observed when we allow for the introduction of noise
into the pathway for perception of taste.
Within the general entropy theory of sensation
(Norwich, 1993), Stevens's law
emerges when Fechner's law is applied to the combination of two sensory
signals (convolution of two probability densities): the external taste
stimulus, C, plus an internal stimulus, CN, which
may be regarded as a constant `noise' or `reference' signal. The procedure is
much simpler here. Let us write equation (7) or (8) for the sum of the two
signals:
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/, which we recognize as
the quasithreshold, we obtain
 | (22) |
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So we see that both Stevens's law and Fechner's law emerge from the entropic representation of taste, Stevens's law holding more precisely for lower concentrations and Fechner's for higher concentrations. However, in the middle range, both laws hold reasonably well.
We may also see that the shift to Stevens's law from Fechner's law permits the system to decrease its threshold. When Fechner's law alone holds sway, the threshold is given by the quasi-threshold value depicted in Figure 2, which is substantially greater than zero. When noise is added to the system, the threshold diminishes to zero, as given by equation (23). In reality, there will still, of course, be a small positive threshold. The above is an example of the principle of stochastic resonance: threshold is reduced as a consequence of the addition of the noise signal.
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Discussion |
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TopAbstractIntroductionMethodsResultsDiscussionConclusionsAppendix: the molar entropy...Glossary of symbols used...References |
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Entropy
Claude Shannon, who formulated the theory currently known as information
theory, defined entropy as a weighted sum of the logarithms of the
probabilities of the outcomes of an event. It was shown
(Norwich, 1984) that if the
magnitude of taste were equated to a general `entropy' or uncertainty,
governed by an unspecified probability density function, one could then derive
from this single definition many of the empirically discovered sensory laws
governing taste: both laws of sensation (Fechner's and Stevens's), the general
function for Weber fraction, the adaptation function for taste, etc. It was
suggested that the receptor might be sensitive to or uncertain about the
density of particles in solution, since this quantity will fluctuate due to
Brownian motion. However, the event remained unspecified—that is, it was
not stipulated what the taste receptor was uncertain about—so there
remained a number of parameters that had to be evaluated by fitting functions
to experimental data.
In the present work, it is hypothesized that the entropy utilized by the taste receptor is the standard physical or thermodynamic entropy. Thus we were able to derive an expression for the law of taste perception largely without adjustable parameters. It was found, however, that the phenomenon of biological threshold was not evidently contained within the physical expression for entropy, and threshold was placed by conjecture.
The new derived law of taste for monovalent, strong salts is given by Equation (7), which is a form of Fechner's law defining not only the logarithmic dependence of taste on concentration of the substance tasted, but also showing how taste will vary for different salts with the same molar concentration, based on differences in the standard molar entropies, S0 = (S0+ + S0-).
Factors affecting the approximations
It is important to recognize that the final step in the derivation of Equation (1) (Appendix), replacing pressure, P, by mole fraction, X, is valid only for dilute (ideal) solutions, and may hold more closely for monovalent than for divalent ions.
One must also allow, in the interpretation of results, for the dilution of tasted solutions by saliva etc. That is, concentration of tastants that actually reach the sensory cells may be somewhat less than the values measured in the laboratory. In some cases, for example equation (13), the dilution factors will tend to cancel; in other cases not.
The hypothetical molal entropies of the fused salt would be greater than the corresponding values for crystalline substance, which could affect all calculations, but particularly will decrease the values of R calculated from equation (20).
How does a taste receptor determine the entropy?
Hitherto, we have given scant consideration to the mechanism of
transduction of the taste signal. Rather, we have emphasized the results that
can be observed if taste is measured simply, linearly by physical entropy.
Entropy is measured in the laboratory by complex methods. It is not expected
that the sensory receptor, which gives rise to the sensation of taste, will
appraise entropy by laboratory techniques, and the mechanism by which it does
so is a matter of speculation. Quite likely, the organism utilizes the
relationship between entropy and free energy, as expressed by the Gibbs
equation for constant temperature,
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The transduction of the taste of sodium chloride is believed to involve the
permeation of sodium ions through amiloride-sensitive sodium channels
(Herness and Gilbertson, 1999;
Geran and Spector, 2000;
Hendricks et al.,
2000). An avenue for exploration, then, may be the relationship
between changes in free energy produced by the sodium chloride tastant to
changes in sodium permeability via amiloride-sensitive sodium channels.
The principle of molar entropy as a measure of the sensation of taste can be expected to apply more widely. For example, it might be expected to hold, mutatis mutandis, for other taste qualities such as sweetness, bitterness, etc. It might also find applicability in the sense of olfaction. However, as the molecules involved become more complex, the associated chemistry becomes more complicated, and published data scarcer.
The entropy principle suggests that the percept of taste originates in the receptor itself, and is modulated at higher centers.
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Conclusions |
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TopAbstractIntroductionMethodsResultsDiscussionConclusionsAppendix: the molar entropy...Glossary of symbols used...References |
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By hypothesis, the negative of the molar entropy of a tastant in solution is taken as a measure of the magnitude of the taste of saltiness. It was shown that under these circumstances, the taste of saltiness becomes a linear function of the logarithm of the concentration of the tastant, in accordance with Fechner's law. To this fundamental hypothesis it was necessary to append a second condition: the line defining Fechner's law must be displaced upward in parallel displacement until the intercept of the line with the x-axis (quasi-threshold) is equal to the standard molar entropy of the salt. The result was a generalized form of Fechner's law defining not only the change in sensation with concentration, but also the difference in sensation (saltiness) between two different salts that have the same molar concentrations. The hypothesis was verified by several techniques using published data. The standard molar entropies were approximated by the entropies of the crystalline form of the salt. The theoretical equations were readily conformed to the data of von Skramlik (von Skramlik, 1926
) on
saltinessmatching. The more complete test utilized only 1:1 salts such as KCl.
Von Skramlik's data not only conformed to the entropy hypothesis, but
permitted the estimation of the value of the gas constant, R. They
theory of 2:1 salts was worked out in part and 2:1 salts such as
CaCl2 also permitted estimation of the gas constant. The mean
estimate of R was close to the accepted value. The entropy hypothesis
was also confirmed using psychophysical data by plotting saltiness against the
logarithm of salt concentration. The value of the quasi-threshold, which could
be read from the graph, again permitted estimation of the value of
R. Adding to the pure sensory signal from the tastant is noise arising within the taste receptor itself. The result of adding this noise was to convert Fechner's law, which governed larger salt concentrations, to Stevens's power law, which was more accurate for lower salt concentrations. The quasi-threshold was replaced by a smaller, more realistic threshold, demonstrating the thesis of stochastic resonance: addition of noise to a system may, under certain conditions, lower the threshold for signal detection. Finally, the issue of signal transduction for sodium chloride was addressed.
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Appendix: the molar entropy of an ion in solution |
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TopAbstractIntroductionMethodsResultsDiscussionConclusionsAppendix: the molar entropy...Glossary of symbols used...References |
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The following is a very brief derivation of equation (1), emphasizing physical content. The derivation is based largely on the first two laws of thermodynamics, the ideal gas law, and the existence of the Gibbs function as a state variable in thermodynamics.
The first law of thermodynamics expresses changes in the internal energy of
a system, dU, as a balance between heat added to the system,
d'Q and P-V work done by the system,
PdV:
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We make the transition from ideal gas to ideal solution, appealing to
Henry's law, by replacing P by the mole fraction of solute,
X (Smith, 1967;
Alberty and Silbey; 1997),
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Glossary of symbols used in the main text |
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TopAbstractIntroductionMethodsResultsDiscussionConclusionsAppendix: the molar entropy...Glossary of symbols used...References |
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Acknowledgments |
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I am very grateful to Professor S. G. Whittington for guiding me along some of the less trodden paths of physical chemistry. The assistance of Brige Paul Chugh in various stages of this project is also greatly appreciated. This work has been supported by means of a grant from the Natural Sciences and Engineering Research Council of Canada.
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References |
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TopAbstractIntroductionMethodsResultsDiscussionConclusionsAppendix: the molar entropy...Glossary of symbols used...References |
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Alberty, R.A. and Silbey, R.J. (1997)Physical Chemistry , 2nd edn. Wiley, New York.
Geran, L.C. and Spector, A.C. (2000) Sodium taste detectability in rats is independent of anion size: the psychophysical characteristics of the transcellular sodium taste transduction pathway. Behav. Neurosci., 114,1229 -1238.[ISI][Medline]
Hendricks, S.J., Stewart, R.E., Heck, G.L., DeSimone, J.A.
and Hill, D.L. (2000) Development of rat chorda
tympani sodium responses: evidence for age-dependent changes in global
amiloride-sensitive Na+ channel kinetics. J.
Neurophysiol., 84,1531
-1544.
Herness, M.S. and Gilbertson, T.A. (1999) Cellular mechanisms of taste transduction.Annu. Rev. Physiol. , 61,873 -900.[ISI][Medline]
Meiselman, H.L., Bose, H.E. and Nykvist, W.F. (1972) Magnitude production and magnitude estimation of taste intensity. Percept. Psychophys.,12 , 249-252.
Norwich, K.H. (1977) On the information received by sensory receptors. Bull. Math. Biol.,39 , 453-461.[Medline]
Norwich, K.H. (1984) The psychophysics of taste from the entropy of the stimulus. Percept. Psychophys., 35,269 -278.[Medline]
Norwich, K.H. (1993) Information, Sensation and Perception. Academic Press, San Diego, CA.
Norwich, K.H. (2000) Physics makes `sense'. Canadian Medical and Biological Engineering Society Conference, October 2000, Proceedings, pp.92 -93.
Pfaffmann, C., Bartoshuk, L.M. and McBurney, D.H. (1971) In Beidler, L.M. (ed.), Handbook of Sensory Physiology: Vol. IV, Chemical Senses 2: Taste. Springer, New York, pp.85 -86.
Smith, N.O. (1967) Chemical Thermodynamics: A Problems Approach. Reinhold, New York.
Stevens, S.S. (1969) Sensory scales of taste intensity. Percept. Psychophys.,6 , 302-308.
von Skramlik, E. (1926) Handbuch der Physiologie der Niederen Sinne. Georg Thieme, Leipzig, pp.458 -460.
Weast, R.C. and Astle, J.J. (eds) (1981)CRC Handbook of Chemistry and Physics . CRC Press, Boca Raton, FL.
Wiesenfeld, K. and Moss, F. (1995) Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDS. Nature, 373,33 -36.[Medline]
Accepted June 25, 2001
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