MOLAR AMOUNT OF WATER

Abstract

When modeling body fluids using physical chemistry, we en-countered a contradiction. We proceeded from the erroneous assumption that the molar amount of water in an aqueous so-lution is the molar amount of H2O molecules (mass divided by the mass of one H2O molecule). Thus, in one kilogram of pure water, we calculated 55.508 moles of water because the molar mass of H2O is 18.01528 g / mol. When calculating the molar fractions as the molar amount of the substance divided by the solution's total molar amount, we thus obtained numerically completely different values than for molalities or molarities. According to the theory, these values should be substitutable. However, it turned out that using these values in the calculati-ons of the solubility of gases in aqueous solutions showed us an error of about 55 mol/kg. Similar errors began to be reported for chemical processes with different numbers of reactants and products (at the same number, the error is annulled algebrai-cally). So is the water molality really about 55 mol/kg? No. This is because water forms bonds with each other, which cluster more H2O molecules into larger particles. From the required molar amount of water, we derived the dissociation constant and enthalpy of this bond. The results are compatible with data from the National Institute of Standards and Technology (NIST) and the data of formation energies of individual substances. Using these constants, it is possible to derive the molar amount of water in aqueous solutions and subsequently make calculati-ons over molar fractions, the results of which begin to coincide with the measured and published experiments.

Full text

85
MEDSOFT 2020
Marek Matejak, Jiri Kofranek MOLAR AMOUNT OF WATER
DOI: 10.35191/medsoft_2020_1_32_85_88
MOLAR AMOUNT OF WATER
Marek Matejak, Jiri Kofranek
Abstract
When modeling body uids using physical chemistry, we en-
countered a contradiction. We proceeded from the erroneous
assumption that the molar amount of water in an aqueous so-
lution is the molar amount of H2O molecules (mass divided by
the mass of one H2O molecule). Thus, in one kilogram of pure
water, we calculated 55.508 moles of water because the molar
mass of H2O is 18.01528 g / mol. When calculating the molar
fractions as the molar amount of the substance divided by the
solution's total molar amount, we thus obtained numerically
completely dierent values than for molalities or molarities.
According to the theory, these values should be substitutable.
However, it turned out that using these values in the calculati-
ons of the solubility of gases in aqueous solutions showed us an
error of about 55 mol/kg. Similar errors began to be reported
for chemical processes with dierent numbers of reactants and
products (at the same number, the error is annulled algebrai-
cally). So is the water molality really about 55 mol/kg? No. This
is because water forms bonds with each other, which cluster
more H2O molecules into larger particles. From the required
molar amount of water, we derived the dissociation constant
and enthalpy of this bond. The results are compatible with data
from the National Institute of Standards and Technology (NIST)
and the data of formation energies of individual substances.
Using these constants, it is possible to derive the molar amount
of water in aqueous solutions and subsequently make calculati-
ons over molar fractions, the results of which begin to coincide
with the measured and published experiments.
1 Introduction
In physical chemistry, there is often the only talk of very dilute
solutions [1], while this is not the case with body uids in phys-
iology. For example, intracellular uid has only approx. 70%
water. Conventional calculations cease to apply here because
the shift from the standard state of an aqueous solution is so
signicant that it is necessary to extend the theory of physical
chemistry to these conditions as well. One of our experiments
is the extension using hypotheses about the molar amount of
water, which we replace the constants of standard molality and
standard molarity.
Our research into the molar amount of aqueous solutions be-
gan unconsciously in April 2015 with a question published on
the website www.researchgate.net: "What is the concentration of
hydroniums (H3O+) or free protons (H+) as the equivalent of pH =
7.4 in the aqueous solution?". Although this question had about
20,000 views on this international scientic network, 38 public
answers, and other private communications followed, its clear
answer was not found here. One way of calculating the result
from the dissociation of water showed a molar fraction of 10-7.4
mol/mol. However, directly from the denition of pH, it should
be a molality of 10-7.4 mol/kg. It was the contradiction of these
physical units at that time that led to the fact that both can be
true only if, under the given conditions, there will be exactly one
mole of all particles in one kilogram of water.
Our laboratory has been dealing with the calculation of the
acidity (pH) of blood and other body uids within the acid-base
balance for many years. Only by integrating several models into
one whole does everything begin to be formalized in such a
way that it is necessary to solve all redundant and opposing re-
lationships so that the resulting mathematical model clearly de-
nes the course of the described variables in time. However, this
formalization requires interconnection ideally up to the level of
a basic theory such as physics or physical chemistry [2]. The use
of physical chemistry for simulations in physiology or medicine
is not very widespread today. Calculations in these elds conti-
nue to remain mainly for empirical relations or for basic physics,
often with a large tolerance for error. This is despite the fact
that modern computer technology can deal with robust phy-
sical systems of equations both algebraically and numerically.
However, from experience with empirical equations, it might
seem that a large number of equations will be associated with
a large number of unknown parameters. However, this may not
be true when using physical relations correctly, because phys-
ics is based on the elimination of "unknown parameters" or, in
extreme cases, even "unknown constants". It does this elimina-
tion precisely through the relations by which it denes these
values. Therefore, it is theoretically possible to create even very
complex and complex physical-mathematical models that need
only a small number of well-known (i.e. derivable or measurab-
le) parameters and constants.
In creating our human physiology models [3-6], we try to
describe in detail body uids: blood plasma, interstitial uid,
intracellular uid, cerebrospinal uid, urine, etc. In describing
these aqueous solutions, we are interested in the relationships
of individual substances and their processes. These processes
are closely related to heat, charge, acidity, water solubility in
water, and other properties described by physical chemistry.
In 2015, we implemented the Modelica library for Physical
Chemistry [7], which we use to calculate these processes in or
between aqueous solutions. At that time, this software library
contained components for the equilibrium of chemical proce-
sses such as chemical reactions, diusion, the solubility of gases
in solutions, electron transfer between dierent media, Donnan
equilibria on the membrane, according to Nernst relations, etc.
One of the other issues arising from implementing this soft-
ware library was the necessary correction in expressing the
solubility of gases in water. The calculated value of Henry's con-
stant for gas solubility in water was not the same as the measu-
red value published using the National Institute of Standards
and Technology (NIST) tables. This problem was described in
detail in 2015 in the dissertation [8] on page 38. At that time, we
compared Henry's constants for dierent gases with their de-
rived values. We found that the coecient by which the values
dier is the same for each gas. But then we didn't know why. By
further investigating the properties of liquid water, we found
that the explanation can be relatively simple. The calculations
depend on the molar fraction of a given gas in water. The molar
fraction of a substance is the ratio of the molar amount (number
of particles) of a given substance to the molar amount of all par-
ticles in a given solution. It didn't take long, and we realized that
if we adjust the molar amount of water, it is possible to agree
with the measured values of dissolved gas. This is because the
molar amount of water is not equal to the molar amount of H2O
molecules. H2O molecules form dynamically weak hydrogen
bonds with each other [9]. These bonds in liquid water form
clusters of H2O molecules, and each such cluster must be consi-
dered one particle at a given point in time.
2 Methods
As a possible hypothesis, we chose the following statement: "In
liquid water, H2O molecules bind to each other in clusters by bonds
so that these bonds do not form cycles and each of these bonds has
the same properties."
The bond cycle is the joining of two H2O molecules that are
already part of one cluster. The assumption that bonds do not
form cycles is relatively strong, and in many articles, on the
contrary, there are structures where these cycles exist. In con-

86
MEDSOFT 2020
Marek Matejak, Jiri Kofranek MOLAR AMOUNT OF WATER
DOI: 10.35191/medsoft_2020_1_32_85_88
trast, we have remained with the idea that the bonds between
individual clusters (with any number of water molecules) are
much more likely to be the same as between individual water
molecules and are independent of each other.
If the bonds are independent of each other, they have the
same properties as the enthalpy and entropy of hydrogen
bonds between H2O molecules. Thus, the dissociation constant
(K) at the junction of two clusters is the same as the dissociation
constant between two free H2O molecules.
Based on these statements, we constructed a mathematical
model that accurately derives the molar fractions (xi) of cluster
sets from a given number of H2O molecules (i). Simultaneously,
it is not important how long this state of specic clusters will
last because the number of individual clusters remains the
same even if the links change rapidly dynamically.
H2O + H2O<-> (H2O)2 K = x1 * x1 / x2
H2O + (H2O)i-1 <-> (H2O)i K = x1 * xi-1 / xi
This chemical reaction denes a geometric series for xi = xi-1 *
x1 * K, which can also be written as xi = x1* (x1*K)i-1. The sum of
all molar fractions in pure water if we neglect OH- and H+ ions,
which are 10-7 mol/mol, expressed as the sum of xi through i
from 1 to innity, is equal to 1.
Sum of all
molar fracons
Assuming that x1*K is positive and less than 1, it is possible to
use the relation for the geometric sequence
.
The size of the individual members decreases exponentially
here (Fig. 1), and therefore the total sum of this innite series
is nite.
In one kilogram of pure water, is 1 / MM (55,508 mol/kg, whe-
re MM is the molar mass of H2O) moles of H2O molecules, while
the number of particles is according to the measured data N:
Number of H2O
molecules in 1kg of
water
So we have two equations for two unknowns, which allows
their direct derivation:
Molar fracon of free H2O
molecule
Dissociaon constant of
hydrogen bonding of water
In conclusion, it is sucient to verify that the product x1 * K
is less than 1. The number of clusters (N) must be less than the
number of H2O molecules (1 / MM) and therefore MM * N <1,
and therefore the product x1 * K = 1 - MM * N <1. Therefore, the
use of the calculation of a geometric series is justied for each
possible measured value of N.
One way to estimate the number of water particles (N) is to
measure the solubility of gases (A) in water and then compare
it with this chemical process's energy equilibrium. From the
balance of chemical potentials of gaseous and dissolved sub-
stance A, it is possible to derive the relationship between the
dierence of formation Gibbs energies (∆disGo) and Henry's
coecient (kH):
Henry's coecient
for molar fracons
From the table values of formation energies for gaseous
(∆fGoA(g)) and dissolved substance in water (∆fGo
A(aq)) it is, the-
refore, possible to determine the value ∆disGo=∆fGo
A(aq)-∆fGo
A(g).
At the same time, it is possible to measure the molar fraction of
substance A in the gas (aA(g)) as well as the molar amount of sub-
stance A that dissolved in water (nA(aq)). Then N can be expressed
from the relation:
Number of parcles
in 1 kg of pure water
Number of parcles
in 1 kg of aqueous
soluon
3 Results
The rst estimate of the number of water particles per kilogram
of pure water is 1 mol (standard molality). For this value, it is
possible to express
According to NIST (National Institute of Standard and Tech-
nology, see https://www.nist.gov/), Henry's constant for the
solubility of CO2 in water is 0.035 mol/kg/bar in pure water,
where 1 kg of water contains 1 mol of all particles. At a partial
pressure of pCO2 = 40 mmHg = 40/760 bar in the intracellular
uid, the concentration of freely dissolved CO2 is measured as
0
100
200
300
400
500
-0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
x
Fig. 1 – Distribution of molar fractions according to the number of H2O molecules i n the cluster. The frequency of a cluster of a given size decrease s exponen-
tially with its size. The larger the cluster, the lower the concentration.

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MEDSOFT 2020
Marek Matejak, Jiri Kofranek MOLAR AMOUNT OF WATER
DOI: 10.35191/medsoft_2020_1_32_85_88
1.24 mmol / L. Thus, the total number of intracellular uid par-
ticles is 0.67 mol / L. This is roughly conrmed by measuring the
volume fraction of water or the measurement of solubility for O2
or even isoosmolarity expressed in the sum of molar fractions of
passively impermeable substances.
We then support these and many other physiological calcu-
lations using a software library designed for physiology [10-13].
Similarly, it is possible to express all body uids' molar densi-
ty and then switch to counting over molar fractions instead of
molality or molarity [14]. In physical chemistry, counting over
molar fractions is directly linked to the calculation of energies
and electrochemical potentials. Therefore, it is then possible
to express the end states of elementary processes as equali-
zing electrochemical potentials. This will allow using physical
chemistry to derive dissociation constants, Henry coecients,
electrical potentials, Nernst voltages, Donnan equilibria on
the cell membrane, osmolarity, and other electrochemical
processes
4 Discussion
Unfortunately, many books on physical chemistry equate mo-
larity, molality, and molar fractions in aqueous solutions (either
directly or through activities or through chemical potential). At
the same time, they are aware that this equality only applies
under specic conditions and at low solute concentrations. In
order to make it all right in terms of physical units, the constants
"standard molality" and "standard molarity" were introduced,
which will convert moles into kilograms or liters. Unfortunately,
these constants have also become part of the denitions. One
of them is the denition of pH. However, today's pH measure-
ment via a hydrogen electrode does not measure molality but
the electrical voltage in redox reactions and thus the activity of
hydrogen ions. The conversion to the molality of these positive-
ly charged water particles' activity should depend on the total
number of particles in one kilogram or in one liter. As a result,
the theoretical balance using the current pH denition cannot
fully coincide with measurements in environments such as in-
tracellular uid (where it is not true that 1 mol of all particles has
one kilogram of solvent or 1 mol of all particles is in one liter of
solution). In principle, the current denition of pH works where
the "standard molality" works, i.e., in solutions where the total
molar density of 1 mol of all particles per 1 kg of solvent applies.
anými experimenty.
Our idea of water is based only on macroscopic properties
that interest us. Although the model derives the abundance of
individual water clusters, only the total number is relevant to us.
Thus, our hypothesis was only partially conrmed, and further
measurements and experiments of already molecular proper-
ties of H2O and their hydrogen bonds are necessary for its full
conrmation.
For example, if we relax from this assumption and allow more
cyclic hydrogen bonds within one cluster, it is possible to re-
formulate the calculation to the molar fraction of the cluster as
xi,j = x1* (x1*K)i-1 *K(int)
j,
assuming that all internal bonds would have the same di-
ssociation constant K(int). The added index j here expresses the
number of internal links that are missing in the original model.
However, when summing these fractions, it is necessary to omit
the rst members because these internal cyclic bonds can exist
only from a certain cluster size. For example, the smallest clus-
ter with one internal bond must have at least 4 H2O molecules,
with two 6, with three 8, four 9, etc. However, these complexities
come into play only when we have conicting data requiring
the theory to be extended by another degree of freedom. So far,
according to the Occam razor pattern, a simpler model where j
= 0 will suce.
5 Conclusion
Water forms bonds with each other that cluster more H2O mo-
lecules into larger particles. We assume that in liquid water, H2O
molecules bind to each other in clusters by bonds so that these
bonds do not form cycles, and each of these bonds has the same
properties as the enthalpy and entropy of the hydrogen bonds
between the H2O molecules. The dissociation constant when
joining two clusters is the same as the dissociation constant
between two free H2O molecules. Based on these assumptions,
we constructed a mathematical model that accurately derives
the molar fractions of cluster sets from a certain number of H2O
molecules. Simultaneously, it does not matter how long this
state of specic clusters lasts because the number of individual
clusters in the steady-state remains the same even if the bonds
change rapidly dynamically.
From the required molar amount of water, we derived the di-
ssociation constant and enthalpy of hydrogen bonds between
H2O molecules to be compatible with data from the National
Institute of Standards and Technology (NIST) and data on
the formation energies of individual substances. Using these
constants, it is possible to derive the molar amount of water
in aqueous solutions and then make calculations over molar
fractions, the results of which begin to coincide with published
experiments.
References
[1.] José J. C . Teixeira-Dias. „Molecula r Physical Chemistr y: AComputer-
-based Approach using Mathematica® and Gaussian“, Sp ringer, 2017
[2.] M. Mateják, „Formalization of Integrative Physiology", PhD Thesis,
Charles University in Prague, 2015.
[3.] Mateják, Marek; Kofránek, Jiří; „Rozsáhlý model fyziologických regu-
lací vModelice“, MEDSOF T 2010. Praha: Agentura Action M, Praha
2010, str. 66- 80. ISSN 1803 81115, 2010
[4.] Mateják, Marek; Kofránek, Jiří; „HumMod–Golem Edition–Rozsáhlý
model fy ziologických systém“, Medsoft 2011, 182-196, 2011
[5.] Kofránek, Jirí; Mateják, Marek; Privitzer, Pavol; Tribula, Martin; Kulhá-
nek, Tomás; Silar, Jan; Pecinovský, Rudolf; „HumMod- Golem Edition:
large scale model of integrative physiology for virtual patient simu-
lators“, Proceedings of the International Conference on Modeling,
Simulation and Visualization Methods (MSV), 1, 2013
[6.] Mateják, Marek; Kofránek, Jiří; „Physiomodel-an integrative physio-
logy in Modelica“, 2015, 37th annual international conference of the
IEEE Engineering in Medi cine and Biology Society (EMBC), 1464-1467,
2015, IEEE
[7. ] M. Mateják, M. Tribula, F. Ježek, aJ. Kofránek , „Free Modelica Library
of Chemical and Electrochemical Processes", in 11th International
Modelica Conference, Versailles, France, 2015, roč. 118, s. 359–366.
[8.] MATEJÁK, MAREK. Formalization of Integrative Physiology.Prague,
2015. 115, 3, 1 CD. Dissertation thesis. Charles Univer sity in Prague,
First Faculty of Medicine, Institute of Pathological Physiology. Super-
visor Doc. MUDr. Jiří Kofránek CSc.
[9.] Luzar, A., & Chandle r, D. „Hydrogen -bond kinetics in liquid water.
Nature“, 379(6560), 55–57. doi:10.1038/379055a0, 1996
[10.] M. Mateják, T. Kulhánek, J. Šilar, P. Privitzer, F. Ježek, aJ. Kofránek,
„Physiolibrary-Modelica library for physiology", in Proceedings of the
10 th International Modelica Conference; March 10-12; 2014; Lund;
Sweden, 2014, s. 499–505.
[11 .] Mateják, Marek; „PHYSIOLOGY IN MODELICA“, MEFANET Journal,
2014
[12 .] Mateják, Marek; Kulhánek, Tomáš; Matoušek, Stanislav; „Adair-
-based hemoglobin equilibrium with oxygen, carbon dioxide and

88
MEDSOFT 2020
Marek Matejak, Jiri Kofranek MOLAR AMOUNT OF WATER
DOI: 10.35191/medsoft_2020_1_32_85_88
hydrogen ion activity“, Scandinavian Journal of Clinical & Laborato-
ry Investigation, 2015, Informa Healthcare
[13 .] Mateják, Marek; Ježek, Filip; Tribula, Martin; Kofránek, Jiří; „Physio-
library 2.3-An Intuitive Tool for Integrative Physiology“,IFAC-Paper-
sOnLine, 48, 1, 699-700, 2015, Elsevier
[14. ] Mateják, Marek , „Modelování tělesných tekutin vjazyku Mo delica“,
Medsoft 2019, 103-112, 2019
Kontakt
Mgr. Marek Matejak, Ph.D.
Laboratory of Biocybernetics
Institute of Pathological Physiology,
First Faculty of Medicine, Charles
University
email: matejak.marek@gmail.com
tel: +420 776 301 395
doc. MUDr. Jiří Kofránek, CSc.
Laboratory of Biocybernetics
Institute of Pathological Physiology,
First Faculty of Medicine, Charles
University
email: kofranek@gmail.com
tel: +420 777 68 68 68