ISSN 01458752, Moscow University Geology Bulletin, 2010, Vol. 65, No. 2, pp. 95–103. © Allerton Press, Inc., 2010.
Original Russian Text © T.G. Makeeva, L.V. Goncharova, V.A. Trofimov, Yu.M. Egorov, 2010, published in Vestnik Moskovskogo Universiteta. Geologiya, 2010, No. 2, pp. 000–
000.
Estimation of the Determination Error of the SolidPhase Density
of Disperse Soils by Different Methods
T. G. Makeevaa, L. V. Goncharovab, V. A. Trofimovc, and Yu. M. Egorovd
a
Geological Faculty, Moscow State University
email: [email protected]
bGeological Faculty, Moscow State University
email: [email protected]
c
Institute of Problems of Complex Development of the Earth’s Interiors, Russian Academy of Sciences
email: [email protected]
dLimited Liability Society Salyut
email: [email protected]
Received June 18, 2009
Abstract—The accuracy and reliability of the determination of the solidphase density (SPD) for disperse
soils by the standard method (AllUnion State Standard 518084) and by the Kalachev rapid method were
estimated. The causes for discrepancies in the SPD determination in clay soils between the two different
methods were determined for the cases of similar and different preanalysis preparations. A justified calcula
tion procedure for the determination of the boundwater density in clay soils, which is adequate for the nature
of complex physical phenomenon, was developed. Corrections for the boundwater density in clay polymin
eral soils with different dispersion properties are suggested.
Key words: disperse soils, solidphase density (SPD), estimation of method accuracy, confidence, bound
water density, boundwater density determination method, correction for boundwater density.
DOI: 10.3103/S0145875210020055
INTRODUCTION
Stolyarov, 2007; Dmitriev and Yarg, 2008; and
Kulyapin, 2009] agree that this is caused by variation
The solidphase density (SPD) of disperse soils,
like the boundwater density, is a useful calculation
4
of the boundwater density. This problem is solved
using different approaches: new methods [Revelis,
1962; Eremenko and Serebryakov, 1987; Kalachev et
al., 1997; and Dovbnya and Kudyakov, 2002], correc
tions to (AllUnion State Standard 518084) [Shlykov
and Trapeznikov, 2002], analysis efforts in different
laboratories to reduce the statistical error [Dmitriev
and Yarg, 2008], and revision of the standard (All
Union State Standard 518084) [Kalachev et al.,
1997].
1
parameter .
The broadly recognized current method of the
determination of the solidphase density of disperse
soils (AllUnion State Standard 518084) is under
broad and sometimes constructive criticism because it
provides imprecise information for clay disperse soils
in SPD determinations, both for drying at 105°C [Vas
ilyev, 1949; Ziangirov, 1964; Kalachev et al., 1997;
Shlykov and Trapeznikov, 2002; Stolyarov, 2007; Dmi
triev and Yarg, 2008, and Makeeva et al., 2006] and
without drying with correction for hygroscopic wet
ness [Shlykov and Trapeznikov, 2002; and Makeeva
Many authors [Vasilyev, 1949; Ziangirov, 1964;
Kalachev et al., 1997; Shlykov and Trapeznikov, 2002;
To solve this complex problem it is necessary to
solve a number of problems: to estimate the accuracy
and reliability of the density determination of the
solidphase dispersed soils by different methods; to
identify the causes for the discrepancy in density
determination of the solidphase disperse soils with
different mineral compositions and disperse proper
1 The
4 In fact, the method is designed to determine the solidphase den
2
et al., 2006] .
3
term “density of the solid particles” is widely used because
disperse soils constitute a threephase system; so this work uses
the term “solidphase soil density.”
2 We use only the definitions for SPD of soils with kerosene (as
most justified).
3 The determination accuracy of solidphase clay soil density is
assessed assuming that the boundwater density is 1.2–
1.4 g/cm3 and that it is constant.
sity and amount of boundwater at the ratio S : L = 1 : 10 with the
weighing accuracy of 0.001 g; while at the ratio S : L = 2 : 10 and
at the weighing accuracy of 0.0001 g it determines the density of
the boundwater, corresponding to the layer charge of the sur
face, and the solidphase density of soil in the disperse systems
with boundwater phase transition of the first kind [Makeeva
et al., 2007, 2008]
95
96
MAKEEVA et al.
ties by different methods for diverse preanalysis prep
arations; to develop a justified method for calculating
the boundwater density; to propose reliable correc
tions for the boundwater density of clay polymineral
soils with different disperse properties; and to establish
the applicability limits and convergence conditions of
the existing methods.
THEORETICAL ANALYSIS
The reduced reliability of the information provided
by the standard method, as Kalachev et al. [1997]
argue, may be due to a number of factors omitted by
the method. One of the most serious omissions is the
disregard of the volume of hygroscopic wetness in the
calculation formula of the standard method in the
density determination of the solidphase clay soil.
Kalachev developed a new method, whose calculation
formula is modified to include a correction for the
boundwater density of soils; also, the ELA2 instru
ment is developed to determine the density of solid
phase rocks. For less dispersed soils (sands and rocks),
experiments showed complete coincidence of results
of determination of the of solidphase density, using
the developed rapid method and standard method
[Kalachev et al., 1997]. It is noteworthy that for more
dispersed soils (loams and clays of diverse mineral
composition and peat), the obtained values of soil
SPD exceed those using the standard method by 0.05–
0.5 g/cm3, with the largest discrepancy being found for
montmorillonite clays and peat. Such a large overpre
diction, as Kalachev et al. [1997] argue, is due to intro
duction of the correction for the boundwater density;
precisely the solidphase soil density is determined. In
our opinion, the fact of SPD overestimation for more
loamy soils may be considered more justified within
the method applicability region, i.e., for disperse sys
tems with the boundwater phase transition of the first
kind, while for water at the transitional state of the sec
ond kind, the standard method gives better results.
Firstly, for disperse systems with the boundwater
phase transition of the first kind, close to the second
kind, Kalachev’s conclusion is incorrect. In such sys
tems, reliable soil SPDs can be extracted neither by
the standard method with preliminary drying at
105°C, nor by the Kalachev method, even using reli
able correction for the boundwater density, due to the
impossibility of reaching the thermodynamic equilib
rium at the Curie point or at the tricritical point,
defined by the Landau–Ginzburg theory [Blinov
et al., 2000]. The thermodynamic equilibrium can be
achieved only at the temperature of the structural
instability, which is higher than the temperature in the
experiment. Then, the equilibrium state is established
for the solidphase density of such systems, and the
reliable SPD value of soils is much higher than the soil
SPDs obtained by these methods (Makeeva, 2009a, b;
5
Makeeva, 2009). Secondly, the recommended cor
rections for the boundwater density were based on the
Olodovskii [1989] studies. The question of how reli
able corrections are for the boundwater densities of
disperse soils of different compositions and properties
is still open.
Data on the boundwater density have been
obtained mainly for monominerals, and this informa
tion is quite contradictory, even for similar mineral
surfaces; while for clay polymineral soils of different
compositions, these data are absent [Makeeva et al.,
2007]. The determination of the boundwater density
in disperse systems by direct methods is very difficult
because of methodical and theoretical aspects. One
way around this contradiction is to develop a justified
method of determination of the boundwater density.
For this, one undoubtedly should establish the nature
and patterns of the variations of the boundwater den
sity in clay soils in different boundwater density and
temperature regimes, create the basis for the theory of
the hydration process of heterogeneous surface of dis
perse systems in different regions of boundwater with
polar ordering of layers, and study different mecha
nisms of phase transitions of boundwater in disperse
soils to identify the method applicability limits.
Despite doubtless advantages, the Kalachev
method still has very limited utility. The pedology
related limitations stem from the absence of data on
estimates of the absolute and relative errors in SPD
determination by this method, as well as from the
absence of reliable data on the correction for the
boundwater densities of disperse soils with different
compositions and properties, and from the unavail
ability of data on the applicability range of the method.
So, the method applicability range is limited to
coarsely dispersed soils (sands and rocks).
When the standard method is used to determine
SPD for one and the same clay soils, but with different
preanalysis preparations (with drying at 105°C and
without it but with correction for the hygroscopic wet
ness), it gives different soil SPDs [Shlykov and Tra
peznikov, 2002]. The largest discrepancies are found
for highly disperse clay differences, namely moderate
and heavy loams, and especially for clays, with the
SPDs of clay soils, determined with correction for
hygroscopic wetness, exceeding those using drying at
105°C [Shlykov and Trapeznikov, 2002]. Our data sug
gest that the standard method (with drying at 105°C
and the Kalachev method (without drying and using
correction for the boundwater density) for identical
5 The
paper considers corrections to the AllUnion State Stan
dard 518084. The corrections refer to conditions of preanaly
sis soil preparation and conditions of the experiment: soil
weighting accuracy and the ratio of the solid to the liquid phase,
as well as two corrections referring to the boundary conditions of
the method. Modification of the AllUnion State Standard
518084 to incorporate a new physical method of determination
of the solidphase density of disperse soils, namely the fast
Kalachev method, is also suggested.
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ESTIMATION OF THE DETERMINATION ERROR OF THE SOLIDPHASE
soils also show the largest discrepancies, again for
more dispersed differences. However, the clay soil
SPDs obtained by the Kalachev method and deter
mined with correction for the boundwater density,
which are assumed to be constant at 1.05 g/sm3, are
less than the soil SPDs determined by the standard
6
method [Makeeva et al., 2006].
These discrepancies stem from the unreliability of
the correction for the boundwater density in clay soils
with differing disperse properties. The conclusion of
Shlykov and Trapeznikov [2002] about the unreliabil
ity of the data obtained by the standard method with
correction for hygroscopic wetness may be considered
more justified than the suggested corrections. When
SPDs of soils at the airdry state are determined with
correction for the hygroscopic wetness according to
the standard method for disperse soils with bound
water phase transition of the first kind, i.e., within the
method applicability limits, this gives unreliable data
because of the absence of reliable corrections for the
boundwater density and because the Curie point can
not be determined in view of the nonuniformity of dis
tribution of different forms of the boundwater in the
pore space, as well as because the reliability cannot be
improved under these conditions of the sample prepa
ration.
Analysis of the literature and experimental data
concerning the determination of solidphase density
by different methods for different preanalysis prepara
tions of clay soils with different disperse properties has
made it possible to conclude that the differences in
determination of the solidphase density of clay soils
with different mineral compositions and disperse
properties by different methods and for different pre
analysis preparations stem from a number of factors:
differences in the amount of bound water in clay soils
with different disperse properties, the absence of reli
able corrections for the boundwater density, differ
ences in the mechanisms of the phase transitions of
bound water, in preanalysis preparations, in the condi
tions of experiments, in the weighing accuracy of the
standard method, and the absence of boundary condi
tions and the area of the applicability region of the
7
existing methods of soil SPD determination.
The differences in SPD determination for more
loamy soils with different disperse properties for simi
6 The
calculations are made under the assumption that the
boundwater density is 1.05 g/cm3 [Olodovskii, 1989] and that it
is constant, as argued by Trofimov and Korolev [2008]. In real
ity, the calculations should use the calculation data of the
boundwater density that is not constant and is determined by
the disperse properties, mineral composition, type and content
of cations and form of their presence in soils, and abundance of
carbonates and organic substance [Makeeva et al., 2007, 2008].
The recommended corrections for the boundwater density of
clay soils are: 0.63 g/cm3 for heavy clays, 0.75 g/cm3 for heavy
loams, and 1.11 g/cm3 for light loams.
7 For a more detailed discussion, refer to [Makeeva, 2009].
MOSCOW UNIVERSITY GEOLOGY BULLETIN
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lar preanalysis preparations (with drying) using the
standard method (with S : L = 1 : 10 and weighing
accuracy of 0.001 g) for disperse systems with bound
water phase transition of the first kind stem from the
differences in the amount of boundwater remaining
after dehydration and from insufficient soil weighing
accuracy. Increasing the soil determination accuracy
from 0.001 to 0.0001 g at S : L = 1 : 10 will improve the
reliability of solidphase soil density determination.
For disperse systems with the boundwater phase tran
sition of the first kind, in a similar manner to the sec
ond kind, it is impossible to obtain reliable values of
the soil SPD because the data derived by the existing
methods are beyond the applicability limits of these
methods. Reliable values of soil SPD for these disperse
systems can be obtained only at the temperature of
structural instability. When the soil is prepared without
drying, with correction for the hygroscopic wetness,
the standard method fails to obtain reliable soil SPDs.
When the soil SPD is determined by the standard
method with drying at 105°C for disperse systems
within the applicability range at the solid to liquid
phase ratio S : L = 2 : 10 and with a weighing accuracy
of 0.0001 g, the solidphase density of the clay soil and
the boundwater density of monolayer adsorption are
determined within the method applicability range (ε =
±0.001 g/cm3, δ = 0.04%) [Makeeva et al., 2007].
When the soil is prepared without drying, with correc
tion for the hygroscopic wetness, and given that a reli
able correction for the boundwater density is applied,
the Kalachev method is used within the method appli
cability limit to determine the solidphase density to
within the method accuracy (ε ± 0.01–0.03 g/cm3,
σ = 0.38%–1.1%) [Makeeva et al., 2006, 2007]. It is
noteworthy that the soil SPDs obtained by the
Kalachev method with a reliable correction for the
boundwater density exceed those determined by the
standard method with drying at 105°C, for one and the
same weighing accuracy (0.01 g). This overprediction
is because the Kalachev method extracts only the soil
solid phase, corresponding to the soil density, while
the standard method extracts both the solid and liquid
phases. Increasing the weighing accuracy to 0.0001 g
within the standard method, while leaving the accu
racy of the Kalachev method unchanged (at 0.01 g),
and applying reliable correction for the boundwater
density within the applicability limits of the methods
reduces these discrepancies to a minimum.
Estimating the accuracy and fixing the applicability
region of the Kalachev method and also introducing
reliable corrections for the boundwater density to the
calculation formula make it possible to obtain reliable
data on the solidphase density of more disperse soils
(clays, loams) according to Kalachev method with
reasonable accuracy and convergence, similar to the
convergence for less disperse soils (sands, rocks),
within the method accuracy of ±0.01–0.02 g/cm3.
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MAKEEVA et al.
MATERIALS AND METHODS
The study method rests upon the experimental and
calculation work. The experiments consisted of a
series of problems: (1) to study the composition and
properties of the natural clay soils; (2) to simulate the
anthropogenic clay soils; (3) to determine the solid
phase density of the natural and anthropogenic clay
soils according to standard method (AllUnion State
Standard 518084) with dehydration [Methods…,
1984]; and 4) to determine the solidphase density of
natural and anthropogenic clay soils according to the
experimental Kalachev method. The calculations
were aimed at estimating the accuracy of disperse soil
SPD determination according to the standard and
experimental (Kalachev) methods.
As the study objects, we chose the clay soils with
different compositions and properties represented by
Kachinskii classification with light clay aQIII, heavy
loam dQIII, and light loam edQIII:
(1) light clay (aQIII, Ufa), sand fraction is 13.39%,
dust fraction is 57.46%, clay fraction is 29.45%;
Wq = 3%, WL = 39%, Wp = 22%, pH = 7.53; Ek =
17.2 mgeq/100 g, S0 = 98.3 cm2/g; 4.1% CaCO3,
0.03% CaSO4, 0.1% soluble salts, 0.52% organic sub
stance, 7.12% Få2O3, and 0.47% FeO;
(2) heavy loam (dQIII, Zvenigorod), the sand fac
tion is 6.26%, dust fraction is 61.08%, clay fraction is
32.66%; Wq = 4%, WL = 36%, Wp = 24%, Ek =
14.6 mgeq/100 g, S0 = 83.5 cm2/g; CaCO3, CaSO4,
no soluble salts and organic substances;
(3) light loam (edQIII, Nizhnii Novgorod), the sand
fraction is 8.42%, dust fraction is 85.55%, clay fraction
is 6.03%; Wq = 2%, WL = 22%, Wp = 16%, pH = 6.42,
Ek = 13.9 mgeq/100 g, S0 = 79.5 cm2/g; 1.13%
CaCO3, no CaSO4, 0.06% soluble salts, 0.04% organic
substance, 5.43% Fe2O3, and 0.44% FeO.
To mimic the anthropogenic disperse soils, the
studied clay soils were saturated with dissolutions of
sulfate salts of copper, zinc, and manganese at three
concentrations (C1, C2, and C3) for the solid to liquid
phase ratio S : L = 1 : 100 under static conditions. The
concentrations in the dissolutions were: C1 = 0.0521,
C2 = 0.2076, and C3 = 1.468 g/l for copper; C1 =
0.0174, C2 = 0.1932, and C3 = 0.9732 g/l for zinc; and
C1 = 0.0075, C2 = 0.0921, and C3 = 0.5035 g/l for man
ganese. For each of the studied soils we plotted an iso
therm of sorbtion q = f(C). It is noteworthy that the
obtained curves can be fitted by the Langmuir equation.
We determined the solidphase density and esti
mated the determination accuracy for natural and
anthropogenic clay soils, and for this we used different
methods: according to AllUnion State Standard
518084 with dehydration at T = 105°C using satura
tion with kerosene, according to experimental
Kalachev method, and using the “calculation”
method for the dissolution concentrations C2 and C3
for copper, zinc, and manganese. The calculation
method essentially consisted of summing the soil den
sity SPD, obtained according to AllUnion State
Standard 518084 with dehydration at T = 105°C
applying the saturation with kerosene at the weighing
accuracy of 0.0001 g, plus the values calculated from
increment of adsorbtion of heavy metals (copper, zinc,
and manganese) according to isotherm of sorbtion per
1 cm3 of soil. The SPD determination by the bottle
method was performed with triple repetition, and the
SPD determination for the anthropogenic clay soils
was controlled using the “calculation” method.
RESULTS AND DISCUSSION
We estimated the absolute and relative errors of the
SPD determination for clay soils according to stan
dard method (AllUnion State Standard 518084)
[Makeeva et al., 2006]. The formula for the determi
nation of the solidphase density has the form
q
ρ = ρ k ,
q + q1 – q2
(1)
where ρ is the solidphase soil density, g/cm3; q is the
weight of the dried soil, g; q1 is the weight of the den
simeter with kerosene, g; q2 is the weight of the den
simeter with kerosene plus soil, g; and ρk is the kero
sene density, g/cm3. To determine the absolute error, it
is necessary to differentiate formula (1). After a bit of
algebra, the formula becomes:
qρ k
qΔr k
ε ρ = ( Δq 1 + Δq 2 )
+ ( q + q1 – q2 ) ( q + q1 – q2 )2
ρk ( q1 – q2 )
+ 2 Δq,
( q + q1 – q2 )
(2)
where Δq, Δq1, and Δq2 are the weighing measurement
errors, and Δρk is the kerosene density measurement
error.
If the kerosene density is exactly known, formula (2)
is rewritten as
qρ k
ε ρ = 2 ( Δq 1 + Δq 2 )
( q + q1 – q2 )
(3)
( q1 – q2 )
+ ρ k 2 Δq.
( q + q1 – q2 )
To estimate the absolute and relative errors of solid
phase density determination by this method, we per
formed a series of calculations for weighings of soils
with different weights (10 and 40 g) with accurate and
inaccurate kerosene densities, as well as for different
measurement accuracies for soil and kerosene (from
0.01 to 0.001 g), with all the calculations assuming that
the kerosene volume is exactly known. The obtained
soil SPDs for different solid to liquid phase ratios
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(S/L = 1 : 10, 2 : 10, and 4 : 10), given different mea
surement accuracies for soil and kerosene weighings
(within 0.01–0.0001 g), are presented in the figure.
The determination error for the disperse soil SPD
is dominated by the kerosene density determination
error. It is noteworthy that increasing the soil weighing
accuracy and kerosene density determination accu
racy reduces these discrepancies to a minimum.
Changing the S : L ratio to 4 : 10 increases the deter
mination accuracy by a factor of 1.3 during saturation
with kerosene and yields a factor of three increase in
the accuracy during SPD determination for soils with
water (Table 1).
We estimated the absolute and relative errors of clay
soil SPD determination according to experimental
Kalachev method [Makeeva et al., 2006]. The calcula
tions are made using the formula [Kalachev et al.,
1997]
Density, g/cm3
2.80
2.75
2.70
2.65
2.60
2.55 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
@
Cu2 Cu3 Zn2 Zn3 Mn2 Mn
Solidphase densities of clay natural and anthropogenic
soils, determined by different methods with error estima
tion: light clay aQIII: experimental soil SPDs (AllUnion
State Standard 518084) at the ratio S : L = 2 : 10 for initial
natural soils with ε = ±0.001 g/cm3, δ = 0.04% and exper
imentalcalculation soil SPDs for anthropogenic soils tak
ing into account the increment of the heavy metal adsorp
tion according to sorption isotherm for the C2 and C3 con
centrations, ε = ±0.02 g/cm3, δ = 0.74% (curve 1);
experimental soil SPDs (AllUnion State Standard 518084)
at the ratio S : L = 2 : 10, for initial natural soils and
anthropogenic soils with ε = ±0.001 g/cm3, δ = 0.04%
(curve 2); experimental soil SPDs according to Kalachev
method, ε = ±0.01 g/cm3, δ = 0.38% (curve 3); experi
mental soil SPDs (AllUnion State Standard 518084) for
the ratio S : L = 4 : 10, ε = ±0.04 g/cm3, δ = 1.5% (curve 4);
experimental soil SPDs (AllUnion State Standard 518084)
for the ratio S : L = 1 : 10, ε = ±0.05 g/cm3, δ = 1.9%
(weighing accuracy is 0.0001 g for curves 1 and 2 and 0.01 g
for curves 3, 4, and 5).
ρs
G–g
(4)
1 + Wg
= ,
V1 P1 – V2 P2 Pt – P1 ⎞
( G – g )W g
P
⋅ – V 1 1 + ⎛ Pt ⎝
P1
P 1 – P 2⎠ ( 1 + W g )ρ bound
where G is the mass of soil together with can, g; g is the
mass of the can (2.88 g); Wg is the hygroscopic wet
ness, in fractions of unity; V1 is the volume of the
largersized reference sample (10 cm3); V2 is the vol
ume of the smallersized reference sample (6 cm3);
P1 is the excess air pressure measured in tests with a
reference sample of volume V1, atm; P2 is the excess air
pressure measured in test with reference sample of vol
ume P2, atm; P1 is air pressure measured in tests of a
soil sample, atm; ρbound is the density of tightly bound
99
water, g/cm3. We note that the quantity in the numer
ator is the mass of the absolutely dry soil, and the
quantity in the denominator is the soil volume V1 mea
sured in device.
Table 1. Estimate of the absolute and relative errors in the determination of solidphase density of disperse soils in the cases
of saturation with kerosene and water according to the standard method (AllUnion State Standard 518084) for different
ratios of solid to liquid phases
Soil weight, g
Kerosene density
accuracy Δρk, g/cm3
Absolute error ε,
g/cm3
Relative error
δ,%
Soil weighing
accuracy Δq, g
10
40
10
40
10
40
10
40
10
40
10
40
Exact, Δρk = 0
Exact, Δρk = 0
0.01
0.01
0.01
0.01
0.001
0.001
0.001
0.001
0.0001
0.0001
0.017
0.006
0.050
0.04
0.0346
0.0342
0.02
0.01
0.005
0.004
0.0005
0.0004
0.6
0.2
1.9
1.5
1.3
1.3
0.7
0.4
0.19
0.15
0.019
0.015
0.01
0.01
0.01
0.01
0.001
0.001
0.01
0.01
0.001
0.001
0.0001
0.0001
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MAKEEVA et al.
Table 2. Estimate of the absolute and relative errors in determination of solidphase density of disperse soils according to
the standard method (AllUnion State Standard 518084) and according to experimental Kalachev method
AllUnion State Standard
518084
Kalachev method
Δq
ΔW
ΔP
ΔV
ε±
δ, %
ε±
δ, %
0.01
0.01
0.001
0.001
0.0001
0.0001
0.001
0.001
0.0001
0.0001
0.00001
0.00001
1
10
1
10
1
10
0.001
0.001
0.001
0.001
0.001
0.001
0.01
0.029
0.003
0.023
0.003
0.023
0.34
1.09
0.11
0.86
0.10
0.86
0.05
0.05
0.005
0.005
0.0005
0.0005
1.9
1.9
0.19
0.19
0.019
0.019
The formulas for the calculation of the absolute (Δ)
and relative (δ) errors in the ρs determination, related
with errors in determination of quantities entering (4),
are:
Δ = dGΔg + dgΔg + dWΔW + ΔP 1 ΔP
+ ΔP 2 ΔP + dP t ΔP + dV 1 ΔV + dV 2 ΔV,
Δ ⋅ 100%,
δ = ρ s ( P 1, P 2, P t, V 1, V 2, G, q, W )
where Δg, ΔW, ΔP, and ΔV are the errors in determi
nation of the corresponding quantities, and
∂
dG = ρ s ,
∂G
∂
dP 2 = ρ s ,
∂P 2
∂
dg = ρ s ,
∂g
∂
dP t = ρ s ,
∂P t
∂ρ ,
dW = s
∂W
∂ρ ,
dV 1 = s
∂V 1
∂ρ ,
dP 1 = s
∂P 1
∂
dV 2 = ρ s .
∂V 2
This calculation assumes all the quantities (except
the boundwater density) to be approximate. We per
formed a series of calculations and estimated the abso
lute and relative errors of determination of solidphase
density of disperse soils with a weighing accuracy from
0.01 to 0.0001 g; also, the accuracy of determination
of hygroscopic wetness was estimated. We estimated
how the parameters themselves and the accuracy of
their determination influence the error of determina
tion of solidphase soil density (Table 2).
If the pressure measurement is the least accurate,
the pressuredetermination errors dominate the errors
of soil SPD determination. Weighingcaused mea
surement errors are less significant, and, at last, the
lowest error is introduced by the inaccuracy of specifi
cation of the volume of reference samples. When the
pressure is determined with the largest uncertainty
(Δp = 10), the errors will amount to ε = ±0.029 g/cm3,
δ = 1.1%; and when the pressure determination
uncertainty is the smallest (Δp = 1), the errors will be
Δp = 0.01 g/cm3, δ = 0.34% for a weighing accuracy up
to 0.01 g and for wetness determination accuracy of
0.001. Increasing the weighing accuracy to 0.001 and
wetness determination accuracy to 0.0001 will
increase the SPD determination accuracy respectively
by a factor of 1.3 at the maximum and by a factor of
3.1 at a minimum; this error will be ε = ±0.02
(0.003) g/cm3, ε = ±0.86% (0.11%) for the maximum
(minimum) pressure determination error. Further
increase of the weighing accuracy up to the fourth sig
nificant digit gives no sensible increase of the soil SPD
determination accuracy, due to a commensurate
countereffect of some other factors.
The calculations showed that the error of soil SPD
determination by the experimental Kalachev method
for minimal (maximal) pressure determination error is
5.6 (1.7) times that of the standard method. The
Kalachev method is a factor of 5 (1.7) more sensitive
than the standard method for minimal (maximal)
pressure determination error ΔP. At the same time, the
accuracy of the SPD determination of disperse soils by
Kalachev method can be increased by a factor of 1.7
over the standard method by using the ELA2M
instrument when the soil weighing accuracy is
increased up to 0.001 g (Table 2). The error estimate of
the determination of solidphase density of clay soils
by different methods for a fixed soil weighing accuracy
of q = 0.01 confirmed the claim of Kalachev et al.
[1997] that “…the new ELA 2 instrument and the
express method are as sensitive as the standard method
and greatly overperform the latter in accuracy and reli
ability of results.” The statement that the data are reli
able will be true if reliable corrections for the bound
water density are introduced within the method appli
cability region.
Analysis of literature data concerning clay soil SPD
determination by different methods [Kalachev et al.,
1997; Shlykov and Trapeznikov, 2002; Olodovskii,
1989] and the obtained data concerning soil SPD
determination by the standard method for different
preanalysis preparations of soil and by the Kalachev
MOSCOW UNIVERSITY GEOLOGY BULLETIN
Vol. 65
No. 2
2010
ESTIMATION OF THE DETERMINATION ERROR OF THE SOLIDPHASE
method, as well as the estimates of the absolute and
relative errors, have made it possible to conclude that
the Olodovskiirecommended corrections for the
boundwater density in clay soils are insufficiently jus
tified. In this regard, a method was developed to calcu
late the boundwater density in polymineral soils
[Makeeva et al., 2007]. The new result underlying the
method is that it determines the boundwater density
to an accuracy of 0.01–0.03 g/cm3 from the binding
energy of water, corresponding to the layer charge of
the heterogeneous surface; for disperse systems with
boundwater phase transition of the first kind, this is
done using the formula
W am zF 8πCε
ρ bound = ,
S
RT
(5)
where Wam is the wetness of one layer of bound water
(Curie point); S is the specific surface area; z is the cat
ion valency; F is the Faraday number; ε is the dielectric
constant of the boundwater; T is the absolute temper
ature; RT = const; C is the concentration of cations
(hydroxonium ions) and anions (hydroxides) of water
101
this case, the boundwater density also corresponds to
the layer charge of the disperse systems [Makeeva
et al., 2008]. The layer charge of the surface can be
determined by other methods, such as from the crys
talchemical formula, from the point of zero charge of
the surface with the help of potentiometer testing,
from values of ζpotential in calculation from Gouy–
Chapman formula, using modern numerical methods
9
of zoning calculation, etc. [Makeeva, 2009].
The obtained results made it possible to calculate,
from an independent formula, the thickness of the
water film of clay soils, saturated with different cat
ions; this formula in its main features coincides with
Debye screening radius ℵ–1. Makeeva [2009] showed
that the adsorption properties of the bound water are
determined by the position of the Fermi level. This is a
principally new proposition on hydration of heteroge
neous surface of disperse systems:
2
2
2
ប ⎛ 3π
N 3
E F = ⎞ ,
2m ⎝ V ⎠
8
within double electric layer (DEL).
Using this formula, we can successfully determine
the boundwater density in disperse systems with suf
ficient accuracy, in contrast to other methods [Ander
son and Low, 1950; Bradley, 1959; Baranova et al.,
1983, Olodovskii, 1989; and Serebryakov, 1988].
In clay soils dominated by montmorillonite (50%
smectite, 40% illite, light clay, aQIII), the boundwater
density is set to 0.63 g/cm3 (ε = ±0.01–0.03). In lighter
clay soils, with predominance of hydromica (light
loam, edQIII (Nizhnii Novgorod), 30% smectite, and
59% illite), the boundwater density is 1.11 g/cm3 (ε =
±0.001–0.03); whereas in heavy loam (dQIII, Zvenig
orod), composed of 40% smectite and 49% illite, the
boundwater density is 0.75 g/cm3 (ε = ±0.01–0.03).
The boundwater density in the polymineral clay soils
depends on the illite/smectite ratio (a) in the clay frac
tion, and it can be described by the regression equation
of the form: ρbound = 0.4201e0.4891a. Such a technical
result cannot be achieved by any other method of
determination of boundwater density; this is because
the method suggested here is adequate for the complex
physical phenomenon under scrutiny. For disperse
systems with boundwater phase transition of the first
kind or close to the second kind, when the binding
energy exceeds the layer charge, all methods fail to
obtain reliable values of the boundwater density; in
where EF is the Fermi energy (activation energy) of the
lattice; m is the electron mass, N is the number of free
electrons; and V is the volume of elementary cell.
It was found that in a layer of bound water when the
temperature and wetness change with a decrease in the
binding energy the concentration of the mobile ions
increases; in this regard, the bound water in disperse
systems can be considered as improper ferroelectrics,
and the phase transitions in such systems can be
described with the help of the Landau theory [Make
eva, 2009]. The question of the boundwater density of
disperse systems is, in fact, the question of the water
film thickness of the bound water. In essence, the solu
tion of this problem dates to theoretical considerations
concerning the effective electron charge on heteroge
neous surfaces [Ginzburg, 2004]. Experimental con
firmation and theoretical justification exist for the
Landau statement that the effective charge equals the
electron charge, as opposed to the arguments of Gin
zburg that e* = (2–3)e, as well as to the tenets of the
Bardeen–Cooper–Schrieffer theory, that the effective
charge is not equal to the free electron charge e, but
rather it is e* = 2e. In this regard, the views of many
researchers on the film thickness of the bound water
and smectite liquid crystals were based on these two
arguments.
8 It
9 The
was found that the boundwater density of disperse systems
does not depend on temperature; rather, it is determined by the
energy of binding to heterogeneous surface, corresponding to
the layer charge, in the absence of translational overlap in the
layer of boundwater for disperse systems with phase transition
of the first kind [Makeeva et al., 2008]. The temperature of the
structural instability of the film of boundwater of clay polymin
eral soils of montmorillonite composition (light clay, aQIII) is
320°C.
MOSCOW UNIVERSITY GEOLOGY BULLETIN
Vol. 65
energy of binding the water molecules created when hydro
gen binds with two oxygen atoms of tetrahedrons of the external
surface of hydromica lattice is two (1.75) times larger than the
energy of adsorbion of molecules, whose atoms come into inter
action simultaneously with two OHgroups of octahedrons of
the internal surface of montmorillonite [Olodovskii, 1984]; and
the boundwater density of monolayer adsorption on hydromica
will be 1.75 times larger than it is on the montmorillonite, as
determined by calculation.
No. 2
2010
102
MAKEEVA et al.
The developed calculation technique [Makeeva
et al., 2007], the established patterns of variations of
boundwater density in disperse systems in different
ranges of boundwater wetness and temperature
[Makeeva et al., 2008], and also the generalized theory
of the hydration processes of heterogeneous surface of
disperse systems, which is based on the basic notions
and laws, formulated in the theoretical physics, accu
rately describe the obtained experimental data and
explain a number of yet unresolved questions, as well
as making it possible to control a number of parame
ters of boundwater density at different interfaces and
in boundwater films, as well as water with a transi
tional state. In contrast to the Landau theory of phase
transitions, we established simple determinable
parameters related to the boundwater density: the
layer charge governs the boundwater density, the
dielectric constant governs the concentration of
mobile ions, and the dielectric losses govern the bind
ing energy [Makeeva, 2009].
The obtained results suggest that the recommended
corrections for the boundwater density in clay soils
are more justified than those proposed by other
authors earlier.
CONCLUSIONS
Estimation of the accuracy and reliability of the
determination of disperse soil SPD, as well the intro
duction of reliable corrections for the boundwater
density within the method applicability limits will
increase the reliability of the determination of disperse
soil SPDs using the Kalachev express method and the
standard method, and will broaden the applicability
limits of the Kalachev express method. The obtained
results may serve as a basis for accomplishing the pat
entability of the Kalachev method and a device for its
implementation, as well as for the creation of recom
mendations for introducing corrections to AllUnion
State Standard 518084 and AllUnion State Standard
2510082.
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