# Empirical Formula to Calculate Ionic Strength of Limnetic and Oligohaline Water on the Basis of Electric Conductivity: Implications for Limnological Monitoring

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{−1}) acts as one of the most important parameters of natural waters. It is indispensable for obtaining ion activities and thus is crucial for describing chemical processes in water solutions. Limnology, I, has many applications, but calculating the partial pressure of CO

_{2}(pCO

_{2}) and the carbonate saturation index (SI) are among the most important examples. The determination of I requires the full ion composition of water to be recognized, and when the concentration of some major ion(s) is/are missing altogether, the I value remains unknown. Because historical and monitoring data are often incomplete, it seems useful to provide a method for the indirect assessment of I. In this paper, we developed and tested an empirical model to estimate I on the basis of electric conductivity at 25 °C (EC). Our model consists of two linear equations: (i) I

_{mod}= 15.231 × 10

^{−6}·EC − 79.191 × 10

^{−6}and (ii) I

_{mod}= 10.647 × 10

^{−6}·EC + 26.373 × 10

^{−4}for EC < 592.6 μS·cm

^{−1}and for EC > 592.6 μS·cm

^{−1}, respectively. We showed that model performance was better than the hitherto used EC–I relationships. We also demonstrated that the model provided an effective tool for limnological monitoring with special emphasis on the assessment of CO

_{2}emissions from lakes.

## 1. Introduction

_{2}in particular. In the latter case, the ionic strength of lake water (I) is strongly required. In the current paper, we propose a refined method to derive the ionic strength (I) of water from the EC values.

^{−1}), the effect of ion pairing is very strong, especially for anions, as, for example, only 9.1% of CO

_{3}

^{2−}, 39% of SO

_{4}

^{2−}and 70% of HCO

_{3}

^{−}exist as free ions; the remaining are involved in more complex entities such as CaCO

_{3}

^{0}, NaSO

_{4}

^{−}, MgHCO

_{3}

^{+}, KSO

_{4}

^{−}, etc. [11]. In fresh water (with I between 0.05 and 10

^{−4}mol·L

^{−1}), the share of free ions is considerably higher and is estimated to be 81–99% for HCO

_{3}

^{−}and SO

_{4}

^{2−}and 46–96% for CO

_{3}

^{2−}[12]. I affects the behavior of ions in solutions and thus regulates dissolution/precipitation-related phenomena. Its fundamental use is to calculate the ion activities or effective concentrations of chemical species that can be involved in calculating chemical equilibria in water solutions and predicting chemical processes in natural waters. There are many potential applications of I in hydrological, limnological and hydrobiological studies; however, the assessment of carbonate saturation (and related decalcification of water), as well as pCO

_{2}estimations (and related assessment of hetero-/autotrophy and diffusion of CO

_{2}from water surfaces to ambient atmosphere), seems to be among the most important.

^{2+}, Mg

^{2+}, K

^{+}, Na

^{+}, as well as HCO

_{3}

^{−}, Cl

^{−}and SO

_{4}

^{2−}, which constitute the most abundant components of natural solutions, and thus contribute greatly to the total I value. In some limnological and hydrobiological studies, where data from many lakes (or sampling sites) are collected, the analyses of these ions act as a laborious and time-consuming task requiring the availability of sophisticated analytical facilities. On the other hand, when historical hydrochemical records are used, one can often find them incomplete (i.e., some ions are missing altogether), which makes I calculations impossible. Therefore, some attempts have been made to obtain I indirectly from electric conductivity (κ), which has been routinely measured during limnological monitoring [13,14,15,16]. This attempt relies on a strong dependence of conductivity on the ionic composition of water and I itself, which is expressed by the formula of McCleskey et al. [17] and Equation (1) is shown below:

_{i}is the ionic molal conductivity, m

_{i}is the speciated molality of the ith ion, λ

^{0}and A are temperature-dependent coefficients, and B is an empirical constant.

_{25}(conductivity standardized to 25 °C), and hereafter, for the sake of simplicity referred to as EC, a few empirical formulae were developed [13,14,16,18,19]. These formulae describe the relationship between I and EC by ordinary linear equations; however, it can demonstrate more complex patterns, and some differences between chemical types of water may exist (Figure S1). This makes it difficult to provide a universal equation that is appropriate for all natural waters. Therefore, the aim of this research was to adjust existing formulae to derive I from EC in lake water and thus provide limnologists and hydrobiologists with a simple tool for basic hydrochemical calculations. For this purpose, data from several European freshwater (salinity < 0.5 ppt; [20]) and oligohaline (salinity 0.5–5.0 ppt; [20]) lakes was collected to calculate the empirical model depicting the relationship between EC and I in lakes. In addition, this paper shows the advantages of using such a method in the analysis of spatial and historical limnological data. The results of this study may have broad implications in limnological and hydrobiological monitoring and, thus, may be useful in limnology as well as lake protection and management.

## 2. Materials and Methods

#### 2.1. Data Collection

^{2+}, Mg

^{2+}, K

^{+}, Na

^{+}, HCO

_{3}

^{−}, SO

_{4}

^{2−}, Cl

^{−}and NO

_{3}

^{−}monitoring survey data from a number of natural glacial and coastal lakes in Poland, Germany, Switzerland, and Sweden. The lakes covered a salinity range from 0.01 to 5.4 ppt. Details on the location of lakes, data collection, and availability are given in Figure 1 and Table S1.

^{−1}) calculated from the ion composition of water (TDS

_{IC}) to the TDS obtained from EC (TDS

_{EC}) was checked. TDS

_{IC}was presumed equal to the sum of the concentration of major ions C

_{i}, which is expressed as follows:

_{EC}was calculated using the formula (Equation (3) [21]), where

_{IC}/TDS

_{EC}was set to −15/+35%. Such a procedure led us to select 864 records to develop model equations.

#### 2.2. Water Analyses

^{−}, SO

_{4}

^{2−}and NO

_{3}

^{−}(881 Compact IC Pro; Metrohm, Herisau, Switzerland), titration with 0.05 M HCl with regard to methyl orange and phenolphthalein for HCO

_{3}

^{−}and CO

_{3}

^{2−}(ISO9963 [24] as well as AAS for Ca

^{2+}, Mg

^{2+}, K

^{+}and Na

^{+}(NovAA300; Analytik Jena GmBH, Jena, Germany) [25]. Analytical quality was verified with certified reference materials (Harbour water, NWHAMIL-20.2) and an ion charge balance.

#### 2.3. Chemical Calculations

^{−1}) was calculated on the basis of molarities m

_{i}and charges z

_{i}of individual ions (Ca

^{2+}, Mg

^{2+}, K

^{+}, Na

^{+}, HCO

_{3}

^{−}, SO

_{4}

^{2−}, Cl

^{−}and NO

_{3}

^{−}) with the formula in Equation (4):

_{i}), approximating the contributions of individual ions in the total conductivity of the mixed electrolyte solution, were assessed on the basis of ionic molar conductivities of the ions (λ

_{i}) and their molalities (M

_{i}; mol·kg

^{−1}) using Equations (5) and (6) [20]

_{2}partial pressures (pCO

_{2aq}; bar) in lake water were determined computationally on the basis of pH and the activity of HCO

_{3}

^{−}[26,27,28,29,30,31]. The mathematical expression for this calculation is as follows:

_{1}indicates the temperature-dependent equilibrium constant in H

_{2}CO

_{3}→H

^{+}+ HCO

_{3}

^{−}system and is obtained from the empirical formula reported by Kelts and Hsü [32]. Because Abril et al. [33] showed that the pCO

_{2}obtained was highly affected by pH, alkalinity, and DOC, calculations were only performed for the samples with circumneutral/alkaline pH and alkalinity > 1000 µmol·L

^{−1}, as recommended by Abril et al. [33]. CO

_{2}saturation was expressed as ΔpCO

_{2}in relation to pCO

_{2}in equilibrium with atmospheric CO

_{2}. Positive ΔpCO

_{2}ΔpCO

_{2}indicated supersaturation, while negative showed undersaturation (and, thus, the absorption of CO

_{2}from ambient air).

_{i}were derived using the Debye–Hückel equation (Equation (8)) and temperature-dependent coefficients A and B, including the ionic radius (r

_{i}) and ion charge (Z

_{i}) from [34]. Thus,

_{calc}) was obtained from the activities of Ca

^{2+}, CO

_{3}

^{2−}(values in brackets in Equation (9)) and the calcite solubility coefficient was obtained at in situ temperature K

_{c}.

_{c}was computed using an empirical equation given by Tchobanoglous et al. [17].

#### 2.4. Model Development

#### 2.5. Model Validation

_{emp}). The validation results were expressed as the relative error (ε%; Equation (10)) between the modeled I value (I

_{mod}) and empirical I value (I

_{emp}) obtained from the analytical concentration data (Equation (4)) using the following formula:

## 3. Results and Discussion

#### 3.1. Data Characterization

^{−1}, translating into an EC of 19 to 8242 μS·cm

^{−1}, respectively. Such a TDS range corresponded to limnetic and oligohaline water [21]. However, the frequency distribution of samples in different TDS/EC intervals was highly and positively skewed (Figure S3). The water between 250 and 818 μS·cm

^{−1}was by far the most abundant group encompassing c.a. 67% of the dataset.

^{−1}), moderately (250 < EC < 750 μS·cm

^{−1}), highly (750 < EC < 2250 μS·cm

^{−1}) and very highly mineralized (EC > 2250 μS·cm

^{−1}) [46]. The first group encompassed Swedish lakes (no. 25–28 in Table S1), the lakes of the Bory Tucholskie area (no. 5–14 in Table S1), as well as Lake Stechlin and Rotsee (no. 23–24 in Table S1). Moderately mineralized lakes were represented by Lake Licheńskie, Lake Łódzko-Dymaczewskie, Lake Dębno, Lake Trześniowskie (no. 1–4 in Table S1), Lake Sarbsko and Rotsee (no. 22 and 24 in Table S1). Highly and very highly mineralized water only occurred in coastal lakes of the Polish Baltic coast (no. 15–21 in Table S1). Weakly mineralized water displayed a similar t

_{i}for different ions, albeit with slightly higher values for Ca

^{2+}(Table 1). Comparable contributions from divalent and monovalent ions to the total conductivity indicated that the EC of lake water was due to the combined effect of Ca

^{2+}, Cl

^{−}, SO

_{4}

^{2−}, HCO

_{3}

^{−}, and Na

^{+}. Moderately mineralized lake water also displayed similar total t

_{i}values for divalent and monovalent ions, but the EC was primarily due to Ca

^{2+}and HCO

_{3}

^{−}with a minor role of other species (Table 1). In highly and very highly mineralized water, monovalent ions by far predominated in shaping conductivity. In the former group, the overall t

_{i}of monovalent ions was 0.65 and, in the latter, 0.84 (Table 1). Respective contributions from the divalent ions ranged from 0.35 in highly mineralized lakes to 0.16 in very highly mineralized water. In general, anions were more influential on conductivity than cations, albeit the role of NO

_{3}was negligible in all lakes studied (Table 1).

#### 3.2. Modeling Results

_{1}) was concordantly positioned with both methods in a narrow range of EC values around 592 µS·cm

^{−1}. From Figure S4, it emerged that in 50% of the subsamples tested, the BP

_{1}obtained was within an <2 µS·cm

^{−1}interval between 592 and 594 µS·cm

^{−1}, and in 80% of the subsamples, values ranged between 590 and 597 µS·cm

^{−1}(i.e., within 7 µS·cm

^{−1}-wide interval). The arithmetic mean value of BP

_{1}from 1000 models was 591.99 µS·cm

^{−1}(Figure S4). However, because the above number was non-existent in the dataset analyzed, BP

_{1}was set to the nearest “real” value, which was 592.6 µS·cm

^{−1}. Given that the precision of EC measurements with digital conductometers is typically around 1% (relative to the mean BP

_{1}value of 592.6 µS·cm

^{−1}, this translates into c.a. 6 µS·cm

^{−1}), the dispersion of results is totally within the instrumental error. The reliability of the estimated BP

_{1}value was further corroborated by the similarity between the mean BP

_{1}of 591.99 µS·cm

^{−1}as well as the median BP

_{1}(593.0 µS·cm

^{−1}) and modal BP

_{1}values (593.3 µS·cm

^{−1}). On the other hand, the second breakpoint, BP

_{2}, was more difficult to find. BIC put it at 4000–4500 µS·cm

^{−1}while, from the Davies test, it emerged that the BP

_{2}was at 600, 1000–1500 and 2500–3000 µS·cm

^{−1}. This variability could, in large part, be related to the fact that the distribution of the EC values in the dataset was highly skewed, with EC > 4000 µS·cm

^{−1}being rather underrepresented. This fact could have affected the model’s stability. Considering the differences in the positioning of the BP

_{2}, a decision was made that the double-segment regression model (i.e., with one breakpoint) was the most appropriate for the dataset analyzed.

^{−106}to 1.65

^{−55}, with 4.49

^{−58}on average. The distribution of p values was highly positively skewed (Sk = 22.82) and leptokurtic (kurtosis = 595.14). The median of the probability distribution Me = 8.98

^{−63}was considerably lower than the average.

^{−1}. The equations are parametrized as follows:

_{mod}= 15.231 × 10

^{−6}·EC − 79.191 × 10

^{−6}for EC < 592.6 µS·cm

^{−1}

_{mod}= 10.647 × 10

^{−6}·EC + 26.373 × 10

^{−4}for EC > 592.6 µS·cm

^{−1}

^{−1}) were different between −8.0 and +16.0%. The mean relative difference between the models was +4.35% (with a standard deviation of 7.13), which indicated that, in relation to segmented regression, the ordinary linear regression tended to overestimate the values of I. The highest positive deviations (>+4%) were for EC < 312 µS·cm

^{−1}as well as for EC > 4500 µS·cm

^{−1}. The highest negative deviations (<−4%) occurred for EC between 500 and 1832 µS·cm

^{−1}.

^{−1}HCO

_{3}

^{−}solution, the EC is around 3870 μS·cm

^{−1}, while the EC of the Ca

^{2+}solution of the same molarity is ca. 8300 μS·cm

^{−1}. Consequently, the changes in EC vs. I due to the increase in concentrations of monovalent ions follow a less steep trend than for divalent ions (Figure 3).

^{−1}) in our dataset was enriched in monovalent ions (Na

^{+}and Cl

^{−}in particular) compared to weakly and moderately mineralized water, and consequently, these ions were the primary carriers of conductivity. The contributions of monovalent ions to the total EC of highly mineralized water were 0.65–0.84 (Table 1). On the contrary, at EC < 750 μS·cm

^{−1}, the transport numbers for divalent ions, t

_{i}, were 0.52–0.54 (Table 1), thus indicating that divalent ions (Ca

^{2+}in particular) were more influential in shaping conductivity. Given the critical role that the ion composition had for EC and I of the solutions, it seems reasonable to conclude that a chemical change from Ca

^{2+}-dominated to Na

^{+}/Cl

^{−}dominated water accompanying an increase in salinity translated into the flattened EC vs. I trend at high conductivity values.

#### 3.3. Model Validation

^{−1}(encompassing most freshwater lakes) as well as for 2500–7500 μS·cm

^{−1}and, for these conductivity ranges, our model gave much better adjustment to the data than other models (Figure 4A–E). On the other hand, the model seems to underpredict the I for EC between 1000 and 2500 μS·cm

^{−1}and for >7500 μS·cm

^{−1}. Slightly better results for these ECs were obtained using the equation shown in [13].

#### 3.4. Potential Implications of the Model

#### 3.4.1. Calculating γ Activity Coefficients for Major Ions

^{2+}, Mg

^{2+}, HCO

_{3}

^{−}, CO

_{3}

^{2−}), for temperatures between 0 and 30 °C and EC ranging from 50 to 8000 μS·cm

^{−1}. The results of these calculations are shown in Figure 5 and Tables S2–S5.

_{3}

^{2−}and showed an up to 4.1% decrease in γ, respectively, at 30 °C. This species also showed the highest differences in γ over the EC range considered. The values of the γ coefficient for CO

_{3}

^{2−}dropped from around 0.90 at 50 μS·cm

^{−1}to 0.36–0.39 at 8000 μS·cm

^{−1}. Divalent cations (Ca

^{2+}, Mg

^{2+}) also demonstrated a considerable decrease in activity resulting from enhanced I; however, the γ at a maximum EC was slightly higher than for CO

_{3}

^{2−}and ranged from 0.41 to 0.45. On the other hand, the activities of HCO

_{3}

^{−}were much less affected by ionic strength. Over our EC range, the activities of bicarbonates declined from 0.97 to 0.76–0.78.

#### 3.4.2. Calculating Carbonate Saturation of Lake Water

_{i}values given in Tables S2–S5 can be applied to hydrological/limnological monitoring. Below, we used these values to calculate SI

_{calc}in lake water based on data from Lake Kierskie, Poland [45]. The calculations were run using different approaches: (i) for the full ion composition of lake water which allowed the true SI

_{calc}values to be approximated, hereafter referred to as SI

_{full}, (ii) using the Langelier SI (LSI) calculator available at: https://www.lenntech.com/ro/index/langelier-explanation.htm (accessed on 3 October 2023) (and applied by [45]) and (iii) using the γ coefficients for HCO

_{3}

^{−}and Ca

^{2+}from Tables S2 and S4, respectively, to obtain modeled SI

_{calc}(SI

_{model}). The idea behind these calculations was to check the recovery of SI

_{full}with different calculation methods.

_{calc}estimations based on modeled γ values (SI

_{model}) better reproduced the SI

_{full}than the LSI. The regression equation for the SI

_{full}–SI

_{model}relationship indicates that SI

_{model}values systematically underestimated the SI

_{full}by c.a. 0.01 (0.01–0.02), while the LSI values were c.a. 0.25 (0.22–0.26) lower than the SI

_{full}.

#### 3.4.3. Screening of Spatial Distribution of pCO_{2} in Lakes

_{3}

^{−}concentrations on the surface water layer. The model developed allowed the screening of one of the largest lakelands in Poland for the distribution of the pCO

_{2}values and potential tendencies for the absorption/emission of CO

_{2}from/to the atmosphere. One lake was excluded from the calculations (Lake Tyrsko) because of its too-low alkalinity (860 µmol·L

^{−1}). From the calculations, it emerged that pCO

_{2}ranged from 0.09 to 7.05 mbar, translating into 3.28–247 μmol CO

_{2}·L

^{−1}. In 20 lakes, the pCO

_{2}obtained was slightly lower than the equilibrium of CO

_{2}concentrations in lake water of approximately 14.171 μmol·L

^{−1}[48] (ΔpCO

_{2}from −11 to −0.2 μmol·L

^{−1}below equilibrium), arguing for their autotrophic character. The autotrophic lakes (i.e., absorbing CO

_{2}from the atmosphere) were primarily located in the Suwałki Lake District. On the other hand, positive ΔpCO

_{2}, indicative of heterotrophy (i.e., emission of CO

_{2}from the ambient atmosphere), occurred in most lakes studied, though it varied in a broad range from 233 to 0.1 μmol·L

^{−1}.

#### 3.4.4. Temporal Changes in pCO_{2} in Lakes

_{2}in lakes at different timescales and seems to be particularly advantageous when using historical data to obtain information on long-term CO

_{2}trends where there is no possibility to measure CO

_{2}directly. Clearly, calculations can only be made for water that has a circumneutral and alkaline pH and alkalinity above 1000 µmol·L

^{−1}, as recommended by [33]. To illustrate this based on t, pH, EC and HCO

_{3}

^{−}values for lake surface water, we have calculated pCO

_{2}for a group of five lakes near Konin (encompassing Lake Gosławskie, Lake Licheńskie, Lake Ślesińskie and Lake Wąsosko-Mikorzyńskie, here referred to as Konin lakes), which since the 1960s, has received heated water from nearby power plants and has thus become thermally polluted [49]. The calculations showed that between 2015 and 2021, the pCO

_{2}values in these lakes ranged between 0.23 and 6.91 mbar and there were no statistically significant differences between the lakes (Figure S8). However, the pCO

_{2}varied greatly throughout each year with maximum values in winter and minimum during the spring and summer. The pCO

_{2}in Konin lakes implies that for the most part of the period studied, lakes were heterotrophic (ΔpCO

_{2}> 0). Autotrophy occurred only sporadically during spring and summer (Figure S8).

_{2}in Lake Suminko and Lake Kierskie varied between 0.19 and 14.3 mbar and from 0.22 to 10.5 mbar, respectively (Figure S9A,B) and thus were slightly higher than in Konin lakes, but displayed similar annual pattern. Lake Suminko displayed a relatively long period of autotrophy between April and September 2009 as well as in April/May 2010 (Figure S9A), while Lake Kierskie was autotrophic between June and September 2016 (Figure S9B). For Lake Kierskie, we had the possibility to compare the modeled pCO

_{2}with the values calculated on the basis of the full chemical composition of lake water samples. The relative error ε% obtained was between −0.24 and −0.95%.

#### 3.4.5. Spatial CO_{2} Distribution in Lakes

_{2}in lake water is often highly irregular [51,52], the selection of site(s) (or a number of sites) should be preceded by a screening of the lake studied to recognize the variability of the gas and its distribution. Our method allows for the relatively rapid collection of data on CO

_{2}distribution throughout a water body. For the sake of demonstration between March 2022 and February 2023, with the monthly resolution, we collected hydrochemical data from three stations across Lake Licheńskie, and one of the thermally polluted Konin lakes [49]. To check the effect of the input of heated, water on CO

_{2}productivity in Lake Licheńskie, we took water samples in front of the mouth of the canal, delivering cooling effluents to the lake as well as in two thermally different parts of the lake (Figure S10). From our results, it emerged that the sites studied indeed showed differences in pCO

_{2}values from 0.2 to 1.9 mbar in the least heated section to 0.6–5.5 mbar at the inflow of PPK water to the lake (Figure S10); however, contrary to what was expected, on a yearly basis, these differences were not statistically significant. Despite the fact that the lake is net heterotrophic, some parts of the lake may temporarily show autotrophy (e.g., site L3 in Figure S10).

## 4. Conclusions

^{−1}. The performance of the model at EC > 1000 µS·cm

^{−1}could be improved by collecting more hydrochemical data from oligohaline water. The method allows for a rapid determination of a number of standard hydrochemical parameters such as ion activity coefficients, pCO

_{2}, and saturation states for different minerals. This method is dedicated to limnological and hydrobiological applications and can be used for the standard monitoring of biogeochemical processes in the lake water column. It should be underlined, however, that the approach proposed should only be used when data on the full ion composition of water samples is unavailable, especially for pCO

_{2}calculations.

## Supplementary Materials

_{2}(difference between pCO

_{2}in lake surface water and atmospheric CO

_{2}concentration) for lakes of NE Poland; Figure S8. Temporal changes in pCO

_{2}in the surface water of the Konin lakes between January 2015 and December 2021. Yellow arrows indicate periods of autotrophy (i.e., when pCO

_{2}< equilibrium pCO

_{2}; Figure S9. Temporal changes in pCO

_{2}in the surface water of Lake Suminko (A) and Lake Kierskie (B); Figure S10. Spatial and temporal changes in pCO

_{2}in the surface water of Lake Licheńskie. the equilibrium CO

_{2}concentrations in Polish lakes marked with a dotted line; Table S1. Location of data collection sites, sampling strategy, and data availability; Table S2. Activity coefficient γ for HCO

_{3}

^{−}for different EC and t; Table S3. Activity coefficient γ for CO

_{3}

^{−}for different EC and t; Table S4. Activity coefficient γ for Ca

^{2+}for different EC and t; Table S5. Activity coefficient γ for Mg

^{2+}for different EC and t. References [53,54,55,56,57,58,59,60,61,62,63,64,65,66,67] are cited in Supplementary Materials.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

_{2}in the group of heated lakes involved in the cooling system of the Konin and Pątnów power plants. Carsten Schubert (EAWAG), Sarine Wollrab (IGB Berlin), Mirosław Żelazny (UJ Kraków), Jacek Tylkowski (ZMŚP Biała Góra), Karina Apolinarska (AMU Poznań), Wojciech Tylmann (UG Gdańsk) are acknowledged for sharing hydrochemical data. A bathymetric map of Lake Licheńskie was used with the courtesy of Adam Choiński (AMU Poznań).

## Conflicts of Interest

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**Figure 1.**Distribution of data collection sites for this study. Consult Table S1 for coordinates and more specific information on data collection and availability. (1−Lake Licheńskie; 2−L. Łódzko-Dymaczewskie; 3−L. Dębno; 4−L. Trześniowskie (Ciecz); 5−L. Ostrowite; 6−L. Jeleń; 7−L. Płęsno; 8−Krzywce Wielkie; 9−Zielone; 10−L. Skrzynka; 11−L. Krzywce Małe; 12−L. Mielnica; 13−L. Głowka; 14−L. Bełczak; 15−L. Resko Przymorskie; 16−L. Jamno; 17−L. Bukowo; 18−L. Kopań; 19−L. Wicko; 20−L. Gardno; 21−L. Łebsko; 22−L. Sarbsko; 23−L. Stechlin; 24−L. Rotsee; 25−L. Edasjön; 26−L. Siggeforasjön; 27−L. Fiolen; 28−L. Tångerdasjön).

**Figure 2.**Relationship between measured EC and modelled I. Data points used for model development marked are as blue dots. Ordinary linear regression model for all data (without the break point) given in green. Stepwise regression model fits for EC < 592.6 μS·cm

^{−1}(I

_{mod}= 15.231 × 10

^{−6}·EC − 79.191 × 10

^{−6}) and for EC > 592.6 μS·cm

^{−1}(I

_{mod}= 10.647 × 10

^{−6}·EC + 26.373 × 10

^{−4}) are in pink and brown, respectively.

**Figure 3.**Relationship between the conductivity and ionic strength of a hypothetical one-component solution for different major ions. The values of EC were calculated using the equation by [20] for a concentration range from 10

^{−7}to 10

^{−2}mol·L

^{−1}. Note that for monovalent ions, the slope of the trend line is lower than for divalent ions.

**Figure 4.**Relative error of the I estimation (ε) vs. EC range considered using the model developed in this study (

**A**) and the models by Ponamperuna et al. [18] (

**B**), Griffin and Jurniak [13] (

**C**) as well as Tchobanoglous et al. [16]. The mean ε values for (

**A**) were −3.4%, 19.7% for (

**B**), −2.7% for (

**C**) and 21.6% for (

**D**). Trendline (thick colored lines) was smoothed using the LOESS method (local polynomials; Cleveland [47]) with the use of the linear function fitted by the method of least squares and moving window encompassing 50% of data points. Graph (

**E**) shows a comparison of the smoothed error distributions of all methods. The colors in panel (

**E**) are the same as in panels (

**A**–

**D**).

**Figure 5.**Ion activity coefficients γ for HCO

_{3}

^{−}(

**A**), CO

_{3}

^{2−}(

**B**), Ca

^{2+}(

**C**) and Mg

^{2+}(

**D**) and different EC and t. Blue and red lines show γ

_{i}values at t = 0 °C and t = 30 °C, respectively. The effect of temperature is negligibly weak.

**Table 1.**Average transport numbers (t

_{i}) for the groups of lakes. The values given show the contributions of particular chemical species into the total electrical conductivity of the solution.

Lake Mineralisation | EC | t_{i} | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

μS·cm^{−1} | Na^{+} | Ca^{2+} | Mg^{2+} | K^{+} | HCO_{3}^{−} | SO_{4}^{2−} | Cl^{−} | NO_{3}^{−} | Dival. Ions | Monoval. Ions | |

Weak | <200 | 0.13 | 0.25 | 0.10 | 0.03 | 0.14 | 0.17 | 0.18 | 0.01 | 0.48 | 0.52 |

Moderate | 200–750 | 0.07 | 0.32 | 0.09 | 0.01 | 0.27 | 0.12 | 0.11 | 0.00 | 0.46 | 0.54 |

High | 750–2250 | 0.17 | 0.16 | 0.09 | 0.02 | 0.12 | 0.10 | 0.34 | 0.00 | 0.65 | 0.35 |

Very high | >2250 | 0.27 | 0.04 | 0.07 | 0.01 | 0.03 | 0.05 | 0.53 | 0.00 | 0.84 | 0.16 |

**Table 2.**Relative error of estimation for I on the basis of EC (ε%) with different empirical formulae. The lowest values indicating the best prediction are marked in bold. The model equations developed in this study provide better perditions throughout most of the conductivity range investigated.

Conductivity [μS·cm ^{−1}] | This Study | Ponnamperuna [18] ^{#} | Griffin and Jurniak [13] ^{&} | Tchobanoglous et al. [16] * |
---|---|---|---|---|

<250 | 1.1 | 13.1 | −8.1 | 14.8 |

250–500 | 2.1 | 8.9 | −11.5 | 10.7 |

500–1000 | −3.5 | 11.0 | −9.8 | 12.7 |

1000–2500 | −12.6 | 11.3 | −9.6 | 13.0 |

2500–5000 | 6.6 | 51.1 | 22.7 | 53.4 |

5000–7500 | −6.9 | 34.5 | 9.3 | 36.6 |

>7500 | −17.8 | 20.4 | −2.2 | 22.3 |

^{#}I = 0.016·EC [mmho];

^{&}I = 0.013·EC [mmho]; * I = 2.5 × 10

^{−5}·TDS [mg·L

^{−1}].

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## Share and Cite

**MDPI and ACS Style**

Woszczyk, M.; Stach, A.; Nowosad, J.; Zawiska, I.; Bigus, K.; Rzodkiewicz, M.
Empirical Formula to Calculate Ionic Strength of Limnetic and Oligohaline Water on the Basis of Electric Conductivity: Implications for Limnological Monitoring. *Water* **2023**, *15*, 3632.
https://doi.org/10.3390/w15203632

**AMA Style**

Woszczyk M, Stach A, Nowosad J, Zawiska I, Bigus K, Rzodkiewicz M.
Empirical Formula to Calculate Ionic Strength of Limnetic and Oligohaline Water on the Basis of Electric Conductivity: Implications for Limnological Monitoring. *Water*. 2023; 15(20):3632.
https://doi.org/10.3390/w15203632

**Chicago/Turabian Style**

Woszczyk, Michał, Alfred Stach, Jakub Nowosad, Izabela Zawiska, Katarzyna Bigus, and Monika Rzodkiewicz.
2023. "Empirical Formula to Calculate Ionic Strength of Limnetic and Oligohaline Water on the Basis of Electric Conductivity: Implications for Limnological Monitoring" *Water* 15, no. 20: 3632.
https://doi.org/10.3390/w15203632