Oxygen solubility

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Oxygen solubility

Description

The oxygen solubility, S_O₂ [µM/kPa] = [(µmol·L^-1)/kPa], expresses the oxygen concentration in solution in equilibrium with the oxygen pressure in a gas phase, as a function of temperature and composition of the solution. The inverse of oxygen solubility is related to the activity of dissolved oxygen. The oxygen solubility in solution, S_O₂(aq), depends on temperature and the concentrations of solutes in solution, whereas the dissolved oxygen concentration at equilibrium with air, c_O₂^*(aq), depends on S_O₂(aq), barometric pressure and temperature. S_O₂(aq) in pure water is 10.56 µM/kPa at 37 °C and 12.56 µM/kPa at 25 °C. At standard barometric pressure (100 kPa), c_O₂^*(aq) is 207.3 µM at 37 °C (19.6 kPa partial oxygen pressure) or 254.7 µM at 25 °C (20.3 kPa partial oxygen pressure). In MiR05 and serum, the corresponding saturation concentrations are lower due to the oxygen solubility factor: 191 and 184 µM at 37 °C or 234 and 227 µM at 25 °C.

Abbreviation: S_O₂ [µM/kPa]

Reference: MiPNet06.03 POS-calibration-SOP

Communicated by Gnaiger E 2010-10-21 (last update 2021-12-19)

Oxygen solubility in gas (g) versus aqueous solution (aq)

At room temperature or 37 °C, the concentration of oxygen is 30- to 40-fold higher in the gas phase compared to the aqueous phase in equilibrium with the gas phase. This explains why even a small gas bubble in an aqueous volume of a respirometer causes enormous errors in the measurement of respiration. And it is the reason why we need red blood cells with hemoglobin, to boost the total oxygen concentration carried in the blood. The ideal gas law plays a central role in elucidating the behavior of gases dissolved in aqueous solution, where the gas interacts with a very different environment compared to the gas phase. Following the track of a few equations derived in this overview provides a deep understanding of the rationale implemented in the oxygen calibration algorithms in HRR (MiPNet06.03 POS-calibration-SOP).

The key quantities in the gas law are:

Gas pressure, p [Pa]: 1 atm = 101.325 kPa; standard pressure is 100 kPa.
Temperature, T [K]: Temperature in °C is converted to absolute temperature as T[K] = T_C[° C] + 273.15.
Amount of substance, n [mol].
Volume, V [m³]: Concentration, c = n·V^-1.
Gas constant, R = 8.314462618 J·mol^-1·K^-1.

Four forms of the gas law are used in the derivations below:

Eq. a:  p·V = n·RT ; the fundamental format: pressure-volume work [Pa·m³]=[J]

Eq. b:  p = c·RT ; pressure [Pa] and concentration: c=n·V^-1  
--------------------- Compare chemical potential: µ = µ° + ln(γ·c/c°)·RT; µ°=0 and γ=1 in the gas phase

Eq. c:  V·n^-1 = RT·p^-1 ; the molar volume, V_m [m³·mol^-1]=[J·mol^-1·Pa^-1]

Eq. d:  c·p^-1 = RT^-1 ; the solubility, S [mol·m^-3·Pa^-1]=[mol·J^-1]

Solubility is defined as concentration per pressure,

Eq. 1:  S_G(g) = c_G/p

The solubility of an ideal gas in the gas phase, S_G(g), can be calculated simply from the ideal gas law, since the activity coefficient of an ideal gas in the gas phase equals one, γ=1. Arranging the ideal gas law (Eq. a) to obtain solubility (Eq. d),

Eq. 2:  S_G(g) = c_G/p = RT^-1

c_G=n_G·V^-1 is the inverse of the molar volume, V_m,G (Eq. c; Cohen 2008 IUPAC Green Book),

Eq. 3:  V_m,G(g) = V·n_G^-1 = RT·p^-1

where the last expression is another form of the ideal gas law (Eq. c). Prior to 1982 the standard pressure has been defined as 101.325 kPa (STPD). Inserting this standard pressure, standard temperature of 273.15 K, and the corresponding value of RT=2271 J/mol into Eq. 3, yields the familiar value for the molar volume of an ideal gas of 22.414 L∙mol^-1. But V_m,G(g) = 22.711 L∙mol^-1 at 100 kPa, compared to V_m,O₂(g) = 22.689 L∙mol^-1 under these standard conditions. The ratio of these molar volumes is 0.999, such that we can practically treat oxygen in the gas phase as an ideal gas.

A distinction has to be made between solubility of an ideal gas in the gas phase, S_G(g), with an activity coefficient equal to one, and oxygen solubility of a real gas in the aqueous phase, such as S_O₂(aq), with an activity coefficient, γ_O₂, obtained from the experimentally determined solubility of dissolved oxygen.

The definition of (relative) activity of oxygen is (concentration basis; Cohen 2008 IUPAC Green Book),

Eq. 4:  a_O₂ = γ_O₂·c_O₂·c°^-1 ; c°=1 mol·L^-1

At equilibrium between the gas phase and aqueous phase at a constant pressure, p, the activity of oxygen is equal in the two phases,

Eq. 5:  a_O₂*(aq) = a_O₂*(g)

c° cancels when inserting Eq. 4 into Eq. 5, and solving for the aqueous activity coefficient,

Eq. 6:  γ_O₂(aq) = γ_O₂(g)·c_O₂*(g)·c_O₂*(aq)^-1

The equilibrium concentrations, c*, in Eq. 6 are obtained from the solubilities (Eq. 1), and the gas pressure cancels in the ratio, since p is equal in both phases,

Eq. 7:  γ_O₂(aq) = γ_O₂(g)·S_O₂(g)·S_O₂(aq)^-1

Oxygen solubilities (Forstner and Gnaiger 1983; MiPNet06.03 POS-calibration-SOP) and the constants defined above are used in these equations to obtain the activity coefficient of dissolved oxygen (Eq. 7) as a function of temperature, summarized in Tab. 1.

The solubility of an ideal gas in the gas phase, S_G(g) [µM·kPa^-1] (Eq. 2), is compared in Tab. 1 to the oxygen solubility in pure water, S_O₂(aq) (for which frequently the abbreviated symbol S_O₂ is sufficient), and in MiR05. The activity coefficient of dissolved oxygen, γ_O₂(aq), is the ratio of the oxygen solubility in gas to aqueous solution (Eq. 7), which is a concentration ratio at equilibrium, c_O₂*(g)/c_O₂*(aq) (Eq. 6; γ_O₂(g)=1). γ_O₂(aq) increases from 20 at 0 °C to 38 at 40 °C (Tab. 1). Importantly, the activities and chemical potentials of oxygen in the gas and aqueous phase are equal at equilibrium, when concentrations in the aquesous phase vary 20- to 40-fold, taking into account that the aqueous phase under experimental conditions with mitochondria or cells is a physiological salt solution (MiR05; Tab. 1), in which the oxygen solubility is further reduced (see Oxygen solubility factor).

The definition of activity in the gas and aqueous phase depends on the definition of the standard state. In this context, it is 'advantageous to choose the standard state of unit activity as that in which the partial pressure of the gas is unity at a given temperature' (Hitchman and Gnaiger 1983).

Numerical examples for oxygen solubility and the activity of dissolved oxygen

Further application of the concept of oxygen solubility introduced above requires the following constants:

(a) The volume fraction of oxygen in dry air is constant at 0.20946.
(b) The saturation water vapour pressure at 25 °C is 3.17 kPa. The partial pressure of oxygen in water vapour saturated air at 25 °C and barometric pressure of 100 kPa equals (100-3.17)·0.20946 = 20.28 kPa. The oxygen solubility in pure water at 25 °C is 12.56 µM/kPa (Tab. 1).
(c) The saturation water vapour pressure at 37 °C is 6.27 kPa. The partial pressure of oxygen in water vapour saturated air at 37 °C and barometric pressure of 100 kPa equals (100-6.27)·0.20946 = 19.63 kPa. The oxygen solubility in pure water at 37 °C is 10.56 µM/kPa (Tab. 1).
(d) The oxygen solubility factor of the medium MiR05 relative to pure water is 0.92 at 25 °C and 37 °C (Tab. 1).

An aqueous solution of pure water in equilibrium with air at standard barometric pressure (100 kPa) is heated from 25 °C to 37 °C, maintaining equilibrium between the aqueous and gaseous phase, achieved by continuous stirring of the aqueous phase in the 'open chamber'. The concentration of dissolved oxygen changes from 254.7 µM at 25 °C to 207.3 µM at 37 °C. The partial pressure of oxygen changes from 20.3 to 19.6 kPa.
An aqueous solution of pure water in equilibrium with air at standard barometric pressure (100 kPa) changed from pure water to a physiological salt solution (MiR05) at 37 °C. The concentration of dissolved oxygen changes from 207.3 µM in pure water to 190.7 µM in MiR05. The partial pressure of oxygen remains constant at 19.6 kPa.
An aqueous solution (MiR05) in equilibrium with air at standard barometric pressure (100 kPa) and 25 °C is heated in a closed system from the initial temperature of 25 °C to 37 °C. If we ignore in a first approximation the thermal volume expansion of water β [K^-1], the concentration remains constant at 234.4 µM, since oxygen cannot be exchanged across the boundaries of the closed system. The actual concentration at 37 °C (233.6 µM) declines relative to β = 0 K^-1 by <0.4 % (β = 0.000207·K^-1 at 20 °C and 0.000385·K^-1 at 40 °C (The Physics Hypertextbook, retrieved 2021-12-19). The partial pressure of oxygen p_O₂ is initially 20.3 kPa at 25 °C (234.4/11.56). p_O₂ increases upon heating by 12 °C to 37 °C in the closed system to p_O₂ = 24.0 kPa (233.6/9.72). The total partial pressure of the gases in solution increases to >100 kPa, which leads to gas bubble formation, if the closed system is maintained at constant barometric pressure, as is the case in a respirometer with a chamber that is closed by a stopper with a titration capillary, through which the excess aqueous volume of thermal expansion escapes and the barometric pressure is kept constant.

References

Bioblast link	Reference	Year
Cohen 2008 IUPAC Green Book	Cohen ER, Cvitas T, Frey JG, Holmström B, Kuchitsu K, Marquardt R, Mills I, Pavese F, Quack M, Stohner J, Strauss HL, Takami M, Thor HL (2008) Quantities, Units and Symbols in Physical Chemistry. IUPAC Green Book 3rd Edition, 2nd Printing, IUPAC & RSC Publishing, Cambridge.	2008
Forstner 1983 POS	Forstner H, Gnaiger E (1983) Calculation of equilibrium oxygen concentration. In: Gnaiger E, Forstner H (eds) Polarographic oxygen sensors. Aquatic and physiological applications. Springer, Berlin, Heidelberg, New York:321-33.	1983
Gnaiger 2020 BEC MitoPathways	Gnaiger E (2020) Mitochondrial pathways and respiratory control. An introduction to OXPHOS analysis. 5^th ed. Bioenerg Commun 2020.2. https://doi.org/10.26124/bec:2020-0002	2020
BEC 2020.1 doi10.26124bec2020-0001.v1	Gnaiger E et al ― MitoEAGLE Task Group (2020) Mitochondrial physiology. Bioenerg Commun 2020.1. https://doi.org/10.26124/bec:2020-0001.v1	2020
Hitchman 1983 POS Membrane	Hitchman ML, Gnaiger E (1983) A thermodynamic consideration of permeability coefficients of membranes. In: Polarographic Oxygen Sensors. Aquatic and Physiological Applications. Gnaiger E, Forstner H (eds), Springer, Berlin, Heidelberg, New York:31-6.	1983
MiPNet06.03 POS-calibration-SOP	O2k Quality Control 1: Polarographic oxygen sensors and accuracy of calibration.	2023-02-06