Free-Base Nicotine Fraction α_fb in Non-Aqueous versus Aqueous Solutions: Electronic Cigarette Fluids Without versus With Dilution with Water

Abstract

An important design aspect of electronic cigarettes (“e-cigarettes”) is the nature of the acid/base chemistry in the e-liquid phase. E-liquids having formulations similar to those of early products are mixes of propylene glycol/glycerol (PG/GL) plus free-base (fb) nicotine and (usually) flavor chemicals that are either rather weak or non-acid/base actors in PG/GL. The fraction of nicotine in the fb form is denoted (α_fb)_e-liquid, with a possible range of 0 < (α_fb)_e-liquid < 1. For e-liquids of an early design, (α_fb)_e-liquid ≈ 1. Because e-cigarette aerosols high in fb nicotine are harsh upon inhalation, many commercial e-liquids now also contain variable levels of an acid additive (e.g., benzoic acid, levulinic acid, etc.) to protonate the nicotine and form dissolved “nicotine salts”: (α_fb)_e-liquid values significantly less than 1 are now common. A framework is developed for predicting α_fb values in a given medium based on the following: (1) acid/nicotine ratios and (2) overall acid + nicotine protonation constant (K_oa) values. This framework is required for understanding (1) e-liquid design in regard to how acid additives affect (α_fb)_e-liquid values, and (2) why (α_fb)_e-liquid values cannot, in general, be measured by any method that involves significant dilution with water.

Graphical Abstract

INTRODUCTION

Electronic cigarette (“e-cigarette”) use has increased greatly during the past decade.¹ Because nicotine is an alkaloid, its behavior in an e-cigarette system and its sensory propeties in the resulting e-cigarette aerosol are affected by the acid/base characteristics of the “e-liquid” used. There is a need to understand nicotine protonation chemistry in e-liquids and their associated aerosol liquids;² measurement efforts have begun.^3–8

Two distinctly different but complementary approaches for probing the acid/base chemistry of e-liquids and their aerosols have emerged: (1) native solution methods (NSMs) and (2) diluted solution methods (DSMs). The explicit goal of NSMs is to understand the acid/base chemistry of an unaltered e-liquid or the associated aerosol liquid. NSMs are best capable of examining the connection between product design and the immediate aspects of nicotine delivery to the user.^3,4 In contrast, DSMs cannot (by definition) directly access the state of the acid/base chemistry of nicotine (or other constiutents) in a native e-liquid or aerosol liquid phase. However, if the dilution water is free of dissolved CO₂³ then DSMs will allow the study of the composition of a given e-liquid mix. For example, a 1:10 dilution of an e-liquid is mostly (~90%) water, so measurement with a glass pH electrode will provide a good approximation of the pH that would be present in a solution of the same resultant mix of acids and nicotine in ~100% water, wherein the needed equilibrium constants are known. In particular, for acids and protonated nicotine, tabulated pK_a (= −log K_a) values in water can then be used to help work out the identities and levels of the acids and nicotine present. To summarize, while the results from a CO₂-excluded DSM for a given e-liquid must not be overinterpreted, those results will provide a valuable characterization for the e-liquid.

Nicotine is a dibasic alkaloid. Whether in propylene glycol (PG), glycerol (GL), PG/GL, diluted PG/GL, or 100% water, nicotine can exist in three forms (Figure 1), including (1) free-base (fb, also known as unprotonated or bound), (2) monoprotonated (mp), and (3) diprotonated (dp). Gaseous nicotine is solely in the fb form. In a liquid phase, the fraction of the nicotine that is in the fb form is denoted α_fb:^2–4,9

$${\alpha }_{\text{fb}}\equiv \frac{[\text{Nic}]}{[\text{Nic}]+\left[{\text{NicH}}^{+}\right]+\left[{\text{NicH}}_2^{2+}\right]}$$ (1)

Figure 1.

Nicotine in its three forms: diprotonated (dp), monoprotonated (mp), and unprotonated/free-base (fb).

The dp species $\left({\text{NicH}}_2^{2+}\right)$ can be important when (1) the level of protonating acid (eq/L) is greater than the level of nicotine (eq/L), and (2) diprotonation is sufficiently favorable in the medium of interest.

The nicotine protonation degree affects nicotine aerosol inhalation harshness^3,4,10 and can also affect the uptake kinetics of aerosolized nicotine.¹⁰ In the absence of a protonating acid, (α_fb)_e-liquid = 1, as in early style PG/GL + nicotine e-liquids.¹¹ At high-nicotine levels such as ~60 mg/mL, (α_fb)_e-liquid ≈ 1 gives an aerosol that is harsh upon inhalation;^3,4,10 adding an acid will lower (α_fb)_e-liquid and moderate harshness. Currently, important carboxylic acids used in e-liquids include benzoic acid and levulinic acid.^4,12 Flavor chemical phenols like vanillin and ethyl vanillin are also acidic; however, these are much weaker in both water and PG/GL than carboxylic acids.⁴ Vanillin and ethyl vanillin have been found at levels as high as 1 to 3% in multiple e-liquid products.^13,14 No other common flavor chemical ingredients are acids nor is vitamin E acetate, as used in vaping cannabis products.^15,16 However, the degradation of ingredients during vaping may produce levels of acids that are relevant for protonation of nicotine, as illustrated by the degradation of sucralose to HCl and other products.¹¹

Duell et al. (2019)⁴ discuss how the incorporation of acids in e-cigarette liquids is a “déjà vu” reprise of the shift, in the mid-1800s for smoked tobacco, from air-cured tobacco to flue-cured tobacco. While the exceedingly complex nature of the liquid matter comprising tobacco smoke particulate material (PM) precludes any easy mathematical framework for predicting α_fb values for such PM, e-liquids are much simpler making such a framework possible. Here, we combine acid/base chemical principles with recent data on nicotine protonation constants in PG/GL⁴ to develop and apply that framework.

RESULTS

General Considerations.

The medium- and temperature-dependent nicotine acidity equilibria are

$${\text{NicH}}_2^{2+}={\text{H}}^{+}+{\text{NicH}}^{+}{K}_{\text{a}}^{{\text{NicH}}_2^{2+}}=K_{\text{a},1}\equiv \frac{\left[{\text{H}}^{+}\right]\left[{\text{NicH}}^{+}\right]}{\left[{\text{NicH}}_2^{2+}\right]}$$ (2)

$${\text{NicH}}^{+}={\text{H}}^{+}+\text{Nic\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{K}_{\text{a}}^{{\text{NicH}}^{+}}=K_{\text{a},2}\equiv \frac{\left[{\text{H}}^{+}\right][\text{Nic}]}{\left[{\text{NicH}}^{+}\right]}$$ (3)

Each bracketed term here is a molar concentration and not a chemical activity: all K values here are medium-specific equilibrium constants, as with ^cK values used in solution chemistry.¹⁷ Manipulation of eq 1 gives

$${\alpha }_{\text{fb}}=\frac{1}{1+\frac{\left[{\text{NicH}}^{+}\right]}{\left[\text{Nic}\right]}+\frac{\left[{\text{NicH}}_2^{2+}\right]}{\left[\text{Nic}\right]}}=\frac{1}{1+\frac{\left[{\text{H}}^{+}\right]}{K_{\text{a}}^{{\text{NicH}}^{+}}}+\frac{{\left[{\text{H}}^{+}\right]}^2}{K_{\text{a}}^{{\text{NicH}}_2^{2+}}{K}_{\text{a}}^{{\text{NicH}}^{+}}}}$$ (4)

wherein

$$\frac{\left[{\text{NicH}}^{+}\right]}{[\text{Nic}]}=\frac{\left[{\text{H}}^{+}\right]}{K_{\text{a}}^{{\text{NicH}}^{+}}}$$ (5)

and

$$\frac{\left[{\text{NicH}}_2^{2+}\right]}{[\text{Nic}]}=\frac{{\left[{\text{H}}^{+}\right]}^2}{K_{\text{a}}^{{\text{NicH}}_2^{2+}}{K}_{\text{a}}^{{\text{NicH}}^{+}}}$$ (6)

If $\left[{\text{NicH}}_2^{2+}\right]$ can be neglected

$${\alpha }_{\text{fb}}\approx \frac{1}{1+\frac{\left[{\text{NicH}}^{+}\right]}{\left[\text{Nic}\right]}}=\frac{1}{1+\frac{\left[{\text{H}}^{+}\right]}{K_{\text{a}}^{{\text{Nich}}^{+}}}}$$ (7)

K_a values in 100% water are known for acids relevant here (e.g., see Table 1). The measurement of [H⁺] values in aqueous solutions is straightforward, and in ~100% water, [H⁺] = 10^−pH. In non-aqueous solvents, K_a values are not generally known, and the measurement of [H⁺] is difficult. Fortunately, as discussed by Duell et al.⁴ and elaborated below, only the overall nicotine protonation constants (and not individual K_a values) are needed to predict α_fb values.

Table 1.

pK_a Values for Acids in 100% Water (Low Ionic Strength)

acid	25 °C	37 °C
nicotine	pKa,1 = 3.10,a pKa,2 = 8.01a	pKa,1 = 2.77,b pKa,2 = 7.65b
benzoic	pKa = 4.20c	pKa = 4.20c
vanillin	pKa = 7.38c	pKa = 7.27c
levulinic	pKa = 4.64d (18 °C)

^aBarlow and Hamilton (1962).¹⁹

^bClayton, Vas et al. (2016).²⁰

^cChristensen and Hansen (1976).²¹

^dKortüm, Vogel, and Andrussow (1961).²²

Because α_fb in a solution mixture depends on the exact nature of the solution, the determination of a given (α_fb)_e-liquid value must proceed by a NSM, as by ¹H nuclear magnetic resonance (¹H NMR) spectroscopy.^3,4 In contrast, a DSM (e.g., with 1:10 dilution)^5,7,8 will unequivocally alter the solution α_fb. Indeed, the fact that solution thermodynamics is affected by the solvent is embodied in the prerequisite that dilution with water be carried out if there is to be an unequivocal interpretation of a pH measurement using a water-calibrated electrode. Defending DSMs, Gholap et al.⁸ state, “…it can be argued that the purpose of the dilution approach is to provide a pH relevant scale for classifying e-liquids based on their free base nicotine yields.” This is true only when the focus is the level of free-base nicotine in respiratory tract fluids after the inhaled e-cigarette aerosol droplets have been deposited and diluted. It is not true if the focus is either the immediate user perceptions of the inhaled nicotine aerosol or the early stages in the processes of nicotine deposition from the inhaled aerosol.¹⁰

DSMs with water-calibrated pH measurements have proceeded based on the “Henderson–Hasselbalch” equation (HHE) for the determination of the [Nic]/[NicH⁺] ratio for use in eq 7. The derivation of the HHE equation is simple. As used here, taking the log of both sides of eq 5 for water gives the HHE for nicotine in 100% water:

$${\text{pH}}_{\text{water}}={\left(\text{p}{K}_{\text{a}}^{{\text{NicH}}^{+}}\right)}_{\text{water}}+\text{log}{\left(\frac{[\text{Nic}]}{\left[{\text{NicH}}^{+}\right]}\right)}_{\text{water}}$$ (8)

$${\left(\frac{[\text{Nic}]}{\left[{\text{NicH}}^{+}\right]}\right)}_{\text{water}}={10}^{\left({\text{pH}}_{\text{water}}-{\left(\text{p}{K}_{\text{a}}^{{\text{NicH}}^{+}}\right)}_{\text{water}}\right)}$$ (9)

As noted above, for a DSM with 1:10 dilution,^5,7 it is reasonable to assume that neither the water-based calibration of the pH electrode response nor the value of $\text{p}{K}_{\text{a}}^{{\text{NicH}}^{+}}$ is substantially affected by the remaining (~10%) PG/GL. That is, the [Nic]/[NicH⁺] ratio in a 1:10 dilution can be estimated based on the response of a water-calibrated pH electrode and the water-based value of $\text{p}{K}_{\text{a}}^{{\text{NicH}}^{+}}$:

$${\left(\frac{[\text{Nic}]}{\left[{\text{NicH}}^{+}\right]}\right)}_{\text{water}\left(1:10\right)}\approx {10}^{\left({\text{pH}}_{\text{water}\left(1:10\right)}-{\left(\text{p}{K}_{\text{a}}^{{\text{NicH}}^{+}}\right)}_{\text{water}}\right)}$$ (10)

However, as suggested above, it is fatally problematic to assume further that the right hand side (RHS) of eq 10 gives [Nic]/[NicH⁺] in the e-liquid. The validity of that circumstance would require that both the pH calibration and the value of $\text{p}{K}_{\text{a}}^{{\text{NicH}}^{+}}$ are the same for water as for ~100% PG/GL or at least that $\left({\text{pH}}_{\text{water}\left(1:10\right)}-{\left(\text{p}{K}_{\text{a}}^{{\text{NicH}}^{+}}\right)}_{\text{water}}\right)=\left(-\text{log}{\left(\left[{\text{H}}^{+}\right]\right)}_{\text{e-liquid}}-{\left(\text{p}{K}_{\text{a}}^{{\text{NicH}}^{+}}\right)}_{\text{e-liquid}}\right)$. As a general matter, this will certainly not be the case: as shown below, pK_a values are greatly affected by the nature of the solvent, and changing the pK_a can greatly alter nicotine protonation. Some special stoichiometric circumstances do allow (α_fb)_{water (1:10)} ≈ (α_fb)_e-liquid: see Table 2.

Table 2.

Special Circumstances Allowing (α_fb)_{water (1:10)} ≈ (α_fb)_e-liquid^a

case 1	no protonating acid is present in the e-liquid, so (αfb)e-liquid = 1, and the nicotine concentration after dilution is high enough to yield a relatively high aqueous pH, giving (αfb)water (1:10) ≈ 1
case 2	a protonating acid is present in the e-liquid and is sufficiently strong to react stoichiometrically with free-base nicotine in both the e-liquid and in water
2a	(eq/L acid/eq/L nicotine) = y < 1: (αfb)e-liquid ≈ (αfb)water (1:10) ≈ 1 − y
2b	(eq/L acid/eq/L nicotine) ≥ 1: (αfb)e-liquid ≈ (αfb)water (1:10) ≈ 0

^aAssumes the use of CO₂-free water.

Protonation by a Single Monoprotic Acid HA: General Equations.

For HA

$$\text{HA}={\text{H}}^{+}+{\text{A}}^{-}{K}_{\text{a}}^{\text{HA}}\equiv \frac{\left[{\text{H}}^{+}\right]\left[{\text{A}}^{-}\right]}{[\text{HA}]}$$ (11)

The first overall nicotine protonation reaction is⁴

$$\text{HA}+\text{Nic}={\text{A}}^{-}+{\text{NicH}}^{+}\frac{\left[{\text{A}}^{-}\right]\left[{\text{NicH}}^{+}\right]}{[\text{HA}][\text{Nic}]}=\frac{K_{\text{a}}^{\text{HA}}}{K_{\text{a}}^{{\text{NicH}}^{+}}}\equiv K_{\text{oa},1}^{\text{HA}}$$ (12)

The second overall protonation reaction is

$$\text{HA}+{\text{NicH}}^{+}={\text{A}}^{-}+{\text{NicH}}_2^{2+}\frac{\left[{\text{A}}^{-}\right]\left[{\text{NicH}}_2^{2+}\right]}{[\text{HA}]\left[{\text{NicH}}^{+}\right]}=\frac{K_{\text{a}}^{\text{HA}}}{K_{\text{a}}^{{\text{NicH}}_2^{2+}}}\equiv K_{\text{oa},2}^{\text{HA}}$$ (13)

Because Debye–Hückel effects can favor ion formation, both $K_{\text{oa},1}^{\text{HA}}$ and $K_{\text{oa},2}^{\text{HA}}$ can increase with increasing total ion concentration; this has likely been observed for $K_{\text{oa},1}^{\text{vanillin}}$.⁴ Comparisons of the protonation chemistry of nicotine in any given solution type by different acids (or in different solution types by the same acid) should be done using K values for the same temperature. The three temperatures relevant here are ~20 °C (lab-bench), 25 °C (many tabulated K values), and 37 °C (the human body).

On the basis of eq 12 and the pK_a values in Table 1, values for K_oa,1 in water are given in Table 3 for benzoic acid and vanillin. While $\text{p}{K}_{\text{a}}^{{\text{NicH}}^{+}}$ and $\text{p}{K}_{\text{a}}^{\text{HA}}$ values in PG/GL mixtures are not known, Duell et al.⁴ measured (α_fb)_e-liquid values at 40 °C (≈ 37 °C), and these were used (with the assumption that $\left[{\text{NicH}}_2^{2+}\right]$ could be neglected) to estimate the K_oa,1 values in PG/GL for benzoic acid and vanillin at 40 °C (≈ 37 °C). They are much smaller than the corresponding values in water. For benzoic acid, for a 43/57 mol % mixture of PG/GL, K_oa,1 = 10^1.41. In JUUL product e-liquids (approximated as ~31/69 wt % PG/GL or ~35/65 mol % PG/GL), on average, K_oa,1 = 10^1.81. It is clear that nicotine protonation by benzoic acid is much more favored in water than in PG/GL, which is why dilution with water creates enormous problems when seeking to determine (α_fb)_e-liquid values: (α_fb)_{water (1:10)} values obtained by DSM methods will be too low (except under the circumstances in Table 2).

Table 3.

Log K_oa,1 Values for Nicotine Protonation by Benzoic Acid and Vanillin in ~100% Water and in Propylene Glycol/Glycerol Solutions

acid	water 25 °C	water 37 °C	~44/56a PG/GL 40 °C	JUUL products 31/69b PG/GL 40 °C
benzoic	3.81	3.45	1.41c	1.81
vanillin	0.63	0.38	−2.07d	NA

^aWeight % fraction.

^bWeight % fraction converted from the NMR-measured molar ratio of 35/65.

^cDuell et al. (2019); 43/57 weight % fraction PG/GL.⁴

^dDuell et al. (2019); 45/55 weight % fraction PG/GL.⁴

In a non-aqueous Nic + HA system, there are six concentration unknowns: [H⁺], [HA], [A⁻], [Nic], [NicH⁺], and $\left[{\text{NicH}}_2^{2+}\right]$. Ion formation by autoprotolysis of PG and GL is assumed to be negligible. When HA is a weak acid (benzoic acid, levulinic acid, acetic acid, etc.), [H⁺] can generally be neglected in the electroneutrality equation (ENE) for the solution, especially in PG/GL, leaving five unknowns. The five equations needed for solving for the speciation values are then (1) the ENE, (2) C_HA = total added HA concentration (mol/L), (3) C_Nic = total added Nic concentration (mol/L), (4) the expression for K_oa,1, and (5) the expression for K_oa,2.

With [H⁺] neglected, the general ENE is

$$\left[{\text{NicH}}^{+}\right]+2\left[{\text{NicH}}_2^{2+}\right]=\left[{\text{A}}^{-}\right]$$ (14)

By definition

$$\begin{gathered} {\alpha }_{\text{fb}}\equiv \frac{[\text{Nic}]}{C_{\text{Nic}}},{\alpha }_{\text{mp}}\equiv \frac{\left[{\text{NicH}}^{+}\right]}{C_{\text{Nic}}},{\alpha }_{\text{dp}}\equiv \frac{\left[{\text{NicH}}_2^{2+}\right]}{C_{\text{Nic}}}, \\ {C}_{\text{Nic}}\equiv [\text{Nic}]+\left[{\text{NicH}}^{+}\right]+\left[{\text{NicH}}_2^{2+}\right] \end{gathered}$$ (15)

Similarly

$${\alpha }_{{\text{A}}^{-}}\equiv \frac{\left[{\text{A}}^{-}\right]}{C_{\text{HA}}}=\frac{\left[{\text{A}}^{-}\right]}{[\text{HA}]+\left[{\text{A}}^{-}\right]}=\frac{1}{\frac{[\text{HA}]}{\left[{\text{A}}^{-}\right]}+1}$$ (16)

Eq 14 can then be represented as

$${\alpha }_{\text{mp}}{C}_{\text{Nic}}+2{\alpha }_{\text{dp}}{C}_{\text{Nic}}={\alpha }_{{\text{A}}^{-}}{C}_{\text{HA}}$$ (17)

$$=\frac{1}{\frac{[\text{HA}]}{\left[{\text{A}}^{-}\right]}+1}{C}_{\text{HA}}$$ (18)

$${\alpha }_{\text{mp}}+2{\alpha }_{\text{dp}}=\frac{1}{\frac{\left[\text{HA}\right]}{\left[{\text{A}}^{-}\right]}+1}\frac{C_{\text{HA}}}{C_{\text{Nic}}}$$ (19)

Since α_dp and α_mp (and α_fb) are available as functions of [HA]/[A⁻] (Table 4), then with input values of C_HA, C_Nic, K_oa,1, and K_oa,2, a numerical solution of eq 19 for the root value of [HA]/[A⁻] can be obtained; the root value is then used to evaluate α_dp, α_mp, and α_fb. The numerical solution can be achieved using a spreadsheet optimization solver that varies log([HA]/[A⁻]) to find LHS – RHS = 0 for eq 19. This is analogous to varying pH = −log [H⁺] in the ENE of an acid/base problem.¹⁷ Varying log([HA]/[A⁻]) rather than [HA]/[A⁻] itself facilitates the numerical process because it speeds the search for the root and automatically excludes negative values of [HA]/[A⁻].

Table 4.

Definitions and Expressions for α_dp, α_mp, and α_fb for Nicotine

Neglecting $\left[{\text{NicH}}_2^{2+}\right]$.

When $\left[{\text{NicH}}_2^{2+}\right]$ is neglected, eq 14 reduces to

$$\left[{\text{NicH}}^{+}\right]\approx \left[{\text{A}}^{-}\right]$$ (20)

This approximation permits the quadratic solution for α_fb given in Table 5, as obtained by Duell et al.⁴ In terms of a mathematical solution based on the ratio [HA]/[A⁻], the solution can equivalently be obtained by writing eq 20 as

$${\alpha }_{\text{mp}}{C}_{\text{Nic}}={\alpha }_{{\text{A}}^{-}}{C}_{\text{HA}}$$ (21)

so that

$${\alpha }_{\text{mp}}=\frac{1}{\frac{[\text{HA}]}{\left[{\text{A}}^{-}\right]}+1}\left(\frac{C_{\text{HA}}}{C_{\text{Nic}}}\right)$$ (22)

The solution proceeds as given in Table 6.

Table 5.

Equation for α_fb in a Solution Containing Nicotine and a Single Monoprotic Acid HA when Neglecting $\left[{\text{NicH}}_2^{2+}\right]$^a

^aFrom Duell et al.;⁴ C_HA/C_Nic = molar concentration ratio of HA to nicotine.

Table 6.

Development of an [HA]/[A⁻]-Based Equation for α_fb in a Solution Containing Nicotine and a Single Monoprotic Acid HA when Neglecting $\left[{\text{NicH}}_2^{2+}\right]$^a

^aC_HA/C_Nic = molar concentration ratio.

Protonation by Two Monoprotic Acids, HA and HA′.

When two monoprotic acids are present, the general ENE (neglecting [H⁺]) is

$${\alpha }_{\text{mp}}{C}_{\text{Nic}}+2{\alpha }_{\text{dp}}{C}_{\text{Nic}}={\alpha }_{{\text{A}}^{-}}{C}_{\text{HA}}+{\alpha }_{{\text{A}}^{\prime -}}{C}_{{\text{HA}}^{\prime }}$$ (23)

$$=\frac{1}{\frac{[\text{HA}]}{\left[{\text{A}}^{-}\right]}+1}{C}_{\text{HA}}+\frac{1}{\frac{\left[{\text{HA}}^{\prime }\right]}{\left[{\text{A}}^{\prime -}\right]}+1}{C}_{{\text{HA}}^{\prime }}$$ (24)

For HA′

$${\text{HA}}^{\prime }={\text{H}}^{+}+{\text{A}}^{\prime -}{K}_{\text{a}}^{{\text{HA}}^{\prime }}\equiv \frac{\left[{\text{H}}^{+}\right]\left[{\text{A}}^{\prime -}\right]}{\left[{\text{HA}}^{\prime }\right]}$$ (25)

As one pair, HA and HA′ might be benzoic acid and levulinic acid, respectively. The proton exchange reaction between the A and A′ systems is

$${\text{HA}}^{\prime }+{\text{A}}^{-}={\text{A}}^{\prime -}+\text{HA\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{[\text{HA}]\left[{\text{A}}^{\prime -}\right]}{\left[{\text{HA}}^{\prime }\right]\left[{\text{A}}^{-}\right]}=\frac{K_{\text{a}}^{{\text{HA}}^{\prime }}}{K_{\text{a}}^{\text{HA}}}\equiv f$$ (26)

where f is the strength factor for HA′ over HA. Then

$$\frac{\left[{\text{HA}}^{\prime }\right]}{\left[{\text{A}}^{\prime -}\right]}=\frac{1}{f}\frac{[\text{HA}]}{\left[{\text{A}}^{-}\right]}$$ (27)

Equation 24 becomes

$${\alpha }_{\text{mp}}{C}_{\text{Nic}}+2{\alpha }_{\text{dp}}{C}_{\text{Nic}}=\frac{1}{\frac{[\text{HA}]}{\left[{\text{A}}^{-}\right]}+1}{C}_{\text{HA}}+\frac{1}{\frac{1}{f}\frac{[\text{HA}]}{\left[{\text{A}}^{-}\right]}+1}{C}_{{\text{HA}}^{\prime }}$$ (28)

$${\alpha }_{\text{mp}}+2{\alpha }_{\text{dp}}=\frac{1}{\frac{[\text{HA}]}{\left[{\text{A}}^{-}\right]}+1}\frac{C_{\text{HA}}}{C_{\text{Nic}}}+\frac{1}{\frac{1}{f}\frac{[\text{HA}]}{\left[{\text{A}}^{-}\right]}+1}\frac{C_{{\text{HA}}^{\prime }}}{C_{\text{Nic}}}$$ (29)

If the two monoprotic acids have similar strengths (f ≈ 1) then the two acids can be lumped as one with a concentration of C_HA + C_HA′. In any case, the Table 4 equations for α_dp, α_mp, and α_fb remain the same. Knowing K_oa,1 and K_oa,2 for HA along with f fully reveals how well HA′ can protonate nicotine: the value of f and the values of K_oa,1 and K_oa,2 for HA can be used to obtain the values of K_oa,1 and K_oa,2 for HA′.

α_fb as a Function of Log K_oa,1 and C_HA/C_Nic.

Figure 1 provides isopleths of α_fb in a log K_oa,1 versus C_HA/C_Nic format for the case of a single monoprotic acid. The calculations were made by assuming K_oa,2 = K_oa,1/10⁵ because $K_{\text{a}}^{{\text{NicH}}_2^{2+}}/K_{\text{a}}^{{\text{NicH}}^{+}}$ in water is ~10⁵ and is likely of a similar magnitude in PG/GL mixtures ($K_{\text{a}}^{{\text{NicH}}_2^{2+}}\gg K_{\text{a}}^{{\text{NicH}}^{+}}$, so the reaction Nic + H⁺ = NicH⁺ is more inherently favored than ${\text{NicH}}^{+}+{\text{H}}^{+}={\text{NicH}}_2^{2+}$). ${\text{NicH}}_2^{2+}$ is negligible for all areas where α_fb ≥ 0.01; for the isopleths shown, α_dp is largest along the α_fb = 0.01 isopleth, yet α_dp is only 0.00098. Along α_fb = 0.1, α_dp= 0.00008. At the upper right corner (log K_oa,1 = 4 and C_HA/C_Nic = 3), α_dp = 0.14, α_mp = 0.86, and α_fb = 0.00005.

Differences Between (α_fb)_e-liquid and (α_fb)_{water (1:10)}.

For 10 JUUL products, Pankow et al.¹⁸ and Duell et al.⁴ found the dominant acid to be benzoic acid with a benzoic acid/nicotine ratio of ~1 (i.e., C_HA/C_Nic ≈ 1). For JUUL products, logK_oa,1 ≈ 1.8 (40 °C),⁴ giving (α_fb)_e-liquid ≈ 0.1 and the point BA_JUUL in Figure 2. However, after a 1:10 dilution with water, with C_HA/C_Nic remaining at ~1, K_oa,1 now moves to ~10^3.45, giving (α_fb)_{water (1:10)} ≈ 0.02 (and point BA_w): the analytical error from presuming (α_fb)_{water (1:10)} = (α_fb)_e-liquid with JUUL products is large.

Figure 2.

Isopleths of α_fb in a log K_oa,1 vs C_HA/C_Nic format assuming K_oa,2 = K_oa,1/10⁵. Additional points: 10 JUUL e-liquid data points (40 °C) from Duell et al.;⁴ BA_JUUL = average (40 °C) for benzoic acid + nicotine in JUUL e-liquid at C_HA/C_Nic = 1; BA_w = benzoic acid + nicotine in water-diluted e-liquid at C_HA/C_Nic = 1, assuming dilution at 37 °C with CO₂-free water; Vanillin_e = vanillin + nicotine in ~1:1 propylene glycol (PG) and glycerol (GL) at C_HA/C_Nic = 1 (40 °C); and Vanillin_w = vanillin + nicotine in water-diluted e-liquid at C_HA/C_Nic = 1, assuming dilution at 37 °C with CO₂-free water.

For vanillin, K_oa,1 in 50/50 mol % (43/57 wt % PG/GL) PG/GL has been measured to be ~10^−2.07 (40 °C).⁴ In a 50/50 PG/GL e-liquid with C_HA/C_Nic = 1.0 for the case of vanillin, (α_fb)_e-liquid ≈ 0.92 at the point Vanillin_e (Figure 2). After a 1:10 dilution with water (with C_HA/C_Nic again remaining at ~1), K_oa,1 moves to ~10^0.38 so that (α_fb)_{water (1:10)} ≈ 0.39 at the point Vanillin_w: the error from presuming (α_fb)_{water (1:10)} = (α_fb)_e-liquid is again large.

CONCLUSIONS

The comprehensive framework for predicting (α_fb)_e-liquid values developed here will be useful in understanding (1) how the manipulation of e-liquid acid/base chemistry can alter consumer perceptions, (2) the chemistry underlying any associated regulatory actions, and (3) the necessity for using native solution methods (NSMs) when determining (α_fb)_e-liquid values.