# Dubowski’s Formula vs Henry's law applied to alcohol sensor 'calibration'

I'm working with some hobby-level gas sensors and I've run across a "calibration" system I don't understand. A general description of this kind of gas sensor is described a bit more in this question and also here. They have some selectivity between gases, but anything that will react with atomic oxygen at elevated temperature in the presence of a catalyst ($$\ce{CO}$$, $$\ce{H2}$$, $$\ce{CH4}$$, etc.) will register.

Since I don't want to mess with explosive gases, the alcohol sensor is a convenient place to begin. Many hobbyists become voluntary test subjects, and heroically subject themselves to alcohol, then breath on the sensors. Since I'm not so brave, I thought I'd try the recommended calibration test setup below. I'm not interested in doing a careful calibration, but just get some kind of ballpark quantitative agreement.

Henry's law suggests a proportionality between the concentration of a dissolved gas and the partial pressure of the gas above the liquid, and the van 't Hoff equation provides for a temperature dependence within some limited range of variation. However it's of the form $$exp(\frac{1}{T}-\frac{1}{T_0})$$ and the equation shown below is only a straight exponential of temperature.

This is described more thoroughly in this excellent answer:

Henry's law works for small concentrations of ideal mixtures at equilibrium. Henry's Law constant varies with temperature according to the Van't Hoff equation:

$$k(T)=k(T_0)exp\left[-C\left(\frac{1}{T}-\frac{1}{T_0}\right)\right]$$

Here $$C$$ is a constant related to the enthalpy of solvation for each gas, and $$T_0$$ represents the standard state, T = 298 K and 1 atm.

Question: Is Dubowski’s formula just a further approximation, or does this method contain something new and/or different that I should be paying attention to?

below: Screenshot from Adafruit's AN4 Using MiCS Sensors for Alcohol Detection.pdf Dubowski's formula is a parameterization of the temperature dependence of the water-vapor partition constant (Henry's law constant). It is based on data accumulated over 63 years (see Ref 1). It is assumed that Henry's law holds, in other words, that the vapour pressure over the solution is proportional to the concentration in the solution as $$\frac{p}{c} = K_{a/w}=$$ constant.

It can be used as a proxy for the concentration of alcohol in the blood although it is not expected to be entirely accurate, as Dubowski comments (Ref 2):

We believe that the conversion of a breath quantity to a blood concentration of ethanol, for forensic purposes, should be abandoned and that the offense of driving while under the influence of alcohol should be statutorily defined in terms of the concentration of ethanol found in the breath in jurisdictions employing breath analysis.

Irrespectively of the details involving breath-alcohol analysis, Henry's law states that the vapor pressure is linearly proportional to the concentration in solution as $$p=K_Hc$$, and is applicable in practice when the alcohol concentration in the blood is very low (it is in general applicable as a limiting law).The following figure shows the partial pressure of ethanol in dilute aqueous solution at $$25 ^\circ C$$ as a function of the mole fraction of alcohol in solution, where the dashed line represents the pressure predicted by Henry's law: For reference, a blood alcohol concentration (BAC) of 0.1%(w/w) translates into a mole fraction of $$\pu{3.9E-4}$$.

The Henry's law constant is obtained by extrapolating to zero concentration of the solute (here alcohol). In general, water-ethanol solutions are highly non-ideal due to hydrogen bonding and would not be expected to follow Henry's law except for very dilute concentrations of one component in the other. As soon as interactions between solute molecules become significant the law can be expected to break down.

The difference in the functional form of the temperature-dependence in Dubowski's formula compared to Van't Hoff's can be ascribed to differences in units: one uses T in Kelvin, the other in degrees Celsius. With proper application of Taylor series formulas they can be shown to be equivalent within narrow T ranges.

References

1. K.M. Dubowski. Breath-alcohol simulators: Scientific basis and actual performance. J. Anal ToxicoL 3" 177-82 (1979).

2. M F Mason, K M Dubowski. Breath-alcohol analysis: uses, methods, and some forensic problems--review and opinion. J. Forensic. Sci. 1976 Jan;21(1):9-41.

• Beautiful, thorough and clear answer, thank you very much!
– uhoh
Mar 23 at 0:15