In the Classroom
The Thermodynamics of Drunk Driving Robert Q. Thompson Department of Chemistry, Oberlin College, Oberlin, OH 44074 The abuse of alcohol, especially on the nation’s roadways, is receiving more and more attention. Groups such as Mothers Against Drunk Driving are on the nightly news, and new motor vehicle laws are tougher on persons driving under the influence (DUI) of alcohol. In the state of Ohio (U.S.A.), a first DUI offense may result in administrative license suspension for 90 days or a one-year suspension if the driver refuses a sobriety test. The license suspension is based on per se laws that state that any driver found with a blood alcohol concentration (BAC) of 0.10% (0.10 g ethanol per 100 mL of blood) or greater is legally intoxicated and guilty of an offense. Some states recently have lowered this limit to 0.08%. It does not matter whether or not the driver is actually influenced by that level of alcohol, and it is likely that many drivers are impaired at lower BACs (1, 2). This legal limit is partly based on a study by the American Medical Association that concluded that half the population shows signs of intoxication at a BAC of 0.10% (2). The lower legal limits have been drawn even more arbitrarily. Implied consent laws state that “if you are arrested by a law enforcement officer who has reasonable grounds to believe that you are driving under the influence of alcohol or drugs of abuse, it is understood that you have consented to a chemical test of your blood, breath, or urine” (3). Hence, refusing to submit to a test results in a severe penalty, and, in most cases, the penalty stands even if the driver is found not guilty of the DUI charge. Usually coordination tests, a check for nystagmus (eye movements), and a test for ethanol in the driver’s breath are made at the site of the arrest. A portable device containing an electrochemical cell that responds to ethanol can provide initial evidence of recent drinking and the BAC. If DUI is indicated, the police officer can take the driver’s license on the spot and then escort the driver to the local police station for further testing and possible jail time. A more accurate breath test can be made by the police or can be requested by the driver. These evidentiary tests usually involve either older instruments, such as the Breathalyzer, or newer, automated, direct-measuring instruments, such as the Intoxilyzer. The Breathalyzer has been described recently in this Journal (4, 5) and is based on the reaction of ethanol with potassium dichromate and subsequent color analysis. The Intoxilyzer is based on the absorption of infrared light by ethanol (6). A direct blood test is best, but is performed in few cases because it requires specially trained personnel to draw the blood and to do the gas chromatographic analysis. All of these breath and blood measurements involve much important chemistry and technology and can be used in the classroom as practical examples of instrumental analysis. Legal limits for DUI are defined in terms of the driver’s blood alcohol concentration (BAC), whereas the portable electrochemical devices, Breathalyzer, and infrared instruments measure breath alcohol concentration (BrAC). Consequently, the most important question to answer is how do BAC and BrAC relate. The forensic science community in the U.S.A. has agreed on a fixed value of 2100 for the ratio of BAC to BrAC, based on early empirical studies by Harger’s group at the Indiana University School of Medicine (7, 8). Yet the forensic science and toxicology literature has seen much debate on the use of this standard. Proponents have argued that the conversion factor is very reli-
532
able and that, in fact, it underpredicts the blood alcohol concentration. Opponents have shown that the ratio varies widely from individual to individual and that many factors, especially temperature, influence the ratio. In vitro studies of water-ethanol solutions and a few large-scale in vivo studies of drinking drivers have been reported, but they do not clear up the matter. Analysis and reanalysis of the data have given fodder to both sides. Because of the debate and the role of science and technology in it, the prediction of blood alcohol levels from breath test measurements is excellent material for teaching aspects of beginning chemistry to majors and nonmajors. Thermodynamics and kinetics principles from general chemistry and concepts from human physiology are used here to develop a fundamental relationship between breath and blood alcohol concentrations. The temperature dependence of this relationship is explored, the theoretical model is compared to empirical equations and values reported in the literature, and ideas about uncertainty in measured values are invoked in understanding the impact of the model on forensic breath testing. Water–Air–Ethanol System
Derivation of the Partition Ratio It may be best to start off with a simpler system than blood–breath–ethanol, namely, the water–air–ethanol system. Aqueous solutions of ethanol (EtOH) of known concentration are used as “simulator solutions” and calibrants for breath testing. For this water–air–ethanol system, the ratio of ethanol concentration (w/v) in pure water to ethanol concentration (w/v) in air is defined as ratio =
g EtOH water Vwater
/
g EtOH air Vair
(1)
where V is volume and g is mass. Dividing through by the molar weight of ethanol gives moles of ethanol in the numerators. Because ethanol vapor in air should behave similarly to an ideal gas, one can substitute PV/RT for moles of gaseous ethanol and get:
ratio =
mol EtOH water Vwater
/
P EtOH air VEtOH RT Vair
where R is the gas constant, T is the absolute (Kelvin) temperature, and P is the pressure of ethanol in atmospheres. Since the volume of ethanol vapor and the volume of air in equilibrium with the solution are the same, those two terms cancel. Moles of ethanol divided by volume is molar concentration. The relation is reduced to
ratio = RT
[EtOH]water P EtOH air
At this point it is helpful to recall Henry’s law for the partitioning of a volatile substance between liquid and gas phases. The equilibrium distribution characterizing the water–air–ethanol system can be depicted by the equation:
Journal of Chemical Education • Vol. 74 No. 5 May 1997
ethanol (aq)
ethanol (air)
In the Classroom
K eq =
P EtOH P EtOH air air = EtOH EtOH activitywater molalityEtOH water γ water =
densitywater P EtOH air [EtOH water] γ EtOH water
where the gaseous component is taken to be an ideal gas, comparison of the vapor pressure to standard pressure of 1 atm is inferred, and γ EtOH is a molal-based, solute activity water coefficient. Substituting this Keq expression into the previous equation for the ratio, we get
ratio =
RT densitywater K eqγ EtOH water
To find the equilibrium constant for the partitioning of alcohol between aqueous solution and air one can substitute for Keq an expression in terms of the standard enthalpy (∆rH°) and standard entropy (∆ rS°) of the partitioning reaction. A final equation results:
RT densitywater
ratio =
γ EtOH water exp
{∆ r H° + T ∆ rS° R′T
(2)
It is interesting to note the inclusion of two forms of the gas constant, one in units of L atm deg{1 mol{1 (R) and one in units of J deg{1 mol{1 (R9). This is the final form of the equation relating temperature, density, activity coefficient, and thermodynamic values to the water–air–ethanol partition ratio that serves as a starting point for discussion of the ratio used in breath testing.
Evaluation of Partition Ratio A blood alcohol concentration at the legal limit in Ohio, 0.10%, is about 0.026 M or 1 mol of ethanol dissolved in about 2500 mol of water. Because the solution is quite dilute and the solute is a nonionic species, the activity coefficient of ethanol is nearly unity. The value for the standard enthalpy of formation for ethanol in the gas phase is {235.1 kJ/mol and for ethanol in aqueous solution is {288.3 kJ/mol (9). The value for the standard entropy for ethanol in the gas phase is +282.6 J/K·mol (corrected to a standard state of 1 atm rather than 1 bar) and for ethanol in aqueous solution is +148.5 J/K·mol (9). Therefore, the standard enthalpy change associated with the partitioning of ethanol between water and air is +53.2 kJ/mol, and the standard entropy is +134.1 J/K·mol. These final values are given for a temperature of 298 K (25 °C), and a good assumption is that they are relatively insensitive to a change in temperature. Most breath test simulators keep the test solutions at a constant 34.0 °C (10), and at this temperature the ratio equals:
Uncertainty in the temperature and the thermodynamic values contributes significantly to the uncertainty in the predicted ratio. If the thermodynamic values have an uncertainty (standard deviation) of ± 1 in the last digit and the temperature is measured to ± 0.2 °C, as it is for simulator solutions in breath testing (11), propagating these uncertainties through the equation above yields a ratio of 2790 ± 120 and a relative standard deviation of 4%. Most of this is due to the uncertainty in the enthalpy of reaction. Commercial breath-alcohol simulators use an aqueous ethanol solution of 1.225 g/L at 34.0 °C to generate a nominal ethanol vapor concentration of 0.100 g / 210 L (12). Two careful studies of vapor–alcohol samples generated by the TOXITESTTM II Alcoholic Breath Simulator (CMI, Inc.) in the laboratory and in the field found that the average vapor concentrations were 0.100 ± 0.0013 g / 210 L (12) and 0.100 ± 0.0027 g / 210 L (13), respectively. Dividing the solution concentration by the vapor concentration gives ratios of 2570 ± 30 and 2570 ± 70. Another simulator, Guth Model 34C (Guth Laboratories, Inc.), gave average vapor concentrations of 0.103 ± 0.0011 g / 210 L for aqueous ethanol solution concentrations of 1.234 g / L (11). The ratio was 2520 ± 30. Jones determined by gas chromatography the partition ratio of ethanol between water and air to be 2133 at 37 °C (14). Equation 2 predicts a value of 2300 ± 100 at 37 °C. The predicted values are consistently higher than the experimental values by a factor of 1.08. This difference could be explained by a nonunity solute activity coefficient, but the values are not significantly different, in any case.
Effect of Temperature As the temperature increases, ethanol tends to exist more and more as a gas rather than as a solute in solution, and the ratio decreases. Temperature appears in both the numerator and denominator of eq 2, but the exponential term in the denominator is by far the dominant factor. Comparing values at 30 °C and 40 °C, the RT term increases 3%, the density of water decreases 0.5%, and the exponential term increases a whopping 96%. Figure 1 shows the trend toward lower predicted values of the ratio at higher temperatures. The predicted values are compared to an empirically derived equation for a water–air–ethanol system, K = 4.15 × 10{5 e0.0658 T, where T is in °C and ratio is the inverse of K (12). The trends of the experimental and theoretical values with change in temperature are quite similar.
3600
. ]. . _
Predicted Values Empirical Values
3200
Ratio
and the form of Henry’s Law is
2800
2400 {1
{1
0.08206 L atm deg mol
{1
307 deg 1.00 atm
0.9944kg / L = 2790
1.00 kg mol{1 exp
{53,200 J mol{1 + 307 deg 134.1 J mol{1deg{1 {1
{1
8.3154 J mol deg
307 deg
2000 304
306
308
310
Temperature (Kelvin)
Figure 1. Temperature dependence of the ratio in the water–air– ethanol system.
Vol. 74 No. 5 May 1997 • Journal of Chemical Education
533
In the Classroom While using a linear temperature coefficient is not strictly correct because the ratio has an exponential dependence on temperature, nevertheless, the percentage change in the ratio per degree Celsius is most often reported. Slopes of 6.5% / °C (14), 6.8% / °C, and 7.1% / °C (12) have been found for water–air–ethanol. Linear regression of the values predicted by eq 2 between 30 and 40 °C shows a 6.5% / °C change in the ratio. Thus, the model correlates well with the experimental data and can correctly predict the relative change in the ratio with temperature. Blood–Breath–Ethanol System Transforming the model for a water–air–ethanol system into one for a blood–breath–ethanol system requires making explicit some of the assumptions in the model and making several changes in the derivation. The focus is on inhaled air, the tiny capillaries of the alveoli in the lungs, and the exchange of ethanol that takes place there. We begin with the equation,
ratio =
g EtOH blood Vblood
/
g EtOH breath Vbreath
g EtOH blood Vblood
/
V α mol EtOH blood P EtOH / airRT VEtOH Vblood air (4)
= RT
Vwater = {Hct (0.72) + (1 – Hct) (0.94)} Vblood Substituting for Vblood in eq 4, equating the moles of ethanol in blood with the moles in the water portion, and replacing moles/volume with concentration gives
ratio = RT
[EtOH]water {Hct (0.72) + (1 – Hct) (0.94)} P EtOH air α
Substitution of the Henry’s law constant and conversion of an activity to a molar concentration, activity coefficient, and a density is the next step in the derivation and leads to the equation,
ratio =
RT {Hct (0.72) + (1 – Hct) (0.94)} densitywater α K eqγ EtOH water
As before, one can substitute for Keq an expression in terms of the standard enthalpy (∆r H°) and standard entropy (∆rS°) of the partitioning reaction. A final equation results:
ratio =
RT {Hct (0.72) + (1 – Hct) (0.94)} densitywater α γ EtOH water exp
{∆ r H° + T ∆ rS° R′ T
mol EtOH blood Vblood P EtOH α air
Ingested ethanol is absorbed through the walls of the stomach and small intestine. The time for complete absorption in fasting subjects can be as long as two hours after the last drink and even longer for nonfasting subjects (1, 19). During the absorption phase the concentration of ethanol in arterial blood is larger than the concentration in venous blood (20–22). This causes a problem in comparing BrAC with BAC because the deep-lung or alveolar air that is captured for the breath sample reflects equilibration with arterial blood, whereas blood samples are usually drawn from an antecubital (arm) vein. Studies have shown that the measured value of the ratio increases as the time since
The temperature of the air changes as it passes through the lungs. While normal body temperature is 37 °C, normal breath temperature is quoted as 34 °C (7), three degrees lower. The range of body temperatures encountered in DUI cases likely stretches from 34 °C (mild hypothermia [16]) to 40 °C (disease state with high fever), so breath temperature may range from 31 °C to 37 °C. Mouth temperature and ambient air temperature may also influence the alveoli–air interfacial temperature, but the use of proper procedures in breath testing eliminates these effects (10, 16). Figure 2. Volume fractions of water and solids in whole blood.
534
(5)
Effect of the Ethanol Absorption–Elimination Cycle
g EtOH air α Vair
In the same way as before the equation can be transformed to
ratio =
Ethanol dissolves almost entirely in the water that makes up most of the blood volume, and ethanol in the lungs distributes between the breath and the aqueous part of the plasma. Plasma is whole blood minus the cellular components. Blood hematocrit (Hct) is the volume fraction of blood that is erythrocytes (red blood cells), and for healthy adults Hct varies from 0.36 to 0.53 (99% confidence limits) (17). In addition, the erythrocytes are 72% aqueous, while the plasma is 94% aqueous on average (18) (see Fig. 2). The volume of aqueous solution or water in which the ethanol is dissolved is related to the volume of blood in the following fashion:
(3)
Changing from Air to Human Breath The model is based entirely on thermodynamic considerations and assumes instantaneous attainment of equilibrium between the blood and the alveolar air. Deep-lung or end-expired air collected in the proper manner most closely approximates a fully equilibrated sample. Hypoventilation or hyperventilation has been found to influence the amount of ethanol expired (5), caused either by incomplete equilibration (kinetics effect) or by a concomitant breath temperature change (thermodynamic effect). Less than full equilibration would tend to decrease the amount of ethanol in the breath by some factor; let’s call it α, where α is always ≤ 1. For the most part, exhaled breath closely resembles ambient air. Of course, the fractions of carbon dioxide, oxygen, and water vapor are altered, and some volatile compounds from the blood are added to the mix, but breath still approximates an ideal gas. Considering this, eq 3 becomes ratio =
Changing from Water to Blood
Journal of Chemical Education • Vol. 74 No. 5 May 1997
In the Classroom Table 1 lists experimental values for the ratio reported in the literature for both controlled tests and the analyNotes Ref. sis of actual case results (field results) field work 26 in the U.S.A. The postabsorptive values range from 2130 to 2370, and the field work 27 average reported value is about 2270 field work 28 ± 200, not significantly different from 29 the predicted ratio. A value of 2300 usually is used in the United King30 dom, based on studies there (31); and a very recent study of 799 persons sus1 pected of DUI in Sweden found and 23 average ratio of 2400 ± 200 (32). 19 Note that the three field studies give a slightly higher mean ratio than the controlled, laboratory results. In The calculation was the field work reported by Harding et al. (26, 27), there was a delay of at least 30 min between the arrest and the breath test and another time lapse (< 1 hr) between the breath test and the blood test. The first delay makes it more likely, but does not assure that the driver was in a postabsorptive state at the time of breath testing, but the second delay could create an error, the direction and magnitude of which depends on the degree and rate of absorption and elimination of ethanol. Harding (26) addressed this potential error in their study, finding “only a weak negative correlation” between the delay time (averaging 36.6 min) and the differences between the pairs of blood alcohol values. Unfortunately, Gullberg does not give any experimental details for his data (28). Another possible explanation of the differences in the reported ratio values is incomplete equilibrium (α < 1), due to sampling of some shallow lung air in actual casework.
Table 1. Literature Reports of in vivo Values of Ratio #Subjects Gender
PostBACa absorptive (g/100 mL)
Ratio ± SD 2370b
395
?
likely
0–0.34 (0.18)
404
?
likely
0–0.42 (0.18)
2360b
~110
?
?
?
2340 ± 50
male
yes
?
2280 ± 240
14
?
yes no
0.04–0.09 (0.06) 0.01–0.09 (0.05)
2340 ± 190 2130 ± 150
21
male
yes
?
2180 ± 170b
149
10 5
5 male
yes
0–0.13 (0.08)
2130 ± 160
?
yes
?
2230 ± 280
aRange
(average). from the slope of the linear regression line for BrAC vs. BAC. ratio = 2100/slope. bCalculated
imbibing increases—that is, as the venous concentration catches up to the arterial concentration (19, 23). The theoretical model assumes complete equilibration between alveolar blood and alveolar air (breath). Consequently, predicted values should be compared only to experimental values obtained from persons in a postabsorptive state.
Evaluation of Partition Ratio Breath temperature is not routinely measured for drivers suspected of DUI; instead, the temperature is assumed to be 34.0 °C. As before, we will use ± 0.2 °C as the standard deviation of the measurement. The blood hematocrit (Hct) and standard deviation for healthy adult females is 0.41 ± 0.03 and for males is 0.47 ± 0.03 (17). The average human hematocrit is about 0.44, and the average fraction of water in whole blood is 0.84. The value of Hct is not recorded in actual case studies. Because of the difference in hematocrit between men and women, ratio values are best calculated for each sex. (Other gender differences concerning drinking and driving have been reported, including firstpass ethanol metabolism in the stomach is much greater in males [24] but overall ethanol elimination is faster in females [25]; and male drivers are by far more often convicted of DUI [2]). The ionic strength of the aqueous portion of blood, the value of which is about 0.15, may have an effect on the thermodynamic values and the activity coefficient for ethanol, but this effect is fairly small because ethanol is uncharged. Under these conditions, with α and γ EtOH water set to unity, and for the average adult male driver, ratio equals:
Effect of Temperature and Water Content of Blood
Linear regression of the predicted values of the ratio between 31 °C and 37 °C shows a 6.5% / °C decrease with increased temperature. Corresponding experimental values are 5.7% / °C (33), 6.5% / °C (14), and 7.3% / °C (7, 16). Thus, the model correlates well with the experimental data and can correctly predict the relative change in the ratio with temperature. A driver with an elevated lung temperature would have the BAC overestimated if the ratio dropped below the fixed 2100 value. This would occur, according to the model, at a temperature of ~36 °C. A person’s breath might reach this temperature in some disease states. The water content of blood varies considerably among individuals. Persons with a higher hematocrit are 0.08206 L atm deg{1mol{1 307 deg 0.47 0.72 + 0.53 0.94 1.00 atm{1 0.9944kg/L more likely to have their BrAC over= 2340 estimated, since the ratio decreases with lower water content (higher {53,200 J mol{1 + 307 deg 134.1 J mol{1deg{1 Hct). Males, on average, would find 1.00 1.00 kg mol{1 exp this to be true more often than fe8.3143 J mol{1deg{1 307 deg males. Heat or exercise can lead to excessive sweating and some degree of For the average adult female driver the predicted ratio dehydration, as well as a slight rise in body temperature. A is 2370, so the average for the population is about 2350. Of person in such a state would have a lower blood water concourse, if the activity coefficient for aqueous ethanol is retent and a lower ratio, and be more likely to have the BAC ally a bit larger than 1, then the value of the ratio would be overestimated. somewhat smaller. On the other hand, if complete equilibUncertainty Due to Physiological Variables rium at the blood–breath interface is not established, then α would be less than 1 and the partition ratio would be The uncertainties associated with eq 5 are listed in somewhat greater. Assuming uncertainty only in the temTable 2. Under current procedures (i.e., without monitoring perature to ±0.2 °C and in the density and thermodynamic breath temperature or hematocrit), the propagated relative values to ±1 in the last reported digit, the standard deviastandard deviation (with contributions from breath temtion for the predicted ratio is ± 100. perature and the water content of cells and plasma, but not
Vol. 74 No. 5 May 1997 • Journal of Chemical Education
535
In the Classroom Table 2. Uncertainties Associated with the Model Parameter
Propagated Rel. Uncertainty in Ratio (%)
Ave. Value
Standard Deviation
{53,200 134.1
± 100 ± 0.1
4.1
34.0 °C
± 1.0 °C
5.5
± 0.2 °C
1.1
Thermodynamic values ∆rH ° ∆rS ° Breath temperature Water content of blood
0.841
Hematocrit Fractions of watera
0.44 0.72, 0.94
± 0.045 ± 0.01
6.4
Hematocrit Fractions of watera
0.44 0.72, 0.94
± 0.01 ± 0.01
2.1
Thermodynamic values and temperature
± 1.0 °C ± 0.2 °C
6.9 4.3
Temperature and water content (physiological variables)
± 1.0 °C Hct ± 0.045
8.5
± 0.2 °C Hct ± 0.045
6.6
± 0.2 °C Hct ± 0.01
2.5
aIn
536
Acknowledgment I thank my colleague Norman Craig for extremely helpful discussions of the solution–vapor– ethanol equilibrium and thermodynamics.
erythrocytes and plasma, respectively.
considering uncertainty in the thermodynamic values) in the predicted ratio is 8.5%. This corresponds to an absolute uncertainty in the ratio of ±200, very similar to the experimental standard deviations reported in Table 1 and in reference 32. The predicted ratio for 95% of the population (mean ± 2 standard deviations) ranges from 1950 to 2750. The fact that a fixed value of 2100 is used to convert BrAC to BAC means that the calculated BAC would be higher than the true value in 11% of the cases. This percentage was calculated using a statistical z-table and assuming that ratio values are normally distributed. In contrast, one report suggests that ratio values in the population may be lognormally distributed (28). Let’s assume that breath temperature is measured to ± 0.2 °C for each person stopped for DUI. The propagated relative standard deviation in the predicted ratio drops to 6.6%. Almost all of this is due to the uncertainty in the water content of blood. Under these conditions, the theoretical ratio for 95% of the population lies in the range (mean ± 2 standard deviations): 2040 to 2660. The calculated BAC would be higher than the true value in 5% of cases, assuming a Gaussian distribution. Simultaneous measurement of breath temperature and BrAC and subsequent temperature correction of the ratio would significantly reduce the number of cases involving inaccurate calculation of BAC from BrAC. If the hematocrit of each subject’s blood could be measured to ± 0.01, then the propagated relative uncertainty in the water content of blood is 2.2% and the propagated uncertainty in the ratio is only 2.5%. Under these conditions, the theoretical ratio for 95% of the population lies in the range (mean ± 2 standard deviations): 2230 to 2470. The calculated BAC, using the fixed 2100:1 ratio, would be higher than the true value in fewer than 0.01% of cases, assuming a Gaussian distribution. Measurement of blood hematocrit for each driver suspected of DUI is certainly warranted. However, it is probably too much to expect in practice. The relationship and uncertainties described here suggest that if measurement of BrAC and subsequent calcula-
tion of BAC using a fixed value of the ratio continues, without concomitant breath temperature and hematocrit measurements, then the fixed value ought to be 1880 rather than 2100. Dubowski’s data (29), with an experimental ratio of 2280 ± 240, suggests a lower fixed value of 1720. Theoretically, this would reduce the fraction of cases that involve overestimates of BAC to less than 1% but would increase dramatically the number of underestimates. In our criminal justice system a person is assumed innocent until proven guilty, and society generally finds a false conviction of an innocent person more egregious than not punishing a lawbreaker. Consequently, overestimates of BAC are to be avoided, even if they result in some underestimates. Use of a smaller conversion factor also would circumvent most of the attacks on calculated BAC values, cogent arguments based on the populations of breath temperature and blood hematocrit values that significantly influence the conversion of BrAC to BAC.
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