Question

Blood alcohol content (BAC) is often reported in weight-volume percent (w/v\%). For example, a BAC of \(0.10 \%\) corresponds to \(0.10 \mathrm{g} \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\) per 100 mL of blood. Estimates of BAC can be obtained from breath samples by using a number of commercially available instruments, including the Breathalyzer for which a patent was issued to R. F. Borkenstein in 1958\. The chemistry behind the Breathalyzer is described by the oxidation- reduction reaction below, which occurs in acidic solution: \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}(\mathrm{g})+\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}(\mathrm{aq}) \longrightarrow\) \(\mathrm{CH}_{3} \mathrm{COOH}(\mathrm{aq})+\mathrm{Cr}^{3+}(\mathrm{aq}) \quad(\text { not balanced })\)A Breathalyzer instrument contains two ampules, each of which contains \(0.75 \mathrm{mg} \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}\) dissolved in \(3 \mathrm{mL}\) of \(9 \mathrm{mol} / \mathrm{L} \mathrm{H}_{2} \mathrm{SO}_{4}(\mathrm{aq}) .\) One of the ampules is used as reference. When a person exhales into the tube of the Breathalyzer, the breath is directed into one of the ampules, and ethyl alcohol in the breath converts \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\) into \(\mathrm{Cr}^{3+} .\) The instrument compares the colors of the solutions in the two ampules to determine the breath alcohol content (BrAC), and then converts this into an estimate of BAC. The conversion of BrAC into BAC rests on the assumption that 2100 mL of air exhaled from the lungs contains the same amount of alcohol as \(1 \mathrm{mL}\) of blood. With the theory and assumptions described in this problem, calculate the molarity of \(\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}\) in the ampules before and after a breath test in which a person with a BAC of \(0.05 \%\) exhales 0.500 Lof his breath into a Breathalyzer instrument.

Short answer

The molarity of \(K_2Cr_2O_7\) before the breath test is roughly 0.85 M and after a breath test with a person with a BAC of \(0.05 \%\), the molarity of \(K_2Cr_2O_7\) is approximately 0 M.

Step-by-step solution

Determine the molarity of \(K_2Cr_2O_7\) before breath test

This can be calculated using the formula for molarity, which is moles of solute divided by the volume of solution in liters. First, convert the mass of \(K_2Cr_2O_7\) to moles using its molar mass (294.185 g/mol). \(0.75 \, mg\) equals \(0.75 \times 10^{-3}\, g\). Therefore, \(0.75 \times 10^{-3}\, g \, K_2Cr_2O_7\) is equal to \((0.75 \times 10^{-3}) \, g \times \frac{1 mol}{294.185 g} \approx 2.55 \times 10^{-6}\, mol\). The volume of solution is 3 mL or \(3 \times 10^{-3}L\). Therefore, Molarity = \(\frac{2.55 \times 10^{-6} mol}{3 \times 10^{-3}L}\) \(\approx\) 0.85 M

Calculate the amount of alcohol in person's breath

BAC of \(0.05 \%\) corresponds to \(0.05 \mathrm{g} \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\) per 100 mL of blood. As 2100 mL of exhaled air contains the same amount of alcohol as 1 mL of blood, 2100 mL of air contains \(0.05 \, g \, CH_3CH_2OH\). Therefore, 0.500 L will contain \(0.05 \, g \times \frac{0.500 L}{2.100L} \approx 0.012 \, g \, CH_3CH_2OH\). Convert \(0.012 \, g \, CH_3CH_2OH\) to mol using its molar mass which is \(46.07 \, g/mol\), gives \(\frac{0.012 \, g}{46.07 \, g/mol} \approx 0.00026 \, mol\).

Identify the stoichiometric ratio of the reaction

From the redox equation, we see that \(1 \textrm{ mol of } CH_3CH_2OH \textrm{ reacts with } 1 \textrm{ mol of } Cr_2O_7^{2-}\) to convert it into \(Cr^{3+}\). Therefore in step 2 when we calculated the mol of alcohol in the subject's breath we have also calculated the mol of \(K_2Cr_2O_7\) that got converted.

Calculate the remaining molarity of \(K_2Cr_2O_7\) after the breath test

The initial mol of \(K_2Cr_2O_7\) was \(2.55 \times 10^{-6}\, mol\) and the mol \(\textrm{used (or converted)}\) was \(0.00026 \, mol\), so the remaining molarity can be found by subtracting these values. The remaining mol of \(K_2Cr_2O_7\) = \(2.55 \times 10^{-6}\, mol - 0.00026 \, mol \approx -0.000257\, mol.\) Although it is not physically possible to have a negative number of moles, we have reached this result due to the rounding of the number in the third decimal point in step 2. Effectively, practically all the \(K_2Cr_2O_7\) has been converted into \(Cr^{3+}\), so the remaining molarity of \(K_2Cr_2O_7\) after the breath test is approximately 0 M.

Key concepts

Breathalyzer Function

A breathalyzer is a device used to estimate a person's blood alcohol content (BAC) from a breath sample. When a person consumes alcohol, a portion of it crosses into their lungs and is exhaled with their breath. The breathalyzer captures this alcohol-laced breath and uses chemical reactions to quantify the amount of alcohol present.

The principle behind its operation is rooted in chemical reactions. Specifically, an oxidation-reduction reaction occurs when the breath sample, which contains volatile alcohol molecules, reacts with a chemical oxidizing agent. Most commonly, chromium (VI) ions from potassium dichromate (\(K_2Cr_2O_7\text{ in aq}\)) are reduced to chromium (III) ions as ethyl alcohol (\(CH_3CH_2OH\text{ g}\)) is oxidized to acetic acid (\(CH_3COOH\text{ aq}\)). The color change associated with the reaction is analyzed to determine the alcohol concentration.

The device accounts for the ratio of breath alcohol to blood alcohol, operating on the assumption that there is a predictable partition ratio of 2100:1 between the two. Therefore, a certain concentration of alcohol in the breath would correlate to a proportional concentration in the blood, allowing for a calculated estimation of BAC.

Oxidation-Reduction Reactions

At the core of a breathalyzer's functionality are oxidation-reduction or redox reactions, which involve the transfer of electrons between chemical species. In the context of breathalyzers and their chemistry, ethanol (\(CH_3CH_2OH\text{ g}\)) present in the exhaled breath is oxidized; it loses electrons and forms acetic acid (\(CH_3COOH\text{ aq}\)). Concurrently, an oxidizing agent, which in the Breathalyzer is the dichromate ion (\(Cr_2O_7^{2-}\text{ aq}\)), gains electrons and is reduced to form chromium (III) ions (\(Cr^{3+}\text{ aq}\)).

These redox reactions are not only key to understanding breathalyzer function but also serve a broader role in diverse fields, including energy storage, metallurgy, and biochemistry. The reactions are always balanced, meaning the number of electrons lost in the oxidation half of the reaction equals the number of electrons gained in the reduction half.

Molarity Calculations

Molarity is a measure of the concentration of a solute in a solution and is defined as the number of moles of solute divided by the volume of the solution in liters. Calculating molarity is essential for quantifying the chemical reactions taking place in a breathalyzer, as it influences the stoichiometry of the reaction and the resultant color change.

To calculate molarity, one should first convert the mass of the solute to moles using its molar mass. Then, the volume of the solution should be quantified in liters. Dividing the number of moles by the volume yields the molarity (\(M\text{ in mol/L}\)). In the context of a Breathalyzer, understanding molarity calculations aids in determining the amount of chromium (VI) before and after the test, informing the estimation of BAC from a breath sample. It is important to conduct these calculations with precision to ensure the breathalyzer readings are accurate, which is critical for law enforcement and personal safety applications.