Question

The blood alcohol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) level can be determined by titrating a sample of blood plasma with an acidic potassium dichromate solution, resulting in the production of \(\mathrm{Cr}^{3+}(a q)\) and carbon dioxide. The reaction can be monitored because the dichromate ion \(\left(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\right)\) is orange in solution, and the \(\mathrm{Cr}^{3+}\) ion is green. The unbalanced redox equation is $$\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}(a q)+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q) \longrightarrow \mathrm{Cr}^{3+}(a q)+\mathrm{CO}_{2}(g)$$ If \(31.05 \mathrm{~mL}\) of \(0.0600 M\) potassium dichromate solution is required to titrate \(30.0 \mathrm{~g}\) blood plasma, determine the mass percent of alcohol in the blood.

Short answer

The mass percent of alcohol in the blood is approximately \(0.857\ \%\).

Step-by-step solution

Balance the redox equation

To balance the given redox equation, first, we should identify the oxidation and reduction half-reactions. The ethanol \(\mathrm{C_{2}H_{5}OH}\) molecule loses electrons (it is oxidized) to form \(\mathrm{CO_{2}}\), while the \(\mathrm{Cr_{2}O_{7}^{2-}}\) ion gains electrons (it is reduced) to form \(\mathrm{Cr^{3+}}\). The balanced half-reactions are: Oxidation half-reaction: \(\mathrm{C_{2}H_{5}OH} \rightarrow \mathrm{CO_{2}} + 12\mathrm{H^{+}} + 12\mathrm{e^-}\) Reduction half-reaction: \(\mathrm{Cr_{2}O_{7}^{2-}} + 14\mathrm{H^{+}} + 6\mathrm{e^-} \rightarrow 2\mathrm{Cr^{3+}} + 7\mathrm{H_{2}O}\) To combine these half-reactions, multiply the oxidation half-reaction by 6 and the reduction half-reaction by 2, then add them together to form the balanced redox equation: \[\begin{aligned} &6\left(\mathrm{C_{2}H_{5}OH} \rightarrow \mathrm{CO_{2}} + 12\mathrm{H^{+}} + 12\mathrm{e^-}\right) \\ + &2\left(\mathrm{Cr_{2}O_{7}^{2-}} + 14\mathrm{H^{+}} + 6\mathrm{e^-} \rightarrow 2\mathrm{Cr^{3+}} + 7\mathrm{H_{2}O}\right) \\ \rightarrow &6\mathrm{C_{2}H_{5}OH} + 2\mathrm{Cr_{2}O_{7}^{2-}} + 52\mathrm{H^{+}} \rightarrow 6\mathrm{CO_{2}} + 4\mathrm{Cr^{3+}} + 14\mathrm{H_{2}O} \end{aligned}\]

Calculate the moles of potassium dichromate

We are given the volume \((31.05\ \mathrm{mL})\) and the concentration \((0.0600\ \mathrm{M})\) of potassium dichromate solution. To determine the moles, use the formula: Moles = Volume × Concentration Moles of potassium dichromate = \(31.05 \times 0.0600\) = \(1.86 \times 10^{-3}\ \mathrm{mol}\)

Convert moles of potassium dichromate to moles of alcohol

According to the balanced redox equation, 2 moles of \(\mathrm{Cr_{2}O_{7}^{2-}}\) react with 6 moles of \(\mathrm{C_{2}H_{5}OH}\). Therefore, the ratio between the moles of potassium dichromate and alcohol is 1:3. To find the moles of alcohol, we can use the following equation: \(\mathrm{moles\ of\ C_{2}H_{5}OH} = 3 \times \mathrm{moles\ of\ Cr_{2}O_{7}^{2-}}\) Moles of alcohol = \(3 \times 1.86 \times 10^{-3}\) = \(5.58 \times 10^{-3}\ \mathrm{mol}\)

Determine the mass of alcohol

To find the mass of alcohol, multiply the moles of alcohol by its molar mass: Mass of alcohol = moles of alcohol × molar mass of alcohol Molar mass of alcohol = \(2\mathrm{(C)} + 6\mathrm{(H)} + 1\mathrm{(O)} = 46.07\ \mathrm{g/mol}\) Mass of alcohol = \(5.58 \times 10^{-3} \times 46.07\) = \(0.257\ \mathrm{g}\)

Calculate the mass percent of alcohol in the blood

The mass percent of alcohol in the blood can be calculated using the following formula: Mass percent of alcohol = \(\frac{\text{mass of alcohol}}{\text{mass of blood plasma}} \times 100\) Mass percent of alcohol = \(\frac{0.257}{30.0} \times 100\) = \(0.857\ \%\) The mass percent of alcohol in the blood is approximately \(0.857\ \%\).

Key concepts

Oxidation-Reduction

Oxidation-reduction reactions, often called redox reactions, involve the transfer of electrons between substances. Simple as that! During this process, one substance gets oxidized while another gets reduced. When a substance is oxidized, it loses electrons. Alternatively, when a substance is reduced, it gains electrons.
In the exercise above, ethanol (\(\mathrm{C}_{2}\mathrm{H}_{5}\mathrm{OH}\)) undergoes oxidation as it transitions to carbon dioxide (\(\mathrm{CO}_{2}\)), losing electrons. On the flip side, the potassium dichromate (\(\mathrm{Cr}_{2}\mathrm{O}_{7}^{2-}\)) is reduced into \(\mathrm{Cr}^{3+}\) by gaining those lost electrons. Overall, redox reactions are vital in chemistry because they involve electron transfer crucial for processes like metabolism and combustion.

Potassium Dichromate

Potassium dichromate, chemically expressed as \(\mathrm{K}_{2}\mathrm{Cr}_{2}\mathrm{O}_{7}\), is commonly used in laboratory settings for redox reactions. It's a strong oxidizing agent, meaning it has a great affinity for gaining electrons in a chemical process.
In solutions, it presents an orange color that makes it visually effective for titrations due to its distinct hue change. In the context of the exercise, potassium dichromate helps detect the alcohol content in blood by getting reduced to \(\mathrm{Cr}^{3+}\), an ion characterized by a green color. This visible change indicates the reaction's progress, allowing analysts to measure the alcohol accurately in the blood.

Blood Alcohol Content

Blood Alcohol Content (BAC) is a measure of the amount of alcohol present in a person's bloodstream. It is usually expressed as a percentage. BAC levels determine intoxication levels and how it affects bodily functions.
In this exercise, BAC is quantified through a simple titration with potassium dichromate. By determining how much potassium dichromate reacts with alcohol in the blood, one can calculate the concentration of alcohol present. The higher the volume of dichromate used, the greater the alcohol concentration in the blood sample.
Monitoring BAC is crucial as it influences legal limits for activities like driving. Knowing how to measure BAC accurately can help ensure safety and adherence to the law.

Balancing Chemical Equations

Balancing chemical equations is a necessary skill in chemistry that ensures the law of conservation of mass is respected. This law states that matter cannot be created or destroyed in an isolated system. A balanced equation reflects equal amounts of each element on both sides of the equation.
When handling redox reactions, correctly balancing oxidation and reduction components is vital. In the exercise, we first separated ethanol oxidation and dichromate reduction into two half-reactions. Each was then balanced thoroughly, before combining them to form a complete and balanced equation.
This practice not only helps to predict the yield of products but also ensures chemical reactions are fully understood, allowing for accurate calculations about reactants and products.