Question

The blood alcohol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) level can be determined by tirrating 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 $$\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}(a q)+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q) \rightarrow \mathrm{Cr}^{3+}(a q)+\mathrm{CO}_{2}(g)$$ If 31.05 \(\mathrm{mL}\) of 0.0600\(M\) potassium dichromate solution is required to titrate 30.0 \(\mathrm{g}\) of blood plasma, determine the mass percent of alcohol in the blood.

Short answer

The mass percent of alcohol in the blood plasma is 0.857%.

Step-by-step solution

Balancing the redox equation

First, we need to balance the given redox equation. We need to balance the atoms and charges on both sides. Our unbalanced equation is: $$\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}(a q)+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q) \rightarrow \mathrm{Cr}^{3+}(a q)+\mathrm{CO}_{2}(g)$$ The balanced redox equation is: $$\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}(a q)+16\mathrm{H}^{+}+3\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q) \rightarrow 2\mathrm{Cr}^{3+}(a q)+11\mathrm{H}_{2}\mathrm{O}(l)+3\mathrm{CO}_{2}(g)$$

Calculate moles of potassium dichromate

We are given the volume (31.05 mL) and concentration (0.0600 M) of potassium dichromate solution. Convert the volume to liters and calculate the moles of potassium dichromate: $$\text{Moles of}\; \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-} = \text{Volume} \times \text{Concentration}$$ $$= 0.03105\; \text{L} \times 0.0600\; \text{M}$$ $$= 0.001863\; \text{moles}$$

Calculate moles of alcohol

From the balanced redox equation, we can see that 1 mole of \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\) reacts with 3 moles of \(\mathrm{C}_{2} \mathrm{H}_{5}\mathrm{OH}\). Using the moles of potassium dichromate, calculate the moles of alcohol: $$\text{Moles of}\; \mathrm{C}_{2} \mathrm{H}_{5}\mathrm{OH} = \frac{3}{1} \times \text{Moles of}\; \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}$$ $$= 3 \times 0.001863\; \text{moles}$$ $$= 0.005589\; \text{moles}$$

Calculate mass of alcohol and mass percent

Now, we will convert the moles of alcohol to mass using the molar mass of alcohol (46.07 g/mol): $$\text{Mass of alcohol} = \text{Moles of alcohol} \times \text{Molar mass of alcohol}$$ $$= 0.005589\; \text{moles} \times 46.07\; \text{g/mol}$$ $$= 0.257\; \text{g}$$ We are given that the blood sample weighs 30.0 g. Calculate the mass percent of alcohol in the blood sample: $$\text{Mass Percent} = \frac{\text{Mass of alcohol}}{\text{Total mass of blood sample}} \times 100$$ $$= \frac{0.257 \; \text{g}}{30.0 \; \text{g}} \times 100$$ $$= 0.857 \%$$ The mass percent of alcohol in the blood plasma is 0.857%.

Key concepts

Blood Alcohol Content

Blood Alcohol Content (BAC) is a measurement of the amount of alcohol in a person's bloodstream. It is often expressed as a percentage. When alcohol is consumed, it is absorbed into the blood and affects the central nervous system. Understanding BAC is important for legal, health, and safety reasons.

• A BAC of 0.08% is typically considered legally impaired in many places.
• Factors affecting BAC include body weight, consumption rate, and the type of alcohol consumed.
• Determining BAC involves measuring the alcohol present in a blood sample, often using chemical reactions.

Titration methods, like those that use potassium dichromate, provide an effective approach to ascertain BAC in controlled environments.

Potassium Dichromate

Potassium Dichromate, \( \text{K}_2\text{Cr}_2\text{O}_7 \), is a bright orange compound commonly used in titration to determine the presence of alcohol. Its chemical properties make it an excellent oxidizing agent, meaning it can remove electrons from other substances.

• In an acidic solution, potassium dichromate oxidizes ethanol \( \left( \text{C}_2\text{H}_5\text{OH} \right)\) to carbon dioxide and water.
• The process involves transforming \( \text{Cr}_2\text{O}_7^{2-} \) to \( \text{Cr}^{3+} \, \) visually indicated by a color change from orange to green.
• This transformation is essential for quantifying the amount of alcohol present in a sample.

The change in color is an obvious indicator of the reaction's progress, making potassium dichromate particularly useful in visual titrations.

Balancing Redox Reactions

Balancing redox reactions is crucial for ensuring that matter and charge are conserved in a chemical reaction. In a redox reaction, one substance is oxidized (loses electrons) and another is reduced (gains electrons).

• Start by splitting the reaction into two half-reactions: one for oxidation and one for reduction.
• Balance the number of atoms in each half-reaction. Don't forget to also balance the charge by adding electrons where necessary.
• Combine the two half-reactions, making sure that the electrons cancel out to give the balanced overall equation.

For example, in our exercise, we balance the dichromate ion \( \left(\text{Cr}_2\text{O}_7^{2-}\right) \) with ethanol \( \left(\text{C}_2\text{H}_5\text{OH}\right)\) to obtain carbon dioxide and water, ensuring both mass and charge balance.

Moles Calculation

Calculating moles is a key step in solving chemical equations and determining how much of a substance reacts or forms. The concept of moles bridges the macroscopic and atomic worlds.

• Moles represent a count of particles, with one mole equating to \(6.022 \times 10^{23}\) particles (Avogadro's number).
• You calculate moles by multiplying the volume of a solution by its molarity: \( \text{Moles} = \text{Volume} \times \text{Concentration}\).
• In our problem, converting the volume of potassium dichromate to liters and multiplying by its molarity gives the moles used in the reaction.

Understanding moles helps to quantify chemical reactions, like how much alcohol reacts with potassium dichromate, leading to results like the mass percent of alcohol.