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 0.286%.

Step-by-step solution

Balance the Redox Equation

We will balance the redox equation using the half-reaction method. In the redox equation, \(\mathrm{Cr_2O_7^{2-}}\) and \(\mathrm{C_2H_5OH}\) are the oxidizing and reducing agents, respectively. We can write the half-reactions as follows: Oxidation Half-Reaction: \(\mathrm{C_2H_5OH \rightarrow CO_2}\) Reduction Half-Reaction: \(\mathrm{Cr_2O_7^{2-}(aq) \rightarrow Cr^{3+}(aq)}\) Balance the half-reactions: For oxidation half-reaction: i) Balance the carbon atoms by adding 2 molecules of CO₂. \(\mathrm{C_2H_5OH \rightarrow 2CO_2}\) ii) Balance the hydrogen atoms by adding 6H₂O to the right side. \(\mathrm{C_2H_5OH \rightarrow 2CO_2 + 6H_2O}\) iii) Balance the oxygen atoms by adding 6H⁺ to the left side. \(\mathrm{C_2H_5OH + 6H^+ \rightarrow 2CO_2 + 6H_2O}\) Now for the reduction half-reaction: i) Balance the chromium atoms by adding 2Cr³⁺. \(\mathrm{Cr_2O_7^{2-}(aq) \rightarrow 2Cr^{3+}(aq)}\) ii) Balance the oxygen atoms by adding 7 water molecules. \(\mathrm{Cr_2O_7^{2-}(aq) + 7H_2O \rightarrow 2Cr^{3+}(aq)}\) iii) Balance the hydrogen atoms and charges by adding 14H⁺. \(\mathrm{Cr_2O_7^{2-}(aq) + 14H^+ \rightarrow 2Cr^{3+}(aq) + 7H_2O}\) Now, we can combine the balanced half-reactions: \(\mathrm{C_2H_5OH(aq) + 6H^+(aq) + Cr_2O_7^{2-}(aq) + 14H^+(aq) \rightarrow 2CO_2(g) + 6H_2O(l) + 2Cr^{3+}(aq) + 7H_2O(l)}\) Simplify the equation: \(\mathrm{C_2H_5OH(aq) + Cr_2O_7^{2-}(aq) + 20H^+(aq) \rightarrow 2CO_2(g) + 2Cr^{3+}(aq) + 13H_2O(l)}\) This is our balanced redox equation.

Use Stoichiometry to Determine the Moles of Ethanol Present

We are given that 31.05 mL of 0.0600 M potassium dichromate solution is required for titration. First, we find the moles of dichromate ions, and then the moles of ethanol using the stoichiometry of the balanced equation. moles of \(\mathrm{Cr_2O_7^{2-}} = \mathrm{Volume} \times \mathrm{Molarity}\) moles of \(\mathrm{Cr_2O_7^{2-}} = (31.05 \times 10^{-3} \mathrm{L}) \times 0.0600 \mathrm{M} = 1.863 \times 10^{-3}\; \mathrm{mol}\) From the balanced equation, we see that 1 mol of \(\mathrm{Cr_2O_7^{2-}}\) reacts with 1 mol of \(\mathrm{C_2H_5OH}\). Thus, the moles of ethanol are the same as the moles of dichromate ions. moles of \(\mathrm{C_2H_5OH} = 1.863 \times 10^{-3}\; \mathrm{mol}\)

Calculate the Mass of Ethanol and Mass Percent

Next, we'll find the mass of ethanol present in the blood plasma and calculate the mass percent. To do this, we will use the molar mass of ethanol (C₂H₅OH), which is approximately \(46.07 \mathrm{g/mol}\). mass of \(\mathrm{C_2H_5OH} = \mathrm{moles} \times \mathrm{molar \, mass}\) mass of \(\mathrm{C_2H_5OH} = (1.863 \times 10^{-3}\; \mathrm{mol}) \times 46.07 \mathrm{g/mol} = 0.0858 \mathrm{g}\) Now, calculate the mass percent of ethanol in the blood plasma: mass percent \(= \frac{\mathrm{mass \, of \, ethanol}}{\mathrm{mass \, of \, blood \, plasma}} \times 100\%\) mass percent \(= \frac{0.0858\; \mathrm{g}}{30.0\; \mathrm{g}} \times 100\% = 0.286\%\) Therefore, the mass percent of alcohol in the blood is approximately 0.286%.

Key concepts

Blood Alcohol Determination

Understanding how to measure blood alcohol concentration is crucial in various fields, such as law enforcement and medical diagnostics. This process typically involves the quantitative determination of ethanol in a blood sample. A common method used is redox titration, which utilizes a chemical reaction to find out how much ethanol is present. In the exercise, an acidic potassium dichromate solution is used. The reaction between ethanol and potassium dichromate changes the color of the solution, making it easy to see when the reaction is complete. The presence of alcohol in the blood is detected and measured through this color change.
Redox titration helps in determining alcohol levels efficiently, providing accurate measurements that are important for various applications.

Potassium Dichromate Solution

Potassium dichromate (\(K_2Cr_2O_7\)) is a strong oxidizing agent utilized in redox titrations due to its ability to change color. In solution, this compound appears orange. During the titration process with ethanol, it oxidizes the ethanol, transforming into trivalent chromium, or \(Cr^{3+}\), which has a green color.
This color shift from orange to green is critical for the titration process, as it signals the endpoint, indicating that all the ethanol has reacted. This visual cue can be easily detected, either by a human observer or an optical instrument, allowing for precise analysis.

Half-Reaction Method

The half-reaction method is a systematic way of simplifying and balancing complex redox reactions. By splitting the full reaction into two parts, the oxidation and reduction processes can be analyzed separately. In this exercise, ethanol (\(C_2H_5OH\)) undergoes oxidation and releases electrons, while \(Cr_2O_7^{2-}\) ions are reduced by accepting those electrons.
First, we balance the individual half-reactions for atoms and charge. This involves adding molecules like \(H_2O\) and \(H^+\) ions to balance oxygen and hydrogen atoms. Then, the balanced half-reactions are recombined to reflect the overall stoichiometry, ensuring all atoms and charges are balanced. This method provides a clear and organized approach to tackling complex redox problems.

Mass Percent Calculation

Calculating mass percent is an important step in quantifying the concentration of a substance within a mixture. In this scenario, it is used to determine the percentage of ethanol in blood plasma.
The calculation begins with finding the mass of ethanol using its moles and molar mass. Then, this mass is divided by the total mass of the blood sample and multiplied by 100 to convert it into a percentage.
\[ \text{mass percent} = \frac{\text{mass of ethanol}}{\text{mass of blood plasma}} \times 100\% \]
This approach provides a straightforward understanding of the composition of a mixture, making it a fundamental tool in chemistry and analytical labs.