Question

A chemical "breathalyzer" test works because ethanol in the breath is oxidized by the dichromate ion (orange) to form acetic acid and chromium(III) ion (green). The balanced reaction is \(3 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q)+2 \mathrm{Cr}_{2} \mathrm{O}_{7}{ }^{2-}(a q)+2 \mathrm{H}^{+}(a q) \longrightarrow\) \(3 \mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}(a q)+4 \mathrm{Cr}^{3+}(a q)+11 \mathrm{H}_{2} \mathrm{O}(l)\) You analyze a breathalyzer test in which \(4.2 \mathrm{mg} \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}\) was reduced. Assuming the volume of the breath was \(0.500 \mathrm{~L}\) at \(30 .{ }^{\circ} \mathrm{C}\) and \(750 . \mathrm{mm} \mathrm{Hg}\), what was the mole percent alcohol of the breath?

Short answer

The mole percent alcohol of the breath can be determined using the balanced chemical reaction, moles of potassium dichromate used and the ideal gas law. Firstly, the mole ratio of ethanol to potassium dichromate is 3:2. Convert the given mass of potassium dichromate to moles to find it is \(1.428 \times 10^{-5}\) mol. Using the mole ratio, the moles of ethanol is \(2.142 \times 10^{-5}\) mol. The total moles of the gas mixture is calculated using the ideal gas law to be \(0.02\) mol. The mole percent of ethanol is then \(\frac{2.142 \times 10^{-5}}{0.02} \times 100 = 0.1071\% \). Therefore, the mole percent alcohol of the breath is approximately 0.1071%.

Step-by-step solution

Write down the balanced reaction and mole ratio

The balanced reaction is given as: \[3 C_{2}H_{5}OH(aq)+2 Cr_{2}O_{7}^{2-}(aq)+16 H^{+}(aq)\longrightarrow 3 HC_{2}H_{3}O_{2}(a q)+4 Cr^{3+}(aq)+11 H_{2}O(l)\] As one can observe from this reaction, 3 moles of ethanol react with 2 moles of dichromate ion. Therefore, the mole ratio of Ethanol to Dichromate is \(3:2\).

Convert mass of K2Cr2O7 to moles

The molar mass of \(K_{2}Cr_{2}O_{7}\) is 294.185 g/mol. Given that we used 4.2 mg (which equals 0.0042 g), we calculate moles as: \[ \frac{0.0042 \, g}{294.185 \, g/mol} = 1.428 \times 10^{-5} mol \]

Calculate moles of Ethanol

Using the mole ratio from step 1 (3:2), moles of C2H5OH (Ethanol) will be: \[ 1.428 \times 10^{-5} mol \times \frac{3}{2} = 2.142 \times 10^{-5} \, mol \]

Calculate total moles of the gaseous mixture

Using the ideal gas law, \(PV=nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature. We can re-arrange for \(n\): \[ n=\frac{PV}{RT} \] Given that \(P=750mmHg\) (we need to convert this to atm for consistency with R, so \(P=750/760=0.987atm\)), \(V=0.500L\), \(R=0.0821 Latm/molK\), and \(T=30 ^{\circ} C (T=303.15K\) when converted from Celsius to Kelvin), we plug these values into the formula to get: \[ n=\frac{0.987atm \cdot 0.500L}{0.0821Latm/molK \cdot 303.15K} = 0.02 mol \]

Compute mole percent of Ethanol

The mole percent is calculated as: \[ \frac{moles \, of \, Ethanol}{total \, moles} \times 100 = \frac{2.142 \times 10^{-5} \, mol}{0.02 \, mol} \times 100 = 0.1071\% \] So the mole percent alcohol of the breath is approximately 0.1071%.

Key concepts

Oxidation-Reduction Reactions

Oxidation-reduction reactions, often known as redox reactions, are a fundamental concept in chemistry that involve the transfer of electrons between chemical species. In a redox reaction, one substance undergoes oxidation, which means it loses electrons, while another substance undergoes reduction, gaining electrons.

In the context of a breathalyzer test, ethanol (2H5OH) in the breath reacts with dichromate ions (2O72-) in an oxidation process, where the ethanol is oxidized to acetic acid (CH3COOH), and the orange dichromate ion is reduced to green chromium(III) ion (Cr3+). This color change is indicative of the presence of ethanol in the breath and the extent of the reaction can be used to estimate the alcohol content of the breath sample.

Mole Ratio Calculation

Mole ratio calculation is an important step in stoichiometry, which is the quantitative relationship between reactants and products in a chemical reaction. By using the coefficients of each compound from the balanced chemical equation, one can determine the exact proportions in which substances react or form products.

In the case of the breathalyzer test, the balanced equation indicates that 3 moles of ethanol (2H5OH) react with 2 moles of dichromate ions (2O72-). This 3:2 mole ratio is essential in calculating the amount of ethanol present in the breath based on the amount of dichromate reduced. Understanding mole ratios is crucial for students when they are solving for quantities in reactions, allowing them to relate the mass of one substance to the molar amount of another.

Ideal Gas Law

The ideal gas law is an equation that relates the pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas. The law is expressed as PV = nRT, where R is the ideal gas constant. The ideal gas law assumes that gases consist of point particles that engage in perfectly elastic collisions and experience no intermolecular forces.

The practical application of the ideal gas law in our example is to determine the total number of moles of gas in the breath sample. This step is essential to eventually calculate the mole percent of ethanol vapor present in the breath. By combining the ideal gas law with the mole ratio from the redox reaction, one can determine the mole percent of alcohol in the breathalyzer test, thus aiding in the measurement of a person's level of intoxication.