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Chapter 22: Problem 151

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

Expert verified
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

01

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 \(3:2\).
02

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: \[ \frac{0.0042 \, g}{294.185 \, g/mol} = 1.428 \times 10^{-5} mol \]
03

Calculate moles of Ethanol

Using the mole ratio from step 1 (3:2), moles of C2H5OH (Ethanol) will be: \[ 1.428 \times 10^{-5} mol \times \frac{3}{2} = 2.142 \times 10^{-5} \, mol \]
04

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: \[ n=\frac{0.987atm \cdot 0.500L}{0.0821Latm/molK \cdot 303.15K} = 0.02 mol \]
05

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%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In the presence of light, chlorine can substitute for one (or more) of the hydrogens in an alkane. For the following reactions, draw the possible monochlorination products. a. 2,2 -dimethylpropane \(+\mathrm{Cl}_{2} \stackrel{\mathrm{hv}}{\longrightarrow}\) b. 1,3 -dimethylcyclobutane \(+\mathrm{Cl}_{2} \stackrel{\text { hv }}{\longrightarrow}\) c. 2,3 -dimethylbutane \(+\mathrm{Cl}_{2} \stackrel{\mathrm{hv}^{-}}{\longrightarrow}\)

For each of the following, fill in the blank with the correct response. All of these fill-in-the-blank problems pertain to material covered in the sections on alkanes, alkenes and alkynes, aromatic hydrocarbons, and hydrocarbon derivatives. a. The first "organic" compound to be synthesized in the laboratory, rather than being isolated from nature, was which was prepared from b. An organic compound whose carbon-carbon bonds are all single bonds is said to be c. The general orientation of the four pairs of electrons around the carbon atoms in alkanes is d. Alkanes in which the carbon atoms form a single unbranched chain are said to be alkanes. e. Structural isomerism occurs when two molecules have the same number of each type of atom but exhibit different arrangements of the between those atoms. f. The systematic names of all saturated hydrocarbons have the ending added to a root name that indicates the number of carbon atoms in the molecule. g. For a branched hydrocarbon, the root name for the hydrocarbon comes from the number of carbon atoms in the continuous chain in the molecule. h. The positions of substituents along the hydrocarbon framework of a molecule are indicated by the of the carbon atom to which the substituents are attached. i. The major use of alkanes has been in reactions, as a source of heat and light. j. With very reactive agents, such as the halogen elements, alkanes undergo \(\longrightarrow\) reactions, whereby a new atom replaces one or more hydrogen atoms of the alkane. k. Alkenes and alkynes are characterized by their ability to undergo rapid, complete reactions, by which other atoms attach themselves to the carbon atoms of the double or triple bond. 1\. Unsaturated fats may be converted to saturated fats by the process ofm. Benzene is the parent member of the group of hydrocarbons called \(\quad\) hydrocarbons. n. An atom or group of atoms that imparts new and characteristic properties to an organic molecule is called \(a\) group. 0\. \(\mathrm{A}\) alcohol is one in which there is only one hydrocarbon group attached to the carbon atom holding the hydroxyl group. p. The simplest alcohol, methanol, is prepared industrially by the hydrogenation of q. Ethanol is commonly prepared by the of certain sugars by yeast. r. Both aldehydes and ketones contain the they differ in where this group occurs along the hydrocarbon chain. s. Aldehydes and ketones can be prepared by of the corresponding alcohol. t. Organic acids, which contain the group, are typically weak acids. u. The typically sweet-smelling compounds called result from the condensation reaction of an organic acid with an

If one hydrogen in a hydrocarbon is replaced by a halogen atom, the number of isomers that exist for the substituted compound depends on the number of types of hydrogen in the original hydrocarbon. Thus there is only one form of chloroethane (all hydrogens in ethane are equivalent), but there are two isomers of propane that arise from the substitution of a methyl hydrogen or a methylene hydrogen. How many isomers can be obtained when one hydrogen in each of the compounds named below is replaced by a chlorine atom? a. \(n\) -pentane c. 2,4 -dimethylpentane b. 2 -methylbutane d. methylcyclobutane

"Super glue" contains methyl cyanoacrylate, C=C(C)C(=O)OC which readily polymerizes upon exposure to traces of water or alcohols on the surfaces to be bonded together. The polymer provides a strong bond between the two surfaces. Draw the structure of the polymer formed by methyl cyanoacrylate.

Draw structural formulas for each of the following alcohols. Indicate whether the alcohol is primary, secondary, or tertiary. a. 1 -butanol c. 2 -methyl-l-butanol b. 2 -butanol d. 2 -methyl-2-butanol
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