Problem 126 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A gaseous alkane is exploded with oxygen. The volume of \(\mathrm{O}_{2}\), for complete combustion to the volume of \(\mathrm{CO}_{2}\) formed is in 7:4 ratio. The molecular formula of alkane is (a) \(\mathrm{CH}_{4}\) (b) \(\mathrm{C}_{3} \mathrm{H}_{8}\) (c) \(\mathrm{C}_{2} \mathrm{H}_{6}\) (d) \(\mathrm{C}_{4} \mathrm{H}_{10}\)

Step-by-Step Solution

Answer

The alkane that has a ratio of 7 volumes of \(O_2\) to 4 volumes of \(CO_2\) upon complete combustion is \(C_2H_6\) or ethane.

Step 1: Write the general combustion reaction for alkanes

The general combustion reaction for an alkane, \(C_nH_{2n+2}\), with oxygen \(O_2\) is: \[C_nH_{2n+2} + (\frac{3n+1}{2})O_2 \rightarrow nCO_2 + (n+1)H_2O\] This reaction assumes complete combustion, where the products are carbon dioxide \(CO_2\) and water \(H_2O\).

Step 2: Apply given volume ratio to the combustion reaction

Since the volume ratio of \(O_2\) to \(CO_2\) is 7:4, for every 7 volumes of \(O_2\), 4 volumes of \(CO_2\) are produced. Using Avogadro's law, which states that equal volumes of gases at the same temperature and pressure contain the same number of molecules, the mole ratio can be applied directly to the reaction coefficients. So the reaction simplifies to: \[(3n+1)/2 = 7/4\] and \[n = 4/7\]

Step 3: Solve for the value of n

Solve the equation \(\frac{3n+1}{2} = \frac{7}{4}\) to find the value of n. \[3n+1 = \frac{7}{2}\] \[3n = \frac{7}{2} - 1\] \[3n = \frac{5}{2}\] \[n = \frac{5}{6}\] This value of n does not correspond to a whole number which is necessary for the number of carbons in an alkane. Thus, we re-evaluate the ratio equivalence in Step 2.

Step 4: Re-evaluate the volume ratio to represent moles

The correct interpretation of Avogadro's law in this context implies that the stoichiometric coefficients of \(O_2\) and \(CO_2\) in the balanced chemical reaction must reflect the 7:4 ratio in volumes. In other words, the coefficients when simplified should maintain a 7:4 ratio. Looking at the possible answers: (a) \(CH_4\) would require 2 \(O_2\) and produce 1 \(CO_2\), a 2:1 ratio. (b) \(C_3H_8\) would require 5 \(O_2\) and produce 3 \(CO_2\), a 5:3 ratio. (c) \(C_2H_6\) would require 3.5 \(O_2\) and produce 2 \(CO_2\), a 7:4 ratio. (d) \(C_4H_{10}\) would require 6.5 \(O_2\) and produce 4 \(CO_2\), which is also not a 7:4 ratio. Of these, only \(C_2H_6\) matches the 7:4 ratio.

Key Concepts

Chemical Stoichiometry

Chemical stoichiometry is a foundational principle in chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. Essentially, it helps us understand the amounts of substances consumed and produced. When dealing with the combustion of alkanes, this principle guides us through calculating the precise amounts of reactants needed for complete combustion.

For example, in the general combustion reaction of an alkane with the molecular formula of \(C_nH_{2n+2}\), the ratio of the number of moles of oxygen (\(O_2\)) to carbon dioxide produced (\(CO_2\)) has a direct correlation with n, the number of carbon atoms in the alkane. Utilizing stoichiometry, we interpret the given volume ratio (7:4 for \(O_2\) to \(CO_2\)) as a clue to find the correct molecular formula of the alkane in question. We need to find the value of n that would balance the equation and reflect this ratio in moles—a key challenge of stoichiometry in this context.

Avogadro's Law

Avogadro's law provides a critical insight into the behavior of gases during chemical reactions, by stating that equal volumes of gases, at the same temperature and pressure, contain an equal number of molecules. The importance of this law in stoichiometry and the combustion of alkanes cannot be understated.

When we consider the volume ratio of \(O_2\) to \(CO_2\) given in the problem, we can, thanks to Avogadro's law, infer the ratio of their molar amounts directly because the volumes of gaseous reactants and products are proportional to the number of moles. Thus, a volume ratio of 7:4 directly translates to the mole ratio of the reactants/products in the balanced equation. This conversion is key to solving the exercise because it simplifies the reasoning to focus solely on the stoichiometric coefficients without additional volume-to-mole calculations.

Molecular Formula Determination

Determining the molecular formula of a compound involves defining the actual number of atoms of each element in one molecule of that compound. For hydrocarbons like alkanes, the determination relies heavily on understanding their general formula along with interpretating combustion reaction data correctly.

As seen in the exercise, the balance between reactants' and products' volumes unveils the alkane's molecular formula. With Avogadro's law and the combustion ratio given, we deduce the molecular formula by matching the whole number that represents n—the number of carbon atoms in one molecule of the alkane. Incorrect values of n, producing non-whole numbers, do not align with an actual alkane structure. Therefore, the molecular formula determination in our exercise has pointed us toward \(C_2H_6\) as the correct answer, tying together all aspects of stoichiometry and Avogadro's law to elucidate the alkane's identity.