Alkanes combine with oxygen to produce carbon dioxide and water according to the following scheme: $$ \mathrm{C}_{n} \mathrm{H}_{2 n+2}+(3 n+1) / 2 \mathrm{O}_{2} \rightarrow(n+1) \mathrm{H}_{2} \mathrm{O}+n \mathrm{CO}_{2} $$ This process is generally referred to as combustion. An important use of this reaction is the quantitative determination of elemental composition (elemental analysis). Typically, a small sample of the compound is completely burned, and the water and carbon dioxide produced are collected and weighed. From the weight of water, the amount of hydrogen in the original compound can be determined. Similarly, the amount of carbon dioxide formed allows us to determine the amount of carbon in the original compound. Oxygen, if present, is usually determined by difference. The determination of the relative molar proportions of carbon and hydrogen in a compound is the first step in deriving its molecular formula. If combustion of \(5.00 \mathrm{mg}\) of a hydrocarbon gives \(16.90\) \(\mathrm{mg}\) of carbon dioxide and \(3.46 \mathrm{mg}\) of water, what are the weight percents of carbon and hydrogen in the sample?

Step by step solution

01

Write the combustion reaction

Consider the general combustion reaction for an alkane: \[- \mathrm{C}_{n} \mathrm{H}_{2 n+2} + \frac{\left(3 n+1\right)}{2} \mathrm{O}_{2} \rightarrow (n+1)\mathrm{H}_{2}\mathrm{O} + n\mathrm{CO}_{2}\]

02

Determine the moles of carbon dioxide produced

Firstly, use the molar mass of carbon dioxide (\(\mathrm{CO}_{2}\)): \(\mathrm{CO}_{2} = 12 + 2(16) = 44 \, \mathrm{g/mol} \) Calculate the moles of \(\mathrm{CO}_{2}\) produced: \[\text{Moles of } \mathrm{CO}_{2} = \frac{16.90 \, \mathrm{mg}}{44 \, \mathrm{g/mol}} = \frac{16.90 \times 10^{-3}}{44} = 0.384 \times 10^{-3} \, \mathrm{mol} \]

03

Determine the moles of water produced

Next, find the molar mass of water (\(\mathrm{H}_{2}\mathrm{O}\)): \( \mathrm{H}_{2}\mathrm{O} = 2(1) + 16 = 18 \, \mathrm{g/mol} \) Calculate the moles of \(\mathrm{H}_{2}\mathrm{O}\) produced: \[ \text{Moles of } \mathrm{H}_{2}\mathrm{O} = \frac{3.46 \, \mathrm{mg}}{18 \, \mathrm{g/mol}} = \frac{3.46 \times 10^{-3}}{18} = 0.192 \times 10^{-3} \, \mathrm{mol} \]

04

Calculate the grams of carbon in the original sample

Use the fact that each mole of \(\mathrm{CO}_{2}\) contains one mole of carbon. Thus, the mass of carbon in \(\mathrm{CO}_{2}\) is: \[\text{Mass of C } = 0.384 \times 10^{-3} \, \mathrm{mol} \times 12 \, \mathrm{g/mol} = 4.61 \times 10^{-3} \, \mathrm{g} = 4.61 \, \mathrm{mg} \]

05

Calculate the grams of hydrogen in the original sample

Each mole of \(\mathrm{H}_{2}\mathrm{O}\) contains 2 moles of hydrogen atoms. Thus, the mass of hydrogen in \(\mathrm{H}_{2}\mathrm{O}\) is: \[ \text{Mass of H} = 0.192 \times 10^{-3} \, \mathrm{mol} \times 2 \times 1 \, \mathrm{g/mol} = 0.384 \times 10^{-3} \, \mathrm{g} = 0.384 \, \mathrm{mg} \]

06

Calculate the weight percent of carbon

Divide the mass of carbon by the total initial mass and multiply by 100 for the weight percent of carbon: \[ \text{Weight percent of C} = \frac{4.61 \%2E \mathrm{mg}}{5.00 \%2E \mathrm{mg}} \times 100 = 92.20\% \]

07

Calculate the weight percent of hydrogen

Divide the mass of hydrogen by the total initial mass and multiply by 100 for the weight percent of hydrogen: \[ \text{Weight percent of H} = \frac{0.384 \%2E \mathrm{mg}}{5.00 \%2E \mathrm{mg}} \times 100 = 7.68\% \]

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Key Concepts

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

Alkanes

Alkanes are a class of hydrocarbons consisting only of single bonds between carbon atoms. They are the simplest type of hydrocarbon and are fully saturated, meaning they contain the maximum number of hydrogen atoms possible for the carbon chain.

• General formula: \(\text{C}_n\text{H}_{2n+2}\)
• Known as paraffins
• Alkanes undergo combustion reactions

In a combustion reaction, alkanes react with oxygen, producing carbon dioxide, water, and releasing energy in the form of heat.

Elemental Analysis

Elemental analysis involves determining the composition of a chemical compound by analyzing the amounts of its constituent elements. This method is critical in understanding the basic structure and formula of organic compounds.
Typically, the compound is completely burned, and the amounts of generated carbon dioxide and water are measured.

• This helps in determining the amounts of carbon and hydrogen in the sample
• Oxygen, if present, is often calculated by difference

Elemental analysis is crucial for deriving the molecular formula of unknown compounds.

Molecular Formula

The molecular formula provides the exact number of each type of atom in a molecule. It is essential for understanding the compound's structure and properties. For example, the molecular formula of an alkane helps identify the total number of carbon and hydrogen atoms.
To determine the molecular formula from combustion data:

• Calculate the amount of carbon and hydrogen from the produced \(\text{CO}_2\) and \(\text{H}_2\text{O}\)
• Use the ratios to establish the empirical formula
• Compare to the molecular formula if molar mass data is available

This approach helps chemists accurately describe the compound's composition.

Quantitative Determination

Quantitative determination in chemistry involves measuring and calculating the precise amounts of substances produced or consumed in a reaction. This is often achieved through gravimetric and volumetric analysis.
In the context of combustion analysis, quantitative determination involves:

• Measuring masses of produced \(\text{CO}_2\) and \(\text{H}_2\text{O}\)
• Calculating the moles of carbon and hydrogen
• Determining the weight percent of elements in the original sample

These quantitative measurements are key to identifying molecular compositions accurately.

Combustion Reaction

A combustion reaction is a chemical process in which a substance reacts rapidly with oxygen, releasing energy in the form of heat and light. For alkanes, the general reaction is:
\(\text{C}_n\text{H}_{2n+2} + \frac{(3n+1)}{2} \text{O}_2 \rightarrow (n+1)\text{H}_2\text{O} + n\text{CO}_2\)
Key features of combustion reactions:

• Exothermic process
• Produces carbon dioxide and water directly
• Used extensively in elemental analysis

Understanding combustion reactions helps in practical applications like energy production and pollution control.