Question

A sample of a hydrocarbon is combusted completely in \(\mathrm{O}_{2}(g)\) to produce \(21.83 \mathrm{~g} \mathrm{CO}_{2}(g), 4.47 \mathrm{~g} \mathrm{H}_{2} \mathrm{O}(g)\), and \(311 \mathrm{~kJ}\) of heat. (a) What is the mass of the hydrocarbon sample that was combusted? (b) What is the empirical formula of the hydrocarbon? (c) Calculate the value of \(\Delta H_{f}^{\circ}\) per empirical-formula unit of the hydrocarbon. (d) Do you think that the hydrocarbon is one of those listed in Appendix C? Explain your answer.

Short answer

The mass of the hydrocarbon sample is 6.4818 g. Its empirical formula is CH. The value of \(\Delta H_{f}^{\circ}\) per empirical-formula unit is 623.9 kJ/mol. It cannot be definitively confirmed or denied if the given hydrocarbon matches with any of those listed in Appendix C based only on this information.

Step-by-step solution

Find the mass of Carbon and Hydrogen in the products

From the combustion reaction, we know that all Carbon in the hydrocarbon will be converted to CO2 and all Hydrogen will be converted to H2O. In 1 mole of CO2, there is 1 mole of Carbon. Mass of Carbon in the CO2 produced: \(21.83 g \frac{12.01 g \ C }{44.01 g \ CO_2} = 5.9822 \ g \ C\) In 1 mole of H2O, there are 2 moles of Hydrogen. Mass of Hydrogen in the H2O produced: \(\frac{4.47 g}{18.02 g\cdot mol^{-1}} \cdot 2(1.008 g\cdot mol^{-1}) = 0.4996 \ g \ H\)

Calculate the mass of the hydrocarbon

To find the mass of the hydrocarbon, we add the mass of Carbon and Hydrogen. Mass of the hydrocarbon = Mass of Carbon + Mass of Hydrogen = 5.9822 g + 0.4996 g = 6.4818 g

Calculate the mole ratio of Carbon to Hydrogen

To find the empirical formula, we need to find the ratio of moles of Carbon to Hydrogen in the hydrocarbon. We can calculate this by dividing the mass of each element by their respective molar masses. Moles of Carbon = \(\frac{5.9822 g}{12.01 g\cdot mol^{-1}} = 0.4983 \ mol\) Moles of Hydrogen = \(\frac{0.4996 g}{1.008 g\cdot mol^{-1}} = 0.4959 \ mol\) Now, divide each value by the smallest value to get the mole ratio: \(\frac{0.4983}{0.4959} : \frac{0.4959}{0.4959}\) This gives us a mole ratio of approximately 1:1.

Determine the empirical formula

Using the mole ratio found in Step 3, the empirical formula of the hydrocarbon is CH.

Calculate the standard heat of formation per empirical-formula unit

To calculate the value of \(\Delta H_{f}^{\circ}\) per empirical-formula unit, we will use the heat produced and the moles of hydrocarbon combusted: \(\Delta H_f^\circ\) per CH unit = \(\frac{311 \mathrm{~kJ}}{0.4983 \ mol} = 623.9 \mathrm{~kJ/mol}\)

Determine if the hydrocarbon is one of those listed in Appendix C

We are given an empirical formula CH and a standard heat of formation of 623.9 kJ/mol. Appendix C will typically contain hydrocarbons like methane, ethene, ethane, etc. However, without further information, we cannot definitively confirm or deny if the given hydrocarbon matches with any of those listed in Appendix C. The empirical formula narrows down the options, but more information would be needed to make a clear determination.

Key concepts

Empirical Formula Calculation

Understanding the empirical formula of a compound is essential for comprehending its basic composition. It represents the simplest ratio of the elements within the compound. To determine the empirical formula, one typically starts by finding the mass of each element in a sample.

In the context of stoichiometry and particularly combustion reactions, this often involves a two-step process: first, deducing the amount of each element present in the combustion products, and second, converting these masses to moles to find the simplest whole-number mole ratio.

For example, when a hydrocarbon burns in oxygen, it produces carbon dioxide and water. From the resulting mass of CO₂ and H₂O, one can back-calculate the mass—and eventually the moles—of carbon and hydrogen in the original hydrocarbon. The key is to understand the stoichiometry of these reactions, where each mole of CO₂ contains one mole of carbon and each mole of H₂O contains two moles of hydrogen.

Key Steps in Empirical Formula Calculation

• Measure masses of the combustion products (e.g., CO₂, H₂O).
• Convert masses to moles based on the molar mass of each element in the products.
• Determine the simplest whole-number ratio between the elements' moles.

Converting these ratios to the empirical formula, as illustrated in the solution to our exercise, provides valuable information about the compound's composition, pivotal for many areas of chemistry.

Combustion Analysis

Combustion analysis is a method used to determine the elemental composition of a substance by burning it and analyzing the resulting products. In organic chemistry, this technique is particularly useful for identifying the amounts of carbon and hydrogen, and sometimes sulfur or nitrogen, in an organic compound.

The process involves complete combustion of the sample in a surplus of oxygen, ensuring all carbon is converted to CO₂ and all hydrogen to H₂O. These products are then collected and quantified, allowing us to calculate the amount of each element present in the original sample.

Utility of Combustion Analysis

• Identifies chemical composition of organic compounds.
• Can be applied to unknown substances to help in determining their structure.
• Enables chemists to calculate empirical and molecular formulas.

By way of the exercise provided, combustion analysis not only offers insights into the makeup of a hydrocarbon but can also lead to further calculations, such as the determination of the hydrocarbon's empirical formula.

Enthalpy of Formation

The enthalpy of formation, represented by \(\Delta H_f^\circ\), measures the energy change involved in the formation of one mole of a substance from its elements in their standard states. For combustion reactions, this concept is crucial as it allows us to appreciate the energetic implications of chemical reactions.

\[\Delta H_f^\circ\] values are determined under standard conditions: a pressure of 1 atm and, typically, a temperature of 25°C. These values can help predict the amount of energy released or absorbed during the formation of compounds.

Calculating \(\Delta H_f^\circ\) from Combustion

• Determine the heat of reaction (energy change) during combustion.
• Divide this energy by the number of moles of the compound combusted to find the enthalpy of formation per mole.

In our exercise, we analyzed the heat produced from the combustion of a known amount of a hydrocarbon to find its \(\Delta H_f^\circ\) per empirical formula unit. Such calculations are pivotal for understanding the energetic aspects of chemical reactions and material stability.