Question

A gaseous hydrocarbon reacts completely with oxygen gas to form carbon dioxide and water vapor. Given the following data, determine \(\Delta H_{f}^{\circ}\) for the hydrocarbon: $$ \begin{aligned} \Delta H_{\mathrm{reacion}}^{\circ} &=-2044.5 \mathrm{kJ} / \mathrm{mol} \text { hydrocarbon } \\ \Delta H_{\mathrm{f}}^{\circ}\left(\mathrm{CO}_{2}\right) &=-393.5 \mathrm{kJ} / \mathrm{mol} \\ \Delta H_{\mathrm{f}}^{\circ}\left(\mathrm{H}_{2} \mathrm{O}\right) &=-242 \mathrm{kJ} / \mathrm{mol} \end{aligned} $$ Density of \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) product mixture at 1.00 \(\mathrm{atm}\) , \(200 . \mathrm{C}=0.751 \mathrm{g} / \mathrm{L}\) . The density of the hydrocarbon is less than the density of Kr at the same conditions.

Short answer

The standard enthalpy of formation for the gaseous hydrocarbon in terms of the number of carbon atoms (x) and hydrogen atoms (y) can be expressed as: \[\Delta H_{f}^{\circ} (C_xH_y) = -393.5x - 121y + 2044.5\] To determine the precise value, more information about the hydrocarbon's composition is needed.

Step-by-step solution

Write the balanced chemical equation for the reaction.

Let the hydrocarbon be represented by \(C_xH_y\). The reaction is as follows: \[C_xH_y (g) + \left( x + \frac{y}{2} \right) O_2 (g) \rightarrow x CO_2 (g) + \frac{y}{2} H_2O(g)\]

Express the standard enthalpy change of the reaction in terms of the standard enthalpies of formation.

We know from Hess's law that: \[\Delta H_{reaction}^{\circ} = \sum n_i \Delta H_{f,i}^{\circ} (products) - \sum n_i \Delta H_{f,i}^{\circ} (reactants)\] From the balanced equation, substitute the values for the enthalpies of formation and the stoichiometric coefficients: \begin{align*} -2044.5\,\text{kJ/mol} &= x \left( -393.5 \,\text{kJ/mol} \right) + \frac{y}{2} \left( -242 \,\text{kJ/mol} \right) - \left( x + \frac{y}{2} \right) \Delta H_{f}^{\circ} (O_2) - \Delta H_{f}^{\circ} (C_xH_y) \\ -2044.5 &= -393.5x - 121y - \Delta H_{f}^{\circ} (C_xH_y) \end{align*} Since \(\Delta H_{f}^{\circ} (O_2) = 0\,\text{kJ/mol}\) for standard state oxygen gas.

Solve for the enthalpy of formation of the hydrocarbon.

We need to find the values of x and y for the hydrocarbon based on the given density information. However, without that information, we can still express the answer in terms of x and y as follows: \[\Delta H_{f}^{\circ} (C_xH_y) = -393.5x - 121y + 2044.5\] The standard enthalpy of formation for the gaseous hydrocarbon is expressed in terms of the number of carbon atoms (x) and hydrogen atoms (y). To find the precise value, further information about the hydrocarbon's composition is required.

Key concepts

Hess's Law

Hess's Law is a foundational principle in chemistry that helps us understand how energy changes during a chemical reaction. It states that the total enthalpy change for a chemical reaction is the same, regardless of the number of steps or pathway taken. This is incredibly useful because it allows chemists to calculate the enthalpy change of a reaction even if it cannot be measured directly.

Enthalpy is a measure of the total energy of a system, including internal energy plus energy associated with the pressure and volume. Using Hess's Law, one can add the enthalpy changes of individual steps in a reaction to find the total enthalpy change. This involves the use of standard enthalpies of formation, which are the changes in enthalpy when one mole of a compound is formed from its elements at their standard states. By summing the enthalpies of the products and subtracting the enthalpies of the reactants, you can determine the overall enthalpy change.

Hydrocarbon Combustion

Hydrocarbon combustion is a type of exothermic reaction where hydrocarbons, typically organic compounds consisting of hydrogen and carbon, react with oxygen. This process often results in the formation of carbon dioxide ( CO_2 ) and water ( H_2O ). Combustion reactions are important both for their energy production (as seen in fuel usage) and for their role in chemical thermodynamics studies.

In a typical combustion reaction, a hydrocarbon molecule reacts completely with oxygen. The reaction releases energy in the form of heat, causing the temperature of the surroundings to increase. This released energy is typically measured as the heat of combustion. The combustion of hydrocarbons is fundamental in energy production and the calculation of energy yields from chemical fuels using their enthalpy changes. It's a critical concept for understanding broader subjects like environmental science, where combustion efficiency and emissions are relevant.

Standard Enthalpy Change

The standard enthalpy change of a reaction is a crucial concept in thermodynamics. It is the heat change associated with a chemical reaction carried out under standard conditions—usually fixed at 1 atmosphere of pressure and a specified temperature, often 25°C (298 K).

This thermodynamic function gives us insight into the energy dynamics of reactions. The standard enthalpy change is denoted by the symbol \(\Delta H^{\circ}_{reaction}\) and represents the difference between the sum of the enthalpies of products and reactants. To find this, you calculate:

• The total enthalpy of formation for the products—all terms multiplied by their stoichiometric coefficients.
• The total enthalpy of formation for the reactants, also adjusted by their coefficients.
• The enthalpy change, \(\Delta H^{\circ}\), is then obtained by subtracting the reactants' total from the products' total.

Standard enthalpy changes are used to estimate the energy changes in many chemical processes and help predict whether a reaction is endothermic or exothermic.

Chemical Reaction Stoichiometry

Chemical reaction stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It determines how much of each reactant is needed to fully convert into products, which is crucial for calculating enthalpy changes and other properties.

A balanced chemical equation is essential in stoichiometry as it provides the necessary ratios. In the context of hydrocarbon combustion, stoichiometry helps us determine how much oxygen is required to combust a given amount of hydrocarbon and how much carbon dioxide and water vapor is produced.

• Each element's number of atoms is balanced across reactants and products.
• Given stoichiometric ratios, one can deduce molar relationships, which are vital in converting known masses or volumes of reactants to moles and further calculations.
• In enthalpy calculations, stoichiometry ensures accurate tallying of enthalpies, as coefficients reflect actual reaction participation levels.

Understanding stoichiometry is fundamental to predicting yields in reactions, optimizing reactant usage, and foreseeing the amounts of byproducts formed.