Question

Challenge Two enthalpy of formation equations, a and b, combine to form the equation for the reaction of nitrogen oxide and oxygen. The product of the reaction is nitrogen dioxide: \(\mathrm{NO}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{g}) \Delta H_{\mathrm{rxn}}^{\circ}=-58.1 \mathrm{kJ}\) a. \(\frac{1}{2} \mathrm{N}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g}) \Delta H_{\mathrm{f}}^{\circ}=91.3 \mathrm{kJ}\) b. \(\frac{1}{2} \mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{g}) \Delta H_{\mathrm{f}}^{\circ}=?\) What is \(\Delta H_{\mathrm{f}}^{\circ}\) for Equation \(\mathrm{b}\) ?

Short answer

The enthalpy of formation \(\Delta H_{\mathrm{f}}^{\circ}\) for Equation b is 33.2 kJ.

Step-by-step solution

Analyze Equation a

Given Equation a: \(\frac{1}{2} \mathrm{N}_{2}(\mathrm{g})+\frac{1}{2}\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}(\mathrm{g}) \Delta H_{\mathrm{f}}^{\circ}=91.3 \mathrm{kJ}\) This equation represents the formation of 1 mole of NO(g) from its elements.

Analyze the overall reaction equation

Given overall reaction equation: \(\mathrm{NO}(\mathrm{g})+\frac{1}{2}\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{g}) \Delta H_{\mathrm{rxn}}^{\circ}=-58.1 \mathrm{kJ}\) This equation shows the formation of 1 mole of NO₂(g) from nitrogen oxide and oxygen.

Combine equations a and b to form the overall reaction equation

To combine Equation a with Equation b to get the overall reaction equation, we need to reverse Equation a (so that we have NO on the reactant side) and add it to Equation b. When reversing an equation, we change the sign of the enthalpy value. Reversed Equation a: \(\mathrm{NO}(\mathrm{g}) \rightarrow \frac{1}{2}\mathrm{N}_{2}(\mathrm{g})+ \frac{1}{2}\mathrm{O}_{2}(\mathrm{g}) \Delta H_{\mathrm{f}}^{\circ}=-91.3 \mathrm{kJ}\)

Add reversed Equation a to the overall reaction equation

Now, we'll add the reversed Equation a to the overall reaction equation: \(\begin{array}{c} \mathrm{NO}(\mathrm{g}) \rightarrow \frac{1}{2}\mathrm{N}_{2}(\mathrm{g})+ \frac{1}{2}\mathrm{O}_{2}(\mathrm{g}) \\ + \\ \mathrm{NO}(\mathrm{g})+\frac{1}{2}\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{g}) \end{array}\) Resulting in: \(\begin{array}{c} \mathrm{NO}(\mathrm{g})+\frac{1}{2}\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{NO}_{2}(\mathrm{g}) \end{array}\) Since the resulting equation matches the overall reaction equation, we can now add the enthalpy values.

Add the enthalpy values

Adding the enthalpy values: \(-91.3\,\mathrm{kJ} + \Delta H_{\mathrm{f}}^{\circ} = -58.1\,\mathrm{kJ}\) We can solve for \(\Delta H_{\mathrm{f}}^{\circ}\) for Equation b: \(\Delta H_{\mathrm{f}}^{\circ} = -58.1\,\mathrm{kJ} + 91.3\,\mathrm{kJ} = 33.2\,\mathrm{kJ}\) Therefore, the enthalpy of formation \(\Delta H_{\mathrm{f}}^{\circ}\) for Equation b is 33.2 kJ.

Key concepts

Understanding Chemical Reactions

Chemical reactions are the processes by which substances, known as reactants, transform into different substances, called products. Every chemical reaction involves a change in the structure of molecules and is governed by the laws of conservation of mass and energy. In the realm of thermochemistry, the study of heat involved in chemical reactions, we often measure the heat change (enthalpy change, \( \Delta H \)) that occurs during these reactions.

To analyze a chemical reaction, like the formation of nitrogen dioxide (\( \mathrm{NO}_2(\mathrm{g}) \) from nitrogen oxide (\( \mathrm{NO}(\mathrm{g}) \) and oxygen (\( \mathrm{O}_2(\mathrm{g}) \) in our exercise, we must carefully examine the balanced chemical equation. This displays the stoichiometry of the reaction—the proportional relationship between the reactants and products based on the balanced equation. Understanding this stoichiometry is critical for accurately calculating the enthalpies of formation which, in essence, provides insight into the energy dynamics of the chemical process involved.

The Role of Thermochemistry in Understanding Reactions

Thermochemistry is a branch of chemistry that deals with the energy changes during chemical reactions. One key concept within thermochemistry is the enthalpy of formation (\( \Delta H_{\mathrm{f}}^\circ \)), which measures the heat change when one mole of a substance is formed from its elements in their standard states. It is a crucial parameter that allows chemists to predict whether a reaction will release or absorb heat.

In our exercise, we determine the enthalpy of formation for a reaction to find out if it is exothermic (releases heat, indicated by a negative \( \Delta H \) value) or endothermic (absorbs heat, indicated by a positive \( \Delta H \) value). The reaction involving the formation of \( \mathrm{NO}_2(\mathrm{g}) \) is shown to have a negative value, which suggests that the reaction is exothermic. By combining the known enthalpies of related reactions, we can calculate unknown variables, like the enthalpy of formation for \( \mathrm{NO}_2(\mathrm{g}) \) in our case.

Stoichiometry: The Quantitative Study of Reactants and Products

Stoichiometry is an area of chemistry that quantitatively studies the relationships between the amounts of reactants consumed and products formed in a chemical reaction. It is based on the balanced chemical equation and the conservation of mass. Precise stoichiometric calculations require balanced equations where the number of atoms of each element is the same on both sides of the reaction.

To solve for unknowns in chemical reactions, like the \( \Delta H_{\mathrm{f}}^\circ \) in the given exercise, stoichiometry tells us how to manipulate and combine equations to align with the reactants and products in question. For example, reversing Equation a turns a product back into reactants, which can then be combined with other equations to achieve the overall reaction. We applied these principles to deduce the enthalpy of formation for \( \mathrm{NO}_2(\mathrm{g}) \), thereby connecting the concepts of stoichiometry and thermochemistry to solve practical problems in chemistry.