Question

Consider the generic reaction: $$\mathrm{A}+2 \mathrm{B} \longrightarrow \mathrm{C}+3 \mathrm{D} \quad \Delta H=155 \mathrm{kJ}$$ Determine the value of \(\Delta H\) for each related reaction: \begin{equation} \begin{array}{l}{\text { a. } 3 \mathrm{A}+6 \mathrm{B} \longrightarrow 3 \mathrm{C}+9 \mathrm{D}} \\ {\text { b. } \mathrm{C}+3 \mathrm{D} \rightarrow \mathrm{A}+2 \mathrm{B}} \\ {\text { c. } \frac{1}{2} \mathrm{C}+\frac{3}{2} \mathrm{D} \longrightarrow \frac{1}{2} \mathrm{A}+\mathrm{B}}\end{array} \end{equation}

Short answer

\(\Delta H\) for reaction a is 465 kJ, for reaction b is -155 kJ, and for reaction c is 77.5 kJ.

Step-by-step solution

Analyzing the Provided Reaction

Look at the stoichiometry in the provided chemical equation. For every mole of A reacting with 2 moles of B, one mole of C and 3 moles of D are produced with an enthalpy change, \( \Delta H \) of 155 kJ.

Adjust \( \Delta H \) for a scaled reaction

Reaction a is the provided reaction scaled up by a factor of 3. Since enthalpy is an extensive property, it scales with the amount of substance. Thus, multiply the given \( \Delta H \) by 3 to obtain the enthalpy change for reaction a.

Calculating \( \Delta H \) for Reaction a

Starting with \( 155 \text{kJ} \) for the original reaction, we find that \( \Delta H \) for reaction a is \( 155 \text{kJ} \times 3 = 465 \text{kJ} \) since it has exactly three times the amount of reactants and products.

Determine \( \Delta H \) for the Reverse Reaction

Reaction b is the reverse of the original reaction. The sign of \( \Delta H \) should be reversed because reversing a reaction changes the direction of heat transfer. Therefore, \( \Delta H \) for reaction b is \( -155 \text{kJ} \).

Adjust \( \Delta H \) for a fraction of the reaction

Reaction c involves one-half of the amounts of C and D forming half the amounts of A and B compared to the original reaction. Thus, the enthalpy change for reaction c will be half the enthalpy change of the original reaction.

Calculating \( \Delta H \) for Reaction c

Taking half of the original \( \Delta H \) gives us \( \frac{155 \text{kJ}}{2} = 77.5 \text{kJ} \) for reaction c.

Key concepts

Chemical Reaction Stoichiometry

The concept of stoichiometry is a fundamental aspect of chemistry that involves the quantitative relationship between reactants and products in a chemical reaction. When we examine a balanced chemical equation, the coefficients tell us the relative amounts of each substance involved.

For instance, in the generic reaction \( \mathrm{A} + 2 \mathrm{B} \longrightarrow \mathrm{C} + 3 \mathrm{D} \) with an enthalpy change (\( \Delta H = 155 \mathrm{kJ} \)), it signifies that 1 mole of A reacts with 2 moles of B to produce 1 mole of C and 3 moles of D, releasing 155 kJ of energy in the form of heat during the process.

In this scenario, if we scale the reaction, such as in option a, where the reactants and products are all multiplied by 3, the stoichiometric relationships also scale accordingly. This means that 3 moles of A react with 6 moles of B to produce 3 moles of C and 9 moles of D, tripling the enthalpy change to 465 kJ.

Enthalpy (\(\Delta H\)) of Reactions

Enthalpy change, denoted as \(\Delta H\), is a measure of the heat change that occurs during a chemical reaction at constant pressure. It is an essential concept when discussing endothermic and exothermic reactions. The sign of \(\Delta H\) indicates whether a reaction absorbs heat from the environment (endothermic, \(\Delta H > 0\)) or releases heat to the environment (exothermic, \(\Delta H < 0\)).

In the given exercise, the positive value of \(\Delta H = 155 \mathrm{kJ}\) indicates an exothermic reaction. However, when the reaction is reversed, as in option b, the enthalpy change sign is flipped to negative (\(-155 \mathrm{kJ}\)), suggesting the reverse reaction is endothermic. It's crucial to note that enthalpy is a state function, meaning it depends only on the initial and final states of the system, not the path taken to get there.

Extensive Property in Chemistry

An extensive property is a property that changes when the size of the sample changes. Enthalpy is an example of an extensive property because it is dependent on the amount of substance present. When we scale a chemical reaction, the enthalpy change scales proportionally with the quantities of reactants and products.

In the scenario of the exercise, for reaction c, where the amounts of reactants and products are halved, the enthalpy change is also halved, resulting in \(\Delta H = 77.5 \mathrm{kJ}\). This concept is crucial in stoichiometric calculations, as adjusting the amounts of substances directly affects the overall energy change in a reaction. It also explains why in reaction a, with three times the amounts, the \(\Delta H\) triples to 465 kJ, demonstrating the nature of enthalpy as an extensive property.