Question

For each generic reaction, determine the value of \(\Delta H _ { 2 }\) in terms of \(\Delta H _ { 1 } .\) $$ \begin{array} { l l } { \text { a. } A + \mathrm { B } \longrightarrow 2 \mathrm { C } } & { \Delta H _ { 1 } } \\ { } & { 2 \mathrm { C } \longrightarrow \mathrm { A } + \mathrm { B } } & { \Delta H _ { 2 } = ? } \end{array} $$ $$ \begin{array} { c l } { \text { b. } A + \frac { 1 } { 2 } B \longrightarrow C } & { \Delta H _ { 1 } } \\ { } & { 2 A + B \longrightarrow 2 C \quad \Delta H _ { 2 } = ? } \end{array} $$ $$ \begin{array} { l l } { \text { c. } A \longrightarrow B + 2 C } & { \Delta H _ { 1 } } \\ { \frac { 1 } { 2 } B + C \longrightarrow \frac { 1 } { 2 } A } & { \Delta H _ { 2 } = ? } \end{array} $$

Short answer

\(\Delta H _ { 2 }\) values are: (a) \(-\Delta H _ { 1 }\), (b) \(2\Delta H _ { 1 }\), (c) \(-\frac { 1 } { 2 }\Delta H _ { 1 }\).

Step-by-step solution

Analyzing Reaction a

The reaction given is a reversible reaction where the reverse reaction is the exact opposite of the forward reaction. According to Hess's Law, the enthalpy change of the reverse reaction is equal in magnitude but opposite in sign to the forward reaction.

Determining \(\Delta H _ { 2 }\) for Reaction a

For reaction a, since the reverse reaction \(2\text{C} \rightarrow \text{A} + \text{B}\) is exactly the opposite process of the forward reaction, \(\Delta H _ { 2 }\) is the negative of \(\Delta H _ { 1 }\), i.e., \(\Delta H _ { 2 } = -\Delta H _ { 1 }\).

Analyzing Reaction b

This reaction is not simply the reverse of the forward reaction. However, we can still use stoichiometry to relate \(\Delta H _ { 1 }\) to \(\Delta H _ { 2 }\). If we multiply the forward reaction by 2 and compare with the given reaction to find \(\Delta H _ { 2 }\), since enthalpies are extensive properties.

Determining \(\Delta H _ { 2 }\) for Reaction b

By multiplying the forward reaction and its enthalpy change by 2, we get the reaction \(2\text{A} + \text{B} \rightarrow 2\text{C}\) with \(\Delta H \) of \(2\Delta H _ { 1 }\), which is the same reaction for which we want to find \(\Delta H _ { 2 }\). Therefore, \(\Delta H _ { 2 } = 2\Delta H _ { 1 }\).

Analyzing Reaction c

For reaction c, the given reaction for \(\Delta H _ { 2 }\) is neither exactly the forward nor reverse reaction. It can be derived from the forward reaction by considering stoichiometry and balancing.

Determining \(\Delta H _ { 2 }\) for Reaction c

We notice that \(\frac { 1 } { 2 }\text{B} + \text{C} \rightarrow \frac { 1 } { 2 }\text{A}\) is half of the forward reaction \(\text{A} \rightarrow \text{B} + 2\text{C}\). Hence \(\Delta H _ { 2 }\) is half of \(\Delta H _ { 1 }\), but because it is in the opposite direction, \(\Delta H _ { 2 }\) is negative: \(\Delta H _ { 2 } = -\frac { 1 } { 2 }\Delta H _ { 1 }\).

Key concepts

Understanding Hess's Law

Hess's Law is a fundamental concept in thermodynamics and chemistry, providing us with an essential tool for calculating enthalpy changes within chemical reactions. It states that the total enthalpy change for a chemical reaction is the same, regardless of the number of steps it takes to get from reactants to products.

This powerful law is based on the principle that enthalpy, as a state function, depends only on the initial and final states of a system and not on the path taken to get from one state to another. In practical terms, this means we can add or subtract known enthalpy changes of reactions to find the enthalpy change for a new reaction.

For example, if a reversible reaction is split into multiple steps and we know the enthalpy changes for each of these steps, Hess's Law enables us to determine the overall enthalpy change simply by summing these values. This is particularly useful when direct measurement of the enthalpy change for a reaction is not possible. Keep in mind, to apply Hess's Law effectively, all reactions must be correctly balanced, which ties into the principle of stoichiometry.

Reversible Reactions and Enthalpy

Reversible reactions are fascinating in the way they can move both forward and backward between their reactants and products. These reactions reach a state of dynamic equilibrium, where the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in concentration of reactants and products over time.

When analyzing reversible reactions, it's crucial to understand that the enthalpy change for the forward reaction has the same magnitude but an opposite sign to that of the reverse reaction. This concept aligns with Hess's Law, which implies that flipping a reaction's direction is like changing the sign of its enthalpy change. This relationship is neatly demonstrated in the solution for Reaction a, where \(\Delta H _ { 2}\) for the reverse reaction is simply \( -\Delta H _ { 1}\).

Understanding this aspect of reversible reactions allows us to control the conditions that will favor the formation of products or reactants and leverages our ability to calculate the unknown enthalpy changes in a given chemical process.

Stoichiometry: The Mathematics of Chemistry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships of the reactants and products in chemical reactions. It enables us to predict the proportions in which chemicals will react and to calculate the amounts of reactants needed or products formed.

Using stoichiometry, we can deduce the enthalpy change for a reaction if we know the enthalpy changes of related reactions. This is because stoichiometric coefficients, which represent the ratio of moles of each substance involved in the reaction, can be used to scale the enthalpy change for a balanced chemical equation. For instance, in Reaction b, doubling both the reactants and products effectively doubles the enthalpy change, leading to the insight that \(\Delta H _ { 2}\) equals \(2\Delta H _ { 1}\).

Stoichiometric calculations also play a critical role when dealing with limiting reagents, theoretical yields, and the efficiency of chemical reactions. Students must grasp these concepts to conduct accurate and meaningful enthalpy change calculations.

Extensive Properties: Enthalpy's Dependency on Amount

Enthalpy is an example of an extensive property, which means its value is dependent on the amount of substance present in the system. As opposed to intensive properties, which do not change with the quantity of substance, extensive properties are additive for internally non-reacting systems.

In the context of enthalpy change calculations, understanding the extensive nature of enthalpy is crucial. For example, if a chemical reaction is scaled up or down, the associated enthalpy change would also increase or decrease proportionally. The solution for Reaction c illustrates this concept, where halving the reaction effectively halves the enthalpy change. However, since the direction of the reaction is reversed, the sign of \(\Delta H _ { 2}\) becomes negative, yielding \(\Delta H _ { 2} = -\frac { 1 } { 2 }\Delta H _ { 1}\).

The practical implications of extensive properties like enthalpy mean that they are especially important when scaling up chemical reactions for industrial processes, where the quantity of heat absorbed or released can affect reaction conditions and product yields.