Question

Consider the twodiagramsbelow. (a) Based on \((t)\), write an equation showing how \(\Delta H_{\mathrm{A}}\) is related to \(\Delta H_{\mathrm{B}}\) and \(\Delta H_{\mathrm{C}}\). How do both diagram (i) and your equation relate to the fact that enthalpy is a state function? (b) Based on (ii), write an equation relating \(\Delta H_{Z}\) to the other enthalpy changes in the diagram. (c) How do these diagrams relate to Hess's law? [Section 5.6]

Short answer

(a) Based on diagram (i), the relationship between \(\Delta H_{\mathrm{A}}\), \(\Delta H_{\mathrm{B}}\), and \(\Delta H_{\mathrm{C}}\) can be written as: \(\Delta H_{\mathrm{A}} = \Delta H_{\mathrm{B}} + \Delta H_{\mathrm{C}}\). This equation, along with the diagram, shows that enthalpy is a state function, as it only depends on the initial and final states and not the path taken. (b) Based on diagram (ii), the relationship between \(\Delta H_{Z}\) and other enthalpy changes is: \(\Delta H_{Z} = \Delta H_{W} + \Delta H_{X} + \Delta H_{Y}\). (c) These diagrams relate to Hess's law as they illustrate the concept that the overall enthalpy change is equal to the sum of the enthalpy changes of individual steps, regardless of the path taken.

Step-by-step solution

a) Relationship between \(\Delta H_{\mathrm{A}}\), \(\Delta H_{\mathrm{B}}\), and \(\Delta H_{\mathrm{C}}\)

Based on diagram (i), we can see that the process A can be decomposed into process B and process C. Since enthalpy is a state function, it is only dependent on the initial and final states and not on the pathway taken. Thus, the sum of the enthalpy changes for processes B and C should be equal to the enthalpy change for process A. Mathematically, this can be written as: \[ \Delta H_{\mathrm{A}} = \Delta H_{\mathrm{B}} + \Delta H_{\mathrm{C}} \]

a) Enthalpy as a state function

Enthalpy is a state function which means that it only depends on the initial and final states of a system and not on the specific path taken. In diagram (i), process A directly goes from the initial state to the final state. Processes B and C, on the other hand, have an intermediate state. Since enthalpy is a state function, we can see that \(\Delta H_{\mathrm{A}}\) is equal to the sum of \(\Delta H_{\mathrm{B}}\) and \(\Delta H_{\mathrm{C}}\), regardless of the path taken (A or B+C).

b) Relationship between \(\Delta H_{Z}\) and other enthalpy changes

Based on diagram (ii), we can write an equation relating the enthalpy changes of processes W, X, Y, and Z. We can see that process Z can be decomposed into processes W, X, and Y. Again, since enthalpy is a state function, the sum of the enthalpy changes for processes W, X, and Y should be equal to the enthalpy change for process Z. Mathematically: \[ \Delta H_{Z} = \Delta H_{W} + \Delta H_{X} + \Delta H_{Y} \]

c) Relating the diagrams to Hess's law

Hess's law states that the overall enthalpy change of a reaction is equal to the sum of the enthalpy changes of the individual steps in the reaction, regardless of the number of steps or the specific path taken. Both diagrams in the exercise illustrate the concept of Hess's law. In Diagram (i), the overall enthalpy change for process A can be determined by summing the enthalpy changes of processes B and C. Similarly, in Diagram (ii), the overall enthalpy change for process Z can be determined by summing the enthalpy changes of processes W, X, and Y. In both cases, the total enthalpy change is independent of the specific path taken, illustrating the concept of Hess's law.

Key concepts

Enthalpy

Enthalpy is a fundamental concept in thermochemistry that deals with the heat content of a chemical system. It is denoted by the symbol \( H \). Enthalpy includes all the internal energy of the system plus the product of the pressure and volume. This makes it a convenient measure for reactions occurring at constant pressure. In chemical reactions, the change in enthalpy \( \Delta H \) is of particular interest.Enthalpy changes are essential because they tell you whether a reaction absorbs or releases energy:

• When \( \Delta H \) is positive, the reaction is endothermic, indicating that it absorbs heat from the surroundings.
• When \( \Delta H \) is negative, the reaction is exothermic, meaning it releases heat to the surroundings.

Understanding enthalpy changes helps predict reaction behavior and is crucial for engineering applications, such as designing energy-efficient processes or evaluating the energy changes in different pathways of chemical reactions.

State Functions

State functions are properties of a system that depend solely on its current state, not on the path it took to reach that state. This means that no matter how you achieve a change in a state function, whether through one step or multiple ones, the total change remains the same.Common state functions include:

• Enthalpy \( H \)
• Energy \( U \)
• Pressure \( P \)
• Volume \( V \)
• Temperature \( T \)

For example, in our context of enthalpy, the state function nature of \( H \) allows you to say that \( \Delta H_A = \Delta H_B + \Delta H_C \) regardless of the intermediates or details of the process. This simplifies complex analyses, as it means thermodynamic calculations can often be made using only initial and final states, without worrying about the specifics of the process itself.

Thermochemistry

Thermochemistry is the branch of chemistry that studies the energy and heat associated with chemical reactions and physical transformations. A major focus is on the transfer and conversion of energy, particularly how energy moves between the system and its surroundings. Key concepts in thermochemistry include:

• Exothermic and endothermic processes: These describe whether energy is released or absorbed.
• Heat capacity and specific heat: These indicate how much energy is required to change the temperature of a substance.
• Standard enthalpy changes: Such as the standard enthalpy of formation, which is the change in enthalpy when one mole of a compound is formed from its elements in their standard states.

Understanding thermochemistry is crucial for managing energy in industrial processes, predicting the feasibility of reactions, and controlling temperature changes in reaction environments. By assessing the enthalpy changes using Hess's law, we gather insights into the energy profile of chemical systems, holding significant practical implications in diverse fields such as environmental science, materials science, and engineering.