4.2 Enthalpy of reactions and Hess's Law

Enthalpy and Hess's Law are key concepts in thermochemistry, helping us understand heat flow in chemical reactions. They're essential for predicting reaction spontaneity and calculating energy changes, connecting the microscopic world of molecules to macroscopic heat effects we can measure.

These tools let us figure out tricky reaction energies by breaking them into simpler steps. This ties into the broader theme of energy conservation and transformation in chemical processes, a cornerstone of thermodynamics and real-world applications in energy science.

Enthalpy in Thermochemistry

Definition and Significance of Enthalpy

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• Enthalpy is a thermodynamic state function represented by the symbol H, which is the sum of the internal energy of a system plus the product of its pressure and volume ($H = U + PV$)
• Enthalpy measures the total heat content of a system at constant pressure
• Changes in enthalpy ($ΔH$) represent the heat absorbed or released by a system during a chemical reaction or physical process
• The enthalpy change of a reaction helps determine the heat exchanged between a system and its surroundings, as well as the spontaneity and direction of chemical reactions

Types of Enthalpy Changes

• Exothermic reactions have a negative enthalpy change ($ΔH < 0$) indicating heat is released from the system to the surroundings (combustion of methane)
• Endothermic reactions have a positive enthalpy change ($ΔH > 0$) indicating heat is absorbed by the system from the surroundings (photosynthesis)
• Standard enthalpy of formation ($ΔH°f$) is the enthalpy change that occurs when one mole of a compound is formed from its constituent elements in their standard states at a specified temperature (usually 298 K) and 1 atm pressure
• By convention, the standard enthalpy of formation for any element in its most stable form at 1 atm and the specified temperature is zero (graphite for carbon, diatomic molecules for gases like $H_2$ and $O_2$)

Calculating Enthalpy Changes

Using Standard Enthalpy of Formation

• The standard enthalpy of reaction ($ΔH°rxn$) can be calculated using the standard enthalpies of formation of the reactants and products according to the equation: $ΔH°rxn = Σ(n × ΔH°f (products)) - Σ(n × ΔH°f (reactants))$, where n is the stoichiometric coefficient of each species
• To calculate the enthalpy change for a reaction, multiply the standard enthalpy of formation of each product by its stoichiometric coefficient and sum these values, then subtract the sum of the standard enthalpies of formation of the reactants multiplied by their stoichiometric coefficients
• Example: For the reaction $CH_4(g) + 2O_2(g) → CO_2(g) + 2H_2O(l)$, $ΔH°rxn = [ΔH°f(CO_2) + 2ΔH°f(H_2O)] - [ΔH°f(CH_4) + 2ΔH°f(O_2)]$

Using Hess's Law

• Hess's Law states that the total enthalpy change for a chemical reaction is independent of the pathway or the number of steps taken to reach the final products, as long as the initial and final states are the same
• Hess's Law is a consequence of the conservation of energy and the state function nature of enthalpy, which means that the enthalpy change of a reaction depends only on the initial and final states of the system, not on the path taken between these states
• To apply Hess's Law, combine the known enthalpy changes of related reactions to determine the enthalpy change of the target reaction by reversing reactions, multiplying reactions by a factor, and/or adding reactions together
• When reversing a reaction, the sign of the enthalpy change is also reversed; when multiplying a reaction by a factor, the enthalpy change is multiplied by the same factor; when adding reactions together, the enthalpy changes of the individual reactions are added together to give the overall enthalpy change

Hess's Law for Enthalpy

Applying Hess's Law

• Identify the target reaction for which the enthalpy change is unknown
• Find a series of known reactions that can be combined to yield the target reaction
• Manipulate the known reactions by reversing, multiplying, or adding them together to obtain the target reaction
• Calculate the enthalpy change of the target reaction by combining the enthalpy changes of the manipulated reactions according to the rules of Hess's Law

Example of Hess's Law Application

• Target reaction: $C(s) + 2H_2(g) → CH_4(g)$
• Known reactions:
  1. $C(s) + O_2(g) → CO_2(g)$, $ΔH_1 = -393.5 kJ/mol$
  2. $CH_4(g) + 2O_2(g) → CO_2(g) + 2H_2O(l)$, $ΔH_2 = -890.4 kJ/mol$
  3. $H_2(g) + \frac{1}{2}O_2(g) → H_2O(l)$, $ΔH_3 = -285.8 kJ/mol$
• Manipulate the known reactions:
  • Reverse reaction 1: $CO_2(g) → C(s) + O_2(g)$, $ΔH_1' = +393.5 kJ/mol$
  • Keep reaction 2 as is
  • Multiply reaction 3 by 2: $2H_2(g) + O_2(g) → 2H_2O(l)$, $ΔH_3' = -571.6 kJ/mol$
• Add the manipulated reactions: $CO_2(g) + 2H_2(g) + O_2(g) → C(s) + CH_4(g) + 2H_2O(l)$, $ΔH = ΔH_1' + ΔH_2 + ΔH_3' = -1068.5 kJ/mol$
• The enthalpy change for the target reaction is $-1068.5 kJ/mol$, indicating an exothermic process

Enthalpy Diagrams

Constructing Enthalpy Diagrams

• An enthalpy diagram is a graphical representation of the enthalpy changes that occur during a series of reactions, with the enthalpy plotted on the vertical axis and the reaction progress on the horizontal axis
• Reactants are typically placed at the top, and products are placed at the bottom; the vertical distance between the reactants and products represents the overall enthalpy change of the reaction ($ΔH$)
• Intermediate species or transition states may be included in the diagram, with their enthalpies plotted relative to the reactants and products; the enthalpy differences between these species represent the enthalpy changes for individual reaction steps
• Exothermic reactions are represented by a downward step in the diagram, while endothermic reactions are represented by an upward step

Using Enthalpy Diagrams to Visualize Hess's Law

• Enthalpy diagrams can be used to visualize Hess's Law, as the overall enthalpy change of a reaction is independent of the pathway taken
• The sum of the individual steps in the diagram will equal the total enthalpy change
• Example: The enthalpy diagram for the formation of methane ($CH_4$) from its elements can be constructed using the known reactions from the previous Hess's Law example, demonstrating that the overall enthalpy change is the same regardless of the pathway taken

Bond Energies and Enthalpy Changes

Relationship Between Bond Energies and Enthalpy Changes

• Bond energy is the amount of energy required to break a specific chemical bond in one mole of a substance in the gas phase, or conversely, the amount of energy released when a bond is formed
• The enthalpy change of a reaction can be estimated using the bond energies of the reactants and products, as the overall enthalpy change is related to the difference between the energy required to break bonds in the reactants and the energy released when new bonds form in the products
• To estimate the enthalpy change using bond energies, sum the bond energies of all the bonds broken in the reactants and subtract the sum of the bond energies of all the bonds formed in the products: $ΔH = Σ(bond energies of bonds broken) - Σ(bond energies of bonds formed)$

Limitations of the Bond Energy Approach

• The bond energy method provides an approximation of the enthalpy change, as it assumes that the bond energies are independent of the molecular environment and that the energy required to break a bond is equal to the energy released when the same bond is formed
• In reality, bond energies can vary slightly depending on the specific molecule and its structure
• Comparing the calculated enthalpy change using bond energies with the experimental value can provide insights into the limitations of the bond energy approach and the factors that influence the actual enthalpy change of a reaction
• Example: The enthalpy change for the combustion of methane ($CH_4 + 2O_2 → CO_2 + 2H_2O$) calculated using bond energies may differ from the experimental value due to the assumptions made in the bond energy approach