3.2 Internal energy and enthalpy

The First Law of Thermodynamics is all about energy conservation. It's like keeping track of your bank account, but with energy instead of money. This law helps us understand how energy moves and changes in different processes.

Internal energy and enthalpy are key players in this energy game. They help us figure out how much heat is gained or lost during chemical reactions and physical changes. Understanding these concepts is crucial for solving real-world problems in chemistry and engineering.

Internal Energy vs Enthalpy

Defining Internal Energy and Enthalpy

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• Internal energy is the total kinetic and potential energy of all particles within a system
  • Extensive property depends on the quantity of matter present (1 mole of water vs 2 moles of water)
• Enthalpy is a thermodynamic property defined as the sum of a system's internal energy and the product of its pressure and volume $H = U + PV$
  • Measure of the total heat content of a system at constant pressure
  • Commonly used in chemistry to describe heat changes in reactions and phase transitions

Comparing Internal Energy and Enthalpy

• Changes in internal energy are related to heat and work exchanged between the system and its surroundings [ΔU = q - w](https://www.fiveableKeyTerm:δu_=_q_-_w)
  • Heat (q) is the transfer of thermal energy between the system and surroundings
  • Work (w) is the energy transfer due to a force acting over a distance (e.g., expansion or compression of a gas)
• Changes in enthalpy are associated with heat exchange at constant pressure $ΔH = q_p$
  • At constant pressure, the change in enthalpy equals the heat exchanged with the surroundings
• Both internal energy and enthalpy are state functions
  • Their values depend only on the current state of the system, not on the path taken to reach that state
  • Allows for the application of Hess's law and the construction of thermochemical cycles

First Law of Thermodynamics Applications

Calculating Changes in Internal Energy

• The first law of thermodynamics states that the change in internal energy of a system (ΔU) is equal to the heat added to the system (q) minus the work done by the system (w): $ΔU = q - w$
  • Sign conventions: heat absorbed by the system (q > 0), heat released by the system (q < 0), work done by the system (w < 0), work done on the system (w > 0)
• For processes involving only pressure-volume work $w = -PΔV$
  • Pressure-volume work occurs when a system expands or contracts against an external pressure
  • Example: expansion of a gas in a piston-cylinder assembly

Calculating Changes in Enthalpy

• For processes occurring at constant pressure, the change in enthalpy (ΔH) is equal to the heat exchanged with the surroundings: $ΔH = q_p$
  • $q_p$ is the heat exchanged at constant pressure
• The change in enthalpy can also be calculated using the equation: $ΔH = ΔU + PΔV$
  • Relates the change in enthalpy to the change in internal energy and the pressure-volume work done
• Example: calculating the enthalpy change for the combustion of methane at constant pressure
  • $CH_4 (g) + 2O_2 (g) → CO_2 (g) + 2H_2O (l)$

Heat, Work, and Energy Changes

Adiabatic and Isothermal Processes

• In an adiabatic process (q = 0), the change in internal energy is equal to the negative of the work done: $ΔU = -w$
  • For an adiabatic process at constant pressure, $ΔH = -w$
  • Example: rapid compression or expansion of a gas in an insulated container
• In an isothermal process (ΔT = 0), the change in internal energy is zero $ΔU = 0$, and the heat exchanged is equal to the work done: $q = w$
  • For an isothermal process at constant pressure, $ΔH = q_p = w$
  • Example: slow expansion or compression of a gas in a heat reservoir

Isobaric and Isochoric Processes

• In an isobaric process (ΔP = 0), the change in enthalpy is equal to the heat exchanged: $ΔH = q_p$
  • The change in internal energy is given by $ΔU = q_p - PΔV$
  • Example: heating a liquid at constant pressure
• In an isochoric process (ΔV = 0), no pressure-volume work is done (w = 0), and the change in internal energy is equal to the heat exchanged: $ΔU = q_v$
  • The change in enthalpy is equal to the change in internal energy: $ΔH = ΔU$
  • Example: heating a gas in a rigid container at constant volume

Enthalpy as a State Function

Path Independence and Hess's Law

• Enthalpy, like internal energy, is a state function
  • Its value depends only on the current state of the system, not on the path taken to reach that state
• The change in enthalpy between two states is independent of the path taken
  • Simplifies thermodynamic calculations and allows for the application of Hess's law
• Hess's law states that the total enthalpy change for a reaction is the sum of the enthalpy changes for the individual steps or processes that make up the overall reaction, regardless of the pathway
  • Example: calculating the enthalpy of formation of a compound using a series of reactions

Thermochemical Cycles

• The state function property of enthalpy enables the construction of thermochemical cycles
  • Used to calculate enthalpy changes for reactions that are difficult to measure directly
• Thermochemical cycles involve a series of reactions or processes that start and end with the same state
  • The sum of the enthalpy changes for each step in the cycle equals zero $ΣΔH = 0$
• Example: using a Born-Haber cycle to calculate the lattice energy of an ionic compound

Enthalpy Changes in Reactions and Transitions

Standard Enthalpies of Formation and Reaction

• The standard enthalpy of formation $ΔH°_f$ is the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states at 1 atm pressure and a specified temperature (usually 298 K)
  • Example: the standard enthalpy of formation of water $H_2O (l)$ is -285.8 kJ/mol
• The standard enthalpy of reaction $ΔH°_{rxn}$ can be calculated using the standard enthalpies of formation of the reactants and products:
  • $ΔH°_{rxn} = Σ(n × ΔH°_f(products)) - Σ(n × ΔH°_f(reactants))$
  • n is the stoichiometric coefficient
  • Example: calculating the enthalpy of combustion of methane using standard enthalpies of formation

Phase Transitions and Heat Capacities

• The enthalpy change for a phase transition, such as melting or vaporization, is called the enthalpy of fusion $ΔH_{fus}$ or enthalpy of vaporization $ΔH_{vap}$, respectively
  • These values are specific to each substance and can be used to calculate the heat required or released during a phase change
  • Example: calculating the energy needed to melt a given mass of ice
• The heat capacity (C) of a substance is the amount of heat required to raise its temperature by one degree Celsius or Kelvin
  • Specific heat capacity (c) is the heat capacity per unit mass
  • Molar heat capacity $C_m$ is the heat capacity per mole of the substance
• The enthalpy change for a temperature change can be calculated using the equation: $ΔH = C × ΔT$
  • C is the heat capacity and ΔT is the change in temperature
  • Example: calculating the heat required to raise the temperature of a sample of water by 20°C