Energy Balance Essentials: Isolated Systems, Enthalpy, and Partial Pressures

Internal energy, thermal energy, and chemical energy

• U = total energy; composed of thermal (kinetic) energy and chemical energy (bond energy).
• Chemical energy is negative in the sense of potential energy wells; bond formation lowers the system's chemical energy.
• Energy balance in a closed system: $\Delta U = q + w = \Delta U<em>{\text{chem}} + \Delta U</em>{\text{thermal}}.$
• Isolated system (no heat or work exchange): $\Delta U<em>{\text{total}} = 0 \Rightarrow \Delta U</em>{\text{thermal}} = -\Delta U_{\text{chem}}.$
• If chemical energy decreases (more negative) due to bond strengthening, the thermal energy increases by the same amount in an isolated system.
• If the system is not isolated, the excess thermal energy can be transferred as heat to the surroundings or as work; the budget remains balanced with the outside world.
• In a fixed-volume, insulated container: $q = 0,\; w = 0 \Rightarrow \Delta U = 0.$ The chemical-to-thermal energy transfer still occurs internally as needed.
• Practical takeaway: the heat you feel inside the system depends on whether heat can leave the system.

Enthalpy and constant-pressure processes

• Enthalpy defined: $h = u + p\,v$ where $u$ is internal energy per unit substance and $v$ is specific volume.
• Change in enthalpy: $\Delta h = \Delta u + p\,\Delta v.$
• At constant pressure with only PV-work (no non-PV work): $q_p = \Delta h$.
• Enthalpy is a state function; heat is a boundary quantity. In many reactions, $\Delta u$ dominates and $\Delta h \approx \Delta u$, with a small correction from $p\Delta v$.
• When volume changes are small or pressure is constant, enthalpy provides a convenient energy accounting at constant pressure.

Partial pressures and open vs closed systems

• For mixtures of gases, total pressure is the sum of individual partial pressures: $P = \sum<em>i P</em>i.$
• For ideal gases, $P<em>i = x</em>i P$ where $x_i$ is the mole fraction of component $i$.
• In open (open to surroundings) experiments, the gas tends to equilibrate with the external pressure, performing or receiving PV-work as it expands or contracts.
• In open systems, a work term ($w$) often appears due to volume change against external pressure; this is automatically accounted for when considering enthalpy.

Worked intuition and example notes

• Exothermic reaction in an insulated container: heat stays in the system; the products (system) heat up; the internal energy increases due to the loss of chemical energy.
• Exothermic reaction in contact with surroundings: heat can leave the system; the thermal energy of the system may decrease or rise less, depending on environment.
• Real breathing example: respiration releases chemical energy, warming the exhaled air; excess heat is transferred to surroundings.
• General energy-tracking rule (isolated vs open):
  • Isolated and fixed volume: $\Delta U = 0\Rightarrow \Delta U<em>{\text{thermal}} = -\Delta U</em>{\text{chem}}.$
  • Open or connected to environment: heat and/or work can move between system and surroundings; enthalpy helps track energy when volume can change under constant external pressure.

Quick example (conceptual, using shown numbers)

• If a reaction changes chemical energy from $-5.03\,\text{eV}$ to $-16.48\,\text{eV}$:
  • $\Delta U_{\text{chem}} = (-16.48) - (-5.03) = -11.45\ \text{eV}.$
  • In an isolated system: $\Delta U<em>{\text{thermal}} = -\Delta U</em>{\text{chem}} = +11.45\ \text{eV}.$
• If the system is not isolated, this 11.45 eV may be carried away as heat to the surroundings or stored as thermal energy, depending on the environment and constraints.

Homework-oriented takeaways

• You will be asked to track energy through a chemical change:
  • Compute $\Delta U<em>{\text{chem}} = U</em>{\text{chem, final}} - U_{\text{chem, initial}}.$
  • Determine whether the accompanying change in thermal energy $\Delta U_{\text{thermal}}$ stays in the system (isolated) or leaves to the surroundings (open).
  • If the container is fixed and insulated: $\Delta U = 0 \Rightarrow \Delta U<em>{\text{thermal}} = -\Delta U</em>{\text{chem}}.$
  • If the container is at constant external pressure: use enthalpy: $q_p = \Delta h = \Delta u + p\,\Delta v.$
• For glucose burning or similar reactions, you’ll need the balanced equation and standard molar volumes at STP to evaluate PV-work contributions when using enthalpy.

Summary formulas

• First Law: $\Delta U = q + w$
• Energy components: $\Delta U = \Delta U<em>{\text{chem}} + \Delta U</em>{\text{thermal}}$
• Isolated, fixed volume: $\Delta U = 0 \Rightarrow \Delta U<em>{\text{thermal}} = -\Delta U</em>{\text{chem}}$
• Enthalpy: $h = u + p\,v;\quad \Delta h = \Delta u + p\Delta v$
• Constant pressure, PV-work only: $q_p = \Delta h$
• Partial pressures for ideal gases: $P = \sum<em>i P</em>i;\quad P<em>i = x</em>i P$