Thermodynamics Notes: State Functions, Internal Energy, and Enthalpy

State functions vs. path functions

• State function: a property that depends only on the state of the system, not on how you got there. Examples: energy (U), pressure (P), volume (V), temperature (T).
• Path function: depends on the process path between initial and final states. Examples: heat (q), work (w).
• The main idea: you can manipulate a system along many paths to reach the same final state because state functions depend only on the state, not the route.

Internal energy, heat, and work

• Internal energy, U, is a state function.
• Change in internal energy:
  \Delta U = U{\text{final}} - U{\text{initial}}.
• First law of thermodynamics (conservation of energy):
  \Delta U = q + w,
  where q is heat added to the system and w is work done on the system (sign convention matters).
• For PV work (work due to volume change):
  w = -P\,\Delta V,
  so the commonly used form is
  \Delta U = q - P\,\Delta V.
• If there is no volume change (\Delta V = 0), then
  \Delta U = q.
• Heat and work signs (typical chemistry convention):
  • Heat absorbed by the system (endothermic) => q > 0.
  • Heat released by the system (exothermic) => q < 0.
  • Work done on the system (compression) => w > 0.
  • Work done by the system on the surroundings (expansion) => w < 0.
• The transcript also uses a practical interpretation with examples:
  • A car engine performing work on the surroundings is exothermic (system loses energy to surroundings).
  • If the surroundings do work on the system (e.g., ice cube melting at room temperature), the process is endothermic (energy flows into the system).

Heat at constant volume vs heat at constant pressure

• Constant volume processes (no PV work):
  \Delta U = qV, where qV is the heat added at constant volume.
• Constant pressure processes (PV work occurs):
  • Heat exchanged at constant pressure equals the enthalpy change:
    q_p = \Delta H.
  • Enthalpy is defined as
    H = U + P V.
  • The relationship between changes:
    \Delta H = \Delta U + \Delta (P V).
• Practical implication: in many lab reactions, the pressure is approximately constant, so measuring heat at constant pressure often gives you the enthalpy change directly.

Enthalpy of reaction and formation

• Enthalpy of reaction (change in enthalpy for a reaction) is defined as the difference between the enthalpies of products and reactants: \Delta H{\text{rxn}} = \sumi \nui H{f,i}^{\text{(products)}} - \sumj \nuj H_{f,j}^{\text{(reactants)}}.
  • Here, $\nu$ are the stoichiometric coefficients (extent of reaction terms).
• Enthalpy of formation ($H_f$) is the standard enthalpy change to form 1 mole of a substance from its elements in their standard states.
• Sign convention for reactions:
  • If the products have lower enthalpy than reactants, $\Delta H_{\text{rxn}}$ is negative (exothermic overall).
  • If products have higher enthalpy than reactants, $\Delta H_{\text{rxn}}$ is positive (endothermic).
• Reversing a reaction reverses the sign of $\Delta H_{\text{rxn}}$:
  • If $\Delta H{\text{rxn}} = X$, then the reverse reaction has $\Delta H{\text{rxn}} = -X$.

A classic combustion example (CH$4$ and O$2$)

• Example reaction (typical: methane combustion):
  \ce{CH4(g) + 2 O2(g) -> CO2(g) + 2 H2O(g)}.
• In the transcript, there is a discussion of the enthalpy change for this reaction and how to express it per mole of CH$4$, per mole of O$2$, etc. The same enthalpy change can be scaled with stoichiometric factors because enthalpy is an extensive property.
• If the enthalpy change for the balanced equation is given as a certain value (e.g., $\Delta H_{\text{rxn}} = -N$ kJ per the stoichiometric units shown), you can express it for any multiple by multiplying both sides by the same factor (extensivity):
  • For 1 mole CH$4$: $\Delta H = -N$ kJ/mol CH$4$.
  • For 2 moles CH$_4$: $\Delta H = -2N$ kJ, etc.
• Important note from the transcript: the energy change for a reaction is determined by the difference in enthalpies of products and reactants; if you reverse the reaction, the sign changes but the magnitude stays the same.

Sodium azide example and practical reading approach

• The transcript discusses a thermochemical problem involving sodium azide (NaN$_3$). It mentions:
  • Two moles of NaN$3$ producing two moles of Na and three moles of H$2$ (note: this is not a correct decomposition pathway for sodium azide in reality; the actual decomposition is typically: \ce{2 NaN3(s) -> 2 Na(s) + 3 N2(g)}. The transcript’s given products include H$_2$ gas, which is inconsistent with standard sodium azide decomposition. This discrepancy should be corrected in study notes.)
• The key teaching point from the transcript:
  • When solving thermochemistry problems, you must identify what is a gas, what is a solid, and how the reaction’s stoichiometry affects the amount of gas produced (gas-phase products drive PV work considerations).
  • Read problems aloud and interpret descriptors literally, as chemistry is described as a language. For example, phrases like “two moles of NaN$3$ produce two moles of Na and three moles of N$2$” (or other products) inform the stoichiometry you should use.
• Important definitions used in the problem-solving workflow:
  • If the system involves only PV work, then
    \Delta U = q - P\,\Delta V.
  • If volume does not change, then \Delta U = q.
  • If the process occurs at constant pressure, the heat exchanged equals the enthalpy change: q_p = \Delta H.

Reading problems effectively and key conventions

• Chemistry is framed as a language; read statements literally to identify stoichiometry, the signs of heat and work, and whether the system absorbs or releases energy.
• Always identify the gas involved in the reaction because gas production affects PV work and enthalpy calculations.
• Sign conventions recap (summary):
  • Heat absorbed by the system: q > 0 (endothermic for the system).
  • Heat released by the system: q < 0 (exothermic for the system).
  • Work done on the system: w > 0 (compression).
  • Work done by the system on the surroundings: w < 0 (expansion).
• Important unit conversion note from the transcript:
  1\ \text{atm} = 760\ \text{Torr},
  and typically
  1\ \text{atm} \approx 101.3\ \text{kPa}.
• Extensivity reminder: changes scale with the amount of substance. If you double the amount of reactants, the enthalpy change doubles accordingly.

Worked outline for solving problems (procedural guidance)

• Step 1: Write down the thermochemical equation and identify the reaction extent and the signs of q and w from the problem statement.
• Step 2: Decide whether the process is at constant volume or constant pressure to determine which relationship to apply:
  • If constant volume: use \Delta U = q_V .
  • If constant pressure: use q_p = \Delta H and relate to PV work if needed.
• Step 3: Use the First Law: \Delta U = q + w with the appropriate sign convention, and/or substitute w = -P\,\Delta V for PV work.
• Step 4: If the problem provides a chemical enthalpy change (e.g., formation enthalpies or a given $\Delta H_{\text{rxn}}$), compute the extent by stoichiometry and scale the enthalpy accordingly.
• Step 5: If the problem asks for solar energy or other energy inputs/outputs, relate to the enthalpy or internal energy change via the appropriate equation and the reaction extent: for an endothermic process, energy input is positive; for an exothermic process, energy output is negative.
• Step 6: For any reversals of reactions, remember: the sign of the enthalpy change reverses, but magnitude remains the same.

Quick reference formulas (LaTeX)

• State functions and changes:
  \Delta U = U{\text{final}} - U{\text{initial}}
• First law (sign convention where w is work done on the system):
  \Delta U = q + w
• PV work contribution:
  w = -P\,\Delta V
  \Delta U = q - P\,\Delta V
• Constant volume process:
  \Delta U = q_V
• Enthalpy:
  H = U + P V
• Enthalpy change:
  \Delta H = \Delta U + \Delta (P V)
• Enthalpy of reaction (formation data):
  \Delta H{\text{rxn}} = \sumi \nui H{f,i}^{\text{(products)}} - \sumj \nuj H_{f,j}^{\text{(reactants)}}
• Stoichiometric scaling (extensivity): if you multiply the reaction by a factor, multiply $\Delta H_{\text{rxn}}$ by the same factor.

Notes on potential ambiguities in the transcript

• The transcript contains a specific chemical example (NaN$3$ decomposition) with a products set that includes H$2$ gas, which is not the standard decomposition. The standard decomposition is:
  \ce{2 NaN3(s) -> 2 Na(s) + 3 N2(g)}.
• When solving problems, rely on the correct balanced equation for stoichiometry and apply the appropriate $\Delta H$ values from formation enthalpies or given thermochemical data.

Summary takeaway

• Internal energy change is a state function linked to heat and work via the first law.
• PV work introduces a dependence on volume change; constant volume simplifies to $\Delta U = q$, while constant pressure links heat to enthalpy change: $q_p = \Delta H$.
• Enthalpy of reaction is the key quantity for chemical processes and scales with the extent of reaction (extensive property).
• Reading problems aloud and tracking signs of heat and work helps in correctly applying the equations and avoiding sign errors.
• Always verify stoichiometry and use correct, balanced equations when applying enthalpy and energy calculations.