Short Notes: Thermodynamics

5.1 Basic Terminology

• System: The portion of the universe under study. The surroundings are everything outside the system. The boundary separates system and surroundings and may be real or imaginary.
• Types of systems:
  • Open system - mass and energy can be exchanged with surroundings.
  • Closed system - energy can be exchanged but mass cannot.
  • Isolated system - neither mass nor energy is exchanged.
• State function: A property that depends only on the current state of the system, not on the path taken (examples: internal energy U, enthalpy H, entropy S). Path function depends on the process path (examples: heat q, work w).
• Intensive and extensive properties:
  • Intensive - independent of amount of substance (e.g., temperature T, pressure p).
  • Extensive - proportional to system size (e.g., volume V, internal energy U, enthalpy H).
• Internal energy U: Sum of all microscopic kinetic and potential energies of particles in the system. SI unit: joule (J).
• Enthalpy H: Defined by the relation $H = U + pV$. At constant pressure, the heat exchanged equals the change in enthalpy for processes with only PV-work.
• Heat (q) and Work (w): Modes of energy transfer across the boundary. Sign convention used here: $q>0$ when heat is absorbed by the system; $w>0$ when work is done on the system.
• Heat capacity:
  • Heat capacity C - amount of heat required to raise temperature by 1 K: $C = \\dfrac{dq}{dT}$ (extensive).
  • Specific heat - heat capacity per unit mass.
  • Molar heat capacity - heat capacity per mole.
  • Heat capacities at constant volume and pressure: $C_V = \\left(\\dfrac{\\partial U}{\\partial T}\\right)_V$, $C_P = \\left(\\dfrac{\\partial H}{\\partial T}\\right)_P$.
  • For an ideal gas: $C_P - C_V = R$, where $R$ is the gas constant.

5.2 First Law of Thermodynamics

• Statement: Energy can be neither created nor destroyed; it can only be transferred or converted. For a closed system the first law is written as

  $\\Delta U = q + w$

• Work:
  • Mechanical PV-work for a quasi-static process: $w = -\\displaystyle\\int_{V_i}^{V_f} p_{\\text{ext}}\\,dV$. The negative sign follows the sign convention: work done by the system is negative.
  • For a reversible process, $w_{\\text{rev}} = -\\displaystyle\\int_{V_i}^{V_f} p_{\\text{int}}\\,dV$.
• Heat at constant pressure: For processes at constant pressure with only PV-work, the heat absorbed equals the change in enthalpy:

  $q_P = \\Delta H$

• Special cases for ideal gases:
  • Isothermal process (constant T): $\\Delta U = 0$ for an ideal gas, therefore $q = -w$.
  • Isobaric process (constant p): $q_P = \\Delta H = n C_P \\Delta T$ for an ideal gas.
  • Isochoric process (constant V): $q_V = \\Delta U = n C_V \\Delta T$ for an ideal gas.
• Sign conventions and units: Energies and heats are usually expressed in joules (J) or kilojoules per mole (kJ mol^-1).

5.3 Thermochemical Equations

• Thermochemical equation: A chemical equation that includes the enthalpy change (ΔH) for the reaction, written with stoichiometric coefficients and the physical states of reactants and products.
• Standard enthalpy of formation ($\\Delta_f H^\\circ$): Enthalpy change when one mole of a compound is formed from its elements in their standard states (standard state usually 1 bar and specified temperature, commonly 298.15 K). Units: kJ mol^-1.
• Manipulating thermochemical equations:
  • Multiplying a chemical equation by a factor multiplies ΔH by the same factor.
  • Reversing an equation changes the sign of ΔH.
  • Adding equations adds their ΔH values (Hess's law, see next section).
• Calculating reaction enthalpy:
  • Using formation enthalpies:

    $\\Delta H^\\circ_{\\text{rxn}} = \\displaystyle\\sum \\Delta_f H^\\circ(\\text{products}) - \\sum \\Delta_f H^\\circ(\\text{reactants})$

  • Using bond energies (approximate method for gas-phase molecules):

    $\\Delta H_{\\text{rxn}} \\approx \\displaystyle\\sum \\text{Bond energies (bonds broken)} - \\sum \\text{Bond energies (bonds formed)}$

    Bond-energy calculations give approximate values because values are averages and do not account for molecular environment precisely.

5.4 Hess's Law

• Statement: The total enthalpy change for a chemical reaction is independent of the route between initial and final states; it depends only on initial and final states. This follows from enthalpy being a state function.
• Practical use: If a reaction can be expressed as the sum of two or more steps, the enthalpy change for the overall reaction is the sum of the enthalpy changes of the steps.
• Common formulae:

  $\\Delta H_{\\text{rxn}} = \\displaystyle\\sum \\Delta_f H^\\circ(\\text{products}) - \\sum \\Delta_f H^\\circ(\\text{reactants})$

  $\\Delta H_{\\text{rxn}} = \\displaystyle\\sum \\text{Bond energies (reactants)} - \\sum \\text{Bond energies (products)}$

• Worked approach (qualitative):
  • Write known thermochemical steps and their ΔH values.
  • Reverse or multiply equations as needed to obtain the target reaction; change signs and scale ΔH accordingly.
  • Cancel species that appear on both sides when adding the equations.
  • Add the ΔH values of the manipulated steps to get ΔH for the target reaction.

5.5 Entropy and Gibbs Energy

• Entropy (S):
  • Macroscopic (thermodynamic) definition for a reversible process:

    $\\Delta S = \\displaystyle\\int\\frac{d q_{\\text{rev}}}{T}$

  • Entropy is a state function; its SI unit is J K^-1 (often quoted per mole: J K^-1 mol^-1).
  • Qualitative rules:
    • Entropy increases with increasing disorder and with the number of accessible microstates.
    • Phase changes: entropy increases on melting or vaporisation (solid → liquid → gas).
    • Mixing two different ideal gases increases entropy (spontaneous mixing).
• Gibbs free energy (G):
  • Definition:

    $G = H - TS$

  • Change in Gibbs free energy:

    $\\Delta G = \\Delta H - T\\Delta S$

  • Spontaneity criteria at constant temperature and pressure:
    • If $\\Delta G < 0$,="" the="" process="" is="" spontaneous="" as="">
    • If $\\Delta G = 0$, the system is at equilibrium.
    • If $\\Delta G > 0$, the process is non-spontaneous as written (spontaneous in the reverse direction).
  • Relation with equilibrium constant:

    $\\Delta G^\\circ = -RT\\ln K$

    $\\Delta G = \\Delta G^\\circ + RT\\ln Q$

    Where $K$ is equilibrium constant, $Q$ is reaction quotient, $R$ is gas constant, and $T$ is temperature in kelvin.

  • Temperature dependence of spontaneity:
    • For reactions with $\\Delta H<0$ and="" $\\delta="" s="">0$ → $\\Delta G$ negative at all T (spontaneous at all temperatures).
    • For reactions with $\\Delta H>0$ and $\\Delta S<0$ →="" $\\delta="" g$="" positive="" at="" all="" t="" (non-spontaneous="" at="" all="">
    • For reactions where signs of $\\Delta H$ and $\\Delta S$ are the same, spontaneity depends on temperature (compare $T$ with $\\Delta H/\\Delta S$).
    • Other useful free energy: Helmholtz free energy $A = U - TS$ is the criterion for spontaneity at constant volume and temperature: $\\Delta A < 0$="" is="" spontaneous="" at="" constant="" v="" and="">

    Summary: The central ideas are the conservation of energy (first law), enthalpy and thermochemical equations for heat changes at constant pressure, Hess's law (additivity of enthalpy changes because enthalpy is a state function), and the roles of entropy and Gibbs free energy in determining spontaneity and equilibrium. Use standard enthalpies of formation, bond energies (approximate), and the relations $\\Delta H = \\Delta U + \\Delta(pV)$, $\\Delta S = \\int d q_{\\text{rev}}/T$, and $\\Delta G = \\Delta H - T\\Delta S$ to analyse and predict thermodynamic behaviour in chemical reactions and processes.