Thermodynamics

1. Thermodynamic Systems & Boundaries

• System: Any macroscopic part of the universe that is under observation.
• Surroundings: Apart from the system, everything else is the surroundings.
• Universe = System + Surroundings.
• Boundary: The wall that separates the system from its surroundings. It can be:
  • Real or Imaginary
  • Rigid or Non-Rigid: Rigid boundaries hold a constant volume. Non-rigid boundaries (e.g., fitted with a piston) can change volume.
  • Permeable or Impermeable: Permeable allows mass exchange; impermeable does not allow mass exchange.
  • Adiabatic or Diathermic: Adiabatic means heat flow is not allowed (q = 0). Diathermic means heat flow is allowed (q ≠ 0).

2. Types of Systems

3. Extensive & Intensive Properties

• Extensive Properties: Properties whose value depends on the amount or size of the matter present in the system (i.e., changes on changing the amount of matter).
  • They are additive. (e.g., 5g + 5g = 10g mass).
  • Examples: Mass, Volume, Moles (n), and All energy terms (Internal Energy, Enthalpy, Kinetic Energy, Potential Energy, Entropy, Gibbs Free Energy, Total Energy).
• Intensive Properties: Properties which do not depend on the amount or size of matter present.
  • They are non-additive.
  • If two identical systems are combined, their intensive properties do not add up (e.g., Molarity M₁ + M₂ is incorrect. Instead, M_net = M₁V₁ + M₂V₂V₁ + V₂).

Key Rules for Intensive Properties:

• The ratio of two extensive properties is an intensive property (e.g., Density = Mass / Volume).
• If an extensive property is defined as "per mole", "per gram", "per liter", or "per ml", it becomes intensive. Thus, properties starting with "Molar" or "Specific" are intensive (e.g., Molar Mass, Molar Volume, Molar Heat Capacity, Molar Conductance, Specific Heat, Specific Volume, Specific Resistance).
• Properties which do not change on the division of a system are intensive (e.g., Pressure P, Temperature T).
• All concentration terms are intensive: Molarity, Molality, Normality, Formality, Mole fraction, Strength, Percentage and parts concentration terms.
• All types of temperatures are intensive: Temperature, Boiling Point, Melting Point, Freezing Point.
• All equilibrium constants are intensive: K_c, K_p, K_w, K_f, K_sp, K_a.

Algebraic Operations:

• If X and Y are Extensive properties:
  X + Y = Ext,    X − Y = Ext,    X × Y = Ext,    X ÷ Y = Intensive.
• If X and Y are Intensive properties:
  X + Y = Int,    X − Y = Int,    X × Y = Int,    X ÷ Y = Int.

4. State Functions vs Path Functions

• State Function : Functions or variables whose value depends on the state of the system, not on the path adopted by the system to attain that state.
  Examples : Pressure (P), Volume (V), Temperature (T), Internal energy (U), Entropy (S), Enthalpy (H), Free energy (G).
• Notation for changes in state variables: small changes = δU, δH, δG... ; large changes = ΔU, ΔH, ΔG...
• Path Function : Quantities which are dependent on the path adopted by the system to achieve a particular state. These are defined when a process is going on.
  Examples : Work (W), Heat (q), Heat Capacity (C).
• Notation for changes in path variables: small change = δq, δW ; large change = q, W.

5. Thermodynamic Processes

• Thermodynamic Process : Steps followed by the system to change its state (e.g., from State 1 (P₁, V₁, T₁) to State 2 (P₂, V₂, T₂)).
• Isothermal Process : Temperature is constant (ΔT = 0, dT = 0).
  For an ideal gas : PV = constant ⇒ P₁V₁ = P₂V₂ (Boyle's Law).
• Isochoric Process : Volume remains constant (ΔV = 0, dV = 0).
  For an ideal gas : P ∝ T ⇒ P₁/T₁ = P₂/T₂ (Gay Lussac's Law).
• Isobaric Process : Pressure remains constant (ΔP = 0, dP = 0).
  For an ideal gas : V ∝ T ⇒ V₁/T₁ = V₂/T₂ (Charle's Law).
• Adiabatic Process : Heat exchange is not allowed (q = 0, dq = 0).
  For an ideal gas : PV^γ = constant (where γ = Poisson's ratio, γ > 1).

• Cyclic Process : When a system undergoes different states by different processes and finally returns to its initial state.
• For a cyclic process, the change in all state functions is zero : ΔU = 0, ΔS = 0, ΔG = 0, ΔH = 0, ΔT = 0, ΔP = 0. Net heat (q) and net work (W) are not zero.

6. Internal Energy (U) & Enthalpy (H)

• Internal Energy (U / E) : Every substance possesses a definite amount of energy depending on composition, temperature, and pressure.
  U = U_t + U_r + U_v + U_e + ... (translational, rotational, vibrational, electronic, etc.)
• Since U is a state function, its absolute value cannot be determined, but its change can be: ΔU = U₂ − U₁.
• Enthalpy (H) : Used when dealing with processes in open vessels (at constant pressure) instead of internal energy.

• General relation between states 1 and 2 : ΔH = ΔU + (P₂V₂ − P₁V₁)

7. Relation Between ΔH and ΔU for Gases

• From ideal gas law PV = nRT, taking changes gives Δ(PV) = ΔnRT + nRΔT.
• For an ideal gas (only temperature changes, no chemical reaction) :

• For a chemical reaction (at constant temperature) :

Δn_g = (Moles of product gases) − (Moles of reactant gases)

• If Δn_g = 0, then ΔH = ΔU. (e.g., C (s) + O₂ (g) → CO₂ (g)).

8. Heat (q)

• Quantity of energy which flows between a system and its surroundings on account of a temperature difference. Heat always flows from high to low temperature.
• IUPAC Sign Convention :
  • Heat flowing into the system : q = +ve
  • Heat flowing out of the system : q = −ve

9. Work & Heat

• Heat and work are both dynamic forms of energy and are boundary phenomena.

Mathematical Expression of Work :

• P_ext is the pressure which is used to compress the gas (contraction).
• P_ext is the pressure against which the gas is doing work (expansion).

Special Cases :

• Case-1 (P_ext = constant) :
  W = −P_ext ∫ dV = −P_ext(V₂ − V₁)
• Case-2 (P_ext ≠ constant) :
  W = − ∫ P_ext dV

Sign Convention :

• W is taken as negative if work is done by the system.
• W is taken as positive if work is done on the system.
• When heat is given by the system to the surrounding, it is given a negative sign (−q).
• When heat is absorbed by the system from the surrounding, it is positive (+q).

10. Zeroth Law of Thermodynamics

• If two systems A and B are each in thermal equilibrium with a third system C, then A and B are in thermal equilibrium with each other.
• Equivalently: if T_A = T_C and T_B = T_C, then T_A = T_B.

11. First Law of Thermodynamics (FLOT)

U₂ = U₁ + q + W  ⇒  U₂ − U₁ = q + W

FLOT on Different Processes :

• Isothermal Process (dT = 0, ΔT = 0) :
  ΔH = 0. For an ideal gas, ΔU = 0 (only when molecular interactions during the process remain the same).
  Since ΔU = q + W = 0 ⇒ +q = −W or −q = +W.
• Exceptions (Isothermal but ΔU ≠ 0, ΔH ≠ 0) : Phase changes like Ice ⇌ water (at M.P.) and water ⇌ Vapour (at B.P.).
• Adiabatic Process (q = 0) :
  ΔU = q + W ⇒ ΔU = W.
  → If work is done on the system, its internal energy increases (T_Final > T_initial).
  → If work is done by the system, its internal energy decreases (T_Final < T_initial).
• Isochoric Process (dV = 0) :
  W = 0 ⇒ ΔU = q_v.
  Heat supplied to the system or by the system is equal to the change in internal energy.
• Isobaric Process (ΔP = 0) :
  ΔU = q + W ⇒ U₂ − U₁ = q − P(V₂ − V₁)
  (U₂ + PV₂) − (U₁ + PV₁) = q_p ⇒ H₂ − H₁ = q_p ⇒ ΔH = q_p.

12. Heat Capacity

• Defined as the quantity of heat required to raise the temperature of the system by one degree.
• Larger the sample, greater is its heat capacity.

Unit is J K⁻¹ or J / °C

Specific Heat Capacity (s) :

• Heat supplied to the system to raise its temp from T₁ to T₂ :
  q = m ∫ s dT. If specific heat capacity is given as constant, Q = ms(T₂ − T₁).

Molar Heat Capacity (C) :

• q = n ∫ C dT. If C is constant, q = nC(T₂ − T₁).

13. Heat Capacity at Constant Pressure & Volume

• For gaseous systems :
  C_p = molar heat capacity at constant pressure.
  C_v = molar heat capacity at constant volume.
• At constant volume (W = 0) : q_v = ΔE ⇒ C_v = (dE/dT)_v per mole (or dU/dT).
• At constant pressure : q_p = ΔH ⇒ C_p = (dH/dT)_p per mole.

• C_p is "generally" larger than C_v because at constant pressure, a part of the heat added may be used in the work of expanding, whereas at constant volume all added heat produces a rise in temperature.
• Exception : Substances generally expand with an increase of temperature at constant pressure, but there may be contraction in exceptional cases (e.g., water between 0° & 4°).

Relation between C_p and C_v :

For 1 mole of an ideal gas :

• H = U + PV = U + RT
• dHdT = dUdT + R

14. Molar Heat Capacities of Gases

γ = C_p/C_v (Poisson's ratio) ; C_p − C_v = R

• For temperature-dependent specific heat, heat exchanged is calculated using integration : q = n ∫ C dT.

15. Mixture of Gases

• For a mixture of non-reacting ideal gases, the effective molar heat capacities and Poisson's ratio (γ_mix) are :

• Since C_p,mix − C_v,mix = R, we know C_v = Rγ − 1. Substituting this into the mixture formula gives:

16. Reversible & Irreversible Processes

• Reversible Process :
  • Completion time is infinite (hypothetical process).
  • Equilibrium is maintained in every step.
  • Both internal and external pressure change, but external pressure is always equal to internal pressure (P_ext = P_gas).
  • If we reverse the process, each step is achieved back.
• Irreversible Process :
  • Finite step process (all processes in nature are irreversible).
  • Fast process where external pressure is constant.
  • Equilibrium is not observed in between.
  • In the case of expansion processes, external pressure is much less than the internal pressure (P_ext << P_gas).

17. Thermodynamic Changes in Isothermal Process

Reversible Isothermal Expansion (from V₁ to V₂) :

• Here, P_ext = P_gas (but is variable).
• dW = −(P_ext − dP) dV = −P_ext dV + dP dV (where dP dV → 0)
• ∫ dW = − ∫_V₁^V₂ P_ext dV = − ∫_V₁^V₂ P_gas dV = − ∫_V₁^V₂ nRTV dV

Since P₁V₁ = P₂V₂, we can also write W_rev = −2.303 nRT log(P₁/P₂).

Irreversible Isothermal Expansion (from V₁ to V₂) :

• Here, P_ext is constant and P_ext << P_gas.
• ∫ dW = − ∫_V₁^V₂ P_ext dV

• Generally, the final pressure of the gas P₂ = P_ext. Using PV = nRT (so P₁/P₂ = V₂/V₁):

18. Work Done : Reversible vs Irreversible Process

• Expansion (V₁ → V₂) :
  • Reversible : Gas expands against an external pressure almost equal to gas pressure (P_ext = P_gas).
  • Irreversible : Gas expands against a pressure much less than the gas pressure (P_ext << P_gas).
  • Magnitude of work done by gas : W_rev > W_irrev. Area under reversible curve is greater.
• Compression :
  • Reversible : Gas is compressed by an external pressure equal to P_gas.
  • Irreversible : Gas is compressed by an external pressure much greater than P_gas (P_ext >> P_gas).
  • Magnitude of work done on gas : W_irrev > W_rev. Area under irreversible path (rectangle) is greater.

19. Free Expansion

• Expansion of an ideal gas into a vacuum where P_ext = 0.

20. Adiabatic Reversible Process

• Conditions : Walls should be adiabatic ; heat exchange q = 0.
• Expansion : Work is done by the system ⇒ Internal energy decreases (ΔU = negative) ⇒ T_final < T_initial.
• Compression : Work is done on the system ⇒ Internal energy increases (ΔU = positive) ⇒ T_final > T_initial.
• Derivation : ΔU = q + W ⇒ ΔU = W
  nC_v dT = −P_gas dV = −(nRT/V) dV
  C_v (dT/T) = −R (dV/V)
  Integrating from T₁, V₁ to T₂, V₂ : ln(T₂/T₁) = −(γ−1) ln(V₂/V₁) ⇒ T₁V₁^γ−1 = T₂V₂^γ−1
• Useful relations : C_p − C_v = R  ;  C_p/C_v = γ  ;  R/C_v = γ − 1

Derived by substituting P = nRT/V or V = nRT/P into the original equations.

21. Adiabatic & Isothermal Expansion

• For reversible adiabatic processes, the following relations hold :

• From the First Law (ΔU = q + W), for an adiabatic process q = 0, therefore ΔU = W.
• Work done (Reversible) : Since ΔU = nC_v(T₂ − T₁) and P₂V₂ − P₁V₁ = nR(T₂ − T₁) :

• Work done (Irreversible) : Expanding against a constant external pressure P_ext (where P_ext = P₂) :

Comparing Isothermal vs Adiabatic Curves :

• For a given expansion, the Isothermal curve lies above the Adiabatic curve.
• Area under the curve gives magnitude of work: |W_iso| > |W_adia|.
• For adiabatic expansion curves of two different gases, the gas with the higher γ (e.g., Monoatomic > Diatomic) has a steeper curve.

22. Slopes in P-V Graphs

• Isothermal slope : PV = K ⇒ P + V(dP/dV) = 0 ⇒ dP/dV = −P/V
• Adiabatic slope : PV^γ = K ⇒ PγV^γ−1 dV + V^γ dP = 0 ⇒ dP/dV = −γ(P/V)

• Comparison of Gases (Adiabatic process) : Steeper curve (higher slope) means higher γ value.
  For γ : Monoatomic (1.67) > Diatomic (1.40) > Polyatomic (1.33).
• Expansion : Steeper slope ⇒ lower final pressure at the same volume. Curve for He (mono) will be lowest, O₂ (di) middle, NH₃ (poly) highest.
• Compression : Steeper slope ⇒ higher final pressure at the same volume. Curve for He (mono) will be highest, O₂ (di) middle, NH₃ (poly) lowest.

23. Thermochemistry & Enthalpy of Reaction

• Heat of Reaction (ΔH_R) : Amount of heat evolved or absorbed when reacting species react completely as represented by a balanced chemical equation.
• ΔH_R = Σ H_{f (Products)} − Σ H_{f (Reactants)}
• The specific name of the enthalpy depends on the process (e.g., Heat of Formation, Combustion, Neutralisation, Solution, Isomerisation).

Endothermic Reactions :

• Heat is absorbed (+). ΔH = H_P − H_R = (+)ve.
• Examples: Dissociation, Fusion, Vapourisation, Sublimation, Photochemical reactions.

Exothermic Reactions :

• Heat is evolved (−). ΔH = H_P − H_R = (−)ve.
• Examples: Combustion, Neutralisation, Bond Formation, Solution process, Dilution process, Respiration.

24. Standard Enthalpy of Formation (ΔH°_f)

• Change in enthalpy when one mole of a substance is formed from its constituent elements in their most stable/reference state at 1 bar pressure and (usually) 298 K.
• The standard enthalpy of formation of elements in their reference (pure) state is assumed to be zero.

• Reference states (ΔH°_f = 0) : H₂(g), O₂(g), Cl₂(g), N₂(g), C(s, graphite), S₈(rhombic), P₄(white), Br₂(l), I₂(s), Na(s), Fe(s), Zn(s), Sn(white), H⁺(aq).
• Non-reference states (ΔH°_f ≠ 0) : H(g), O(g), Br₂(g), OH⁻(aq), C(s, diamond), Zn(l).

Identifying True Formation Reactions :

• The reaction enthalpy ΔH_R is equal to ΔH_f ONLY if exactly 1 mole of product forms from reference-state elements.
• Examples where ΔH_R = ΔH°_f :
  • C(graphite) + O₂(g) → CO₂(g)
  • 12 H₂(g) + 12 Cl₂(g) → HCl(g)
  • 12 H₂(g) + 12 Br₂(l) → HBr(g)
  • 18 S₈(s) + O₂(g) → SO₂(g)
  • 32 O₂(g) → O₃(g)
• Examples where ΔH_R ≠ ΔH°_f :
  • C(diamond) + O₂(g) → CO₂(g) (Diamond is not the reference state of Carbon)
  • 2 H₂(g) + O₂(g) → 2 H₂O(l) (Forms 2 moles instead of 1 mole)

25. Standard State & Reference State

• Standard state : 1 bar and 298 K.
• Reference state : the most stable form of an element at 1 bar and the specified temperature.

Standard Enthalpy of Formation (Δ_fH°) :

• Enthalpy change when one mole of a substance is formed from its constituent elements in their reference states.

• Δ_fH° ≠ 0 for : H(g), O(g), Zn(l), Br₂(g), OH⁻(aq), C(s, diamond) — these are not reference states.

26. Enthalpy of Formation — Key Relation

• Δ_RH° equals Δ_fH° of the product only when reactants are elements in their standard states.

27. Enthalpy (Heat) of Combustion

• The decrease in enthalpy when one mole of a substance is completely oxidised.
• Organic compounds (containing C, H, O) burn to CO₂(g) and H₂O(l).
• The final oxidation state of the central atom must be maximum (C → +4, S → +6, N → +5, P → +5).

Combustion of C(s) :

• C(s) + ½ O₂(g) → ⁺²CO(g) ; Δ_RH ≠ (Δ_cH) of C(s)  — carbon not in max. oxidation state.
• C(s) + O₂(g) → ⁺⁴CO₂(g) ; Δ_RH = (Δ_cH) of C(s)

Combustion of H₂(g) :

• H₂(g) + ½ O₂(g) → H₂O(l) ; Δ_RH = (Δ_cH) of H₂(g)

Key relations :

28. Bomb Calorimeter

• Used to measure heat of combustion at constant volume.
• During combustion, energy is released; this energy is absorbed by the bomb and the surrounding water.
• Components: insulated outer container filled with water, a stirrer, a thermometer, an oxygen inlet, and a central steel bomb containing the sample compound.

Key formulas :

C = heat capacity of the calorimeter (bomb + water).

Δn_g = (moles of gaseous products) − (moles of gaseous reactants).

29. Kirchhoff's Equation

• Relates the change in reaction enthalpy with temperature.

ΔC_p = ΣC_p(products) − ΣC_p(reactants). If C_p values are constant, ΔH_T₂ = ΔH_T₁ + ΔC_p(T₂ − T₁).

30. Enthalpy (Heat) of Solution

• Heat evolved or absorbed when one mole of solute is dissolved in excess (infinite) solvent so that further dilution causes no heat change.

• H₂SO₄(l) + aq → H₂SO₄(aq) ; ΔH = −20.2 kcal
• KCl(s) + aq → KCl(aq) ; ΔH = +4.4 kcal
• Heat of an ideal solution is taken as zero.

31. Enthalpy (Heat) of Dilution

• Heat evolved or absorbed when a solution containing one mole of solute is diluted from one concentration to another.

• HCl(g) + 10H₂O → HCl·10H₂O ; ΔH₁ = −69.01 kJ/mol
• HCl(g) + 25H₂O → HCl·25H₂O ; ΔH₂ = −72.03 kJ/mol
• HCl(g) + 40H₂O → HCl·40H₂O ; ΔH₃ = −72.79 kJ/mol
• HCl(g) + ∞H₂O → HCl·∞H₂O ; ΔH_∞ = −74.85 kJ/mol
• e.g. Dilution 10 → 40 mol H₂O : ΔH = −72.79 − (−69.01) = −3.78 kJ/mol

Relation between ΔH_sol, lattice enthalpy & hydration enthalpy :

32. Enthalpy (Heat) of Neutralisation

• Heat evolved when one gram equivalent of acid is completely neutralised by one gram equivalent of base in dilute solution, i.e., when one mole of H₂O is formed from H⁺(aq) and OH⁻(aq).
• ΔH°_{neutralisation} is always negative.

• This value (−57.3 kJ) holds for strong acid + strong base (complete ionisation, no extra energy cost).

Effect of weak acid / weak base :

• Part of the heat is consumed in ionising the weak electrolyte, so |ΔH_neut| < 57.3 kJ.

ΔH°_diss (dissociation/ionisation enthalpy) is positive for a weak electrolyte, making the net neutralisation heat less exothermic.

33. Enthalpy of Phase Transition

Enthalpy of Fusion :

• Enthalpy change when 1 mole of a substance converts from solid to liquid at its melting point.

e.g. H₂O(s) → H₂O(l) ; ΔH = 1.44 kcal mol⁻¹

Enthalpy of Vaporisation :

• Enthalpy change when 1 mole of a substance converts from liquid to gas at its boiling point.

e.g. H₂O(l) → H₂O(g) ; ΔH = 10.5 kcal mol⁻¹

Enthalpy of Sublimation :

• Enthalpy change when 1 mole of a solid converts directly into vapour at a temperature below its melting point.

e.g. I₂(s) → I₂(g) ; ΔH = 14.9 kcal mol⁻¹

34. Enthalpy of Atomisation

• Enthalpy change when 1 mole of a compound (solid, liquid or gas) is dissociated into its constituent atoms in the gaseous state.
• Restriction : (i) 1 mole of the compound, (ii) products must be gaseous atoms.

Examples :

• H₂(g) → 2 H(g) ; ΔH_R = ΔH_atom(H₂(g))
• O₃(g) → 3 O(g) ; ΔH_R = ΔH_atom(O₃(g))
• CH₄(g) → C(g) + 4 H(g) ; ΔH_R = ΔH_atom(CH₄(g))
• Br₂(l) → 2 Br(g) ; ΔH_R = ΔH_atom(Br₂(l))
• I₂(s) → 2 I(g) ; ΔH_R = ΔH_atom(I₂(s))

35. Bond Dissociation Energy (BDE)

• Energy required to break 1 mole of a particular bond between atoms in the gaseous state, separating the atoms into gaseous state, at 1 atm and specified temperature.

e.g. H–H(g) → 2 H(g) ; ΔH = 433 kJ mol⁻¹
H–I(g) → H(g) + I(g) ; ΔH = 299 kJ mol⁻¹

Atomisation vs BDE :

• In BDE: both reactants and products are in gaseous state.
• In atomisation: products are gaseous, but reactant may or may not be gaseous.

Br₂(l) → 2 Br(g) ; ΔH_atom  vs  Br–Br(g) → 2 Br(g) ; E_Br–Br  ⇒  ΔH_atom > E_Br–Br

• Similarly for I₂: ΔH_atom(I₂(s)) > BDE(I–I)

Single bond vs multiple bonds of same type :

• If a molecule has only one bond of a particular type, BDE is an absolute value.
• If a molecule has more than one bond of the same type, BDE is an average value.

e.g. In H₂: H–H(g) → 2 H(g) — single bond, absolute value.

In H₂O: first O–H bond breaks with energy a, second with energy b.
ε_O–H = a + b2

In CH₄: four C–H bonds break with energies a, b, c, d.
ε_C–H = a + b + c + d4

36. Using Bond Energies to Find ΔH_R

37. Hess's Law of Constant Heat Summation

• A consequence of the first law of thermodynamics.
• "The enthalpy change in a chemical or physical process is the same whether the process is carried out in one step or in several steps."
• This is because enthalpy change is independent of the path followed.

Example :

C(graphite) + O₂(g) → CO₂(g) ; ΔH

• Step 1 : C(graphite) + 12 O₂(g) → CO(g) ; ΔH₁
• Step 2 : CO(g) + 12 O₂(g) → CO₂(g) ; ΔH₂

38. Resonance Energy (RE)

• The theoretical ΔH_f is calculated assuming a Kekulé-type structure (no delocalisation).
• The experimental ΔH_f differs because of resonance stabilisation.

• RE is always negative (the actual molecule is more stable than the hypothetical structure).

39. Born–Haber Cycle

• Used to calculate lattice enthalpy (LE) of an ionic solid indirectly, since it cannot be measured directly.

For NaCl :

• Na(s) + 12 Cl₂(g) → NaCl(s)   [ΔH_f]
• Steps: Na(s) →^ΔH_sub Na(g) →^IE Na⁺(g)
• 12 Cl₂(g) →^{12 BE} Cl(g) →^ΔH_eg Cl⁻(g)
• Na⁺(g) + Cl⁻(g) →^LE NaCl(s)

For Al₂O₃ :

• 2 Al(s) + 32 O₂(g) → Al₂O₃(s)
• Steps involve: 2 ΔH_sub(Al), then 2 I₁ + 2 I₂ + 2 I₃ for aluminium; 32 O₂ → 3 O(g), then 3 ΔH_eg1 and 3 ΔH_eg2 for oxygen; finally lattice enthalpy LE to form the solid.

40. Limitations of the First Law

• The First Law cannot explain why some processes occur on their own and others do not.
• It cannot explain the direction of heat flow — it can only calculate q, W, ΔU, etc.
• It cannot predict the spontaneity of a process.

41. Proposed Criteria for Spontaneity

1. Enthalpy change (ΔH_sys) :

• "Processes which are exothermic (ΔH_sys < 0) are always spontaneous" — this is NOT always true (counter-examples exist).

2. Entropy change of system (ΔS_sys) :

• Entropy (S) is a measure of randomness / disorder.
• Liquid (less random, S₁) → Gas (more random, S₂) : S₂ − S₁ > 0 ⇒ ΔS_sys > 0.

3. Entropy change of Universe (ΔS_univ) :

42. Second Law of Thermodynamics

• For any irreversible (spontaneous) process, the entropy of the universe is increasing.
• Transfer of heat from a cold body to a hot body cannot occur without an external agent performing work.
• Any process occurring on its own is thermodynamically irreversible.

43. Entropy — Definition & Properties

• Entropy (S) = degree of randomness or disorderliness of a system.
• Change in entropy is the capacity for spontaneous changes in a system.
• Entropy is a state function.
• Entropy is an extensive property.
• ΔS is also a state function.

(Between the same two states, ΔS is the same whether the path is reversible or irreversible, because S is a state function.)

44. Qualitative Comparison of Entropy

i) Phase :

• Solid → Liquid : ΔS > 0
• Liquid → Gas : ΔS > 0
• Gas → Liquid : ΔS < 0
• Liquid → Solid : ΔS < 0
• Solid → Gas (sublimation) : ΔS > 0

ii) Volume constant, Temperature increased :

• Temperature ↑ ⇒ Entropy ↑ ⇒ ΔS > 0

iii) Temperature constant, Volume increased :

• Volume ↑ ⇒ Entropy ↑ ⇒ ΔS > 0

iv) For a chemical reaction (gaseous moles rule) :

• aA(g) + bB(g) → cC(g) + dD(g)

• Δn_g > 0 ⇒ ΔS > 0
• Δn_g < 0 ⇒ ΔS < 0

• Note : Even when Δn_g = 0, heteronuclear diatomic molecules have more entropy than homonuclear ones.
  e.g. H₂(g) + Cl₂(g) → 2 HCl(g) ; Δn_g = 0, yet ΔS > 0.

v) Different gases — Atomicity rule :

• More atomicity ⇒ more entropy.
• SO₂ (atomicity 3) > O₂ (atomicity 2)
• CH₄ (atomicity 5) > O₃ (atomicity 3)

vi) Same atomicity — Molecular weight rule :

• Higher molecular weight ⇒ higher entropy.
• SO₂ > CO₂ (both atomicity 3, but M.Wt of SO₂ > CO₂)

vii) Isomers of hydrocarbons :

• Branched isomers have less entropy than their straight-chain isomers.
• e.g. For C₅H₁₂ isomers : S(n-pentane) > S(isopentane) > S(neopentane)

viii) Mixing / Dissolution :

• During mixing of solute in solvent, entropy increases : ΔS_solution > 0.

45. Quantitative Estimation of Entropy

Entropy change of system :

dq_rev = heat exchanged by the system (reversible path) ; T = temperature at which heat is exchanged.

Entropy change of surroundings :

46. Third Law of Thermodynamics

• As the temperature of a system approaches absolute zero, the entropy of a perfect crystal approaches zero.

• According to the 3^rd Law, it is impossible to reach 0 K.
• It provides a reference point for absolute entropy.
• It helps in the calculation of absolute entropy of a substance.
• Fails for imperfect crystals and amorphous solids.

47. Entropy Change on Heating / Cooling a Substance

m = mass, s = specific heat (J g⁻¹ K⁻¹), n = moles, C = C_p or C_v as appropriate.

• For a reversible process where T_surr always equals T_sys: ΔS_surr = −ΔS_sys, so ΔS_total = 0.
• For an irreversible heating using a reservoir at fixed T_res:
  ΔS_surr = −q_sysT_res = −m s (T₂ − T₁)T_res   and   ΔS_total > 0.

48. Entropy Change — Phase Transformations

Melting (Fusion) :

• Solid → Liquid at the melting point T_trans

Boiling (Vaporisation) :

• Liquid → Vapour at the boiling point T_trans

Sublimation :

• Solid → Vapour at the sublimation temperature T_trans

49. Reversible Phase Change — ΔS_univ = 0

• When a phase change occurs at the transition temperature with surroundings at the same temperature, the process is reversible.

50. Entropy of the Universe — Criterion of Spontaneity

• ΔS_surr = −q_sysT_surr   (surroundings are large, so T_surr = const; the process in the surroundings is always reversible).
• At constant P: q_sys = ΔH_sys, so ΔS_surr = −ΔH_sysT_surr

• ΔS_sys or ΔS_surr alone is not responsible for spontaneity — only ΔS_univ decides.

51. Entropy Change for an Ideal Gas (General Formula)

This is a state function — valid for any process connecting (T₁, V₁) and (T₂, V₂).

a) Isothermal Process (T₂ = T₁) :

• For a reversible isothermal expansion: ΔS_surr = −n R ln(V₂/V₁), so ΔS_total = 0.
• For free expansion into vacuum: q = 0, so ΔS_surr = 0 and ΔS_univ = ΔS_sys.

b) Reversible Adiabatic Process :

• q_rev = 0, so ΔS_sys = 0 (isentropic). Both temperature and volume change, but the two terms in the general formula cancel.

c) Isochoric Process (V₂ = V₁) :

• ΔS_surr = −n C_v (T₂ − T₁)T_surr

d) Isobaric Process (P = const, V ∝ T) :

• At constant P: V₂/V₁ = T₂/T₁.

• ΔS_surr = −n C_p (T₂ − T₁)T_surr

52. Isothermal Process (T₁ = T₂)

• For an ideal gas: ΔU = 0 ⇒ q = −w
• Temperature factor vanishes (ln 1 = 0).

Reversible :

• q_sys = −W = nRT ln(V₂/V₁) ⇒ ΔS_surr = − q_sysT = − nRT ln(V₂/V₁)T = −nR ln V₂V₁

Irreversible :

• ΔS_sys = nR ln V₂V₁   (same states ⇒ same ΔS_sys, since entropy is a state function)
• ΔS_surr = − P_ext(V₂ − V₁)T

53. Adiabatic Process — Reversible

• q = 0 ⇒ dq_rev = 0

• Proof: ΔS_sys = nR ln V₂V₁ + nC_v ln T₂T₁
• Using TV^γ−1 = const ⇒ T₂T₁ = (V₁V₂)^γ−1
• = nR ln V₂V₁ + n(γ−1)C_v ln V₁V₂
• Since (γ−1)C_v = RC_v · C_v = R ⇒ nR ln V₂V₁ + nR ln V₁V₂ = 0

• q = 0 (no heat reaches the surroundings) ⇒ ΔS_surr = 0

54. Adiabatic Process — Irreversible

• ΔS_surr = 0 (no heat exchange with surroundings).
• ΔS_sys ≠ 0 for an irreversible adiabatic process (final state differs from reversible case).

• Entropy is a state function: (ΔS_sys)_rev from state 1→2 = (ΔS_sys)_irrev from state 1→2 (same endpoints).
• But in irreversible adiabatic, the final state (state 3) is different from the reversible final state (state 2), so ΔS_irr ≠ ΔS_rev.

Work in adiabatic processes :

• Since q = 0: ΔU = W. As ΔU is a state function, if initial and final states are the same, W_rev = W_irrev.
• However, in general the irreversible adiabatic reaches a different final state, so W is not a state function.

55. Gibbs Energy (G)

• Gibbs energy is defined at constant temperature and pressure to predict spontaneity in terms of the system alone.
• It measures the capacity of a system to do useful (non-PV) work; it is that part of total energy which can be converted into useful work.

Derivation from ΔS_univ :

• ΔS_univ = ΔS_sys + ΔS_surr = ΔS_sys − ΔH_sysT
• Multiply by −T:   −TΔS_univ = ΔH_sys − TΔS_sys

56. Effect of Temperature on Spontaneity (Four Cases)

• Case 3: At ΔG = 0, T = ΔHΔS = maximum temperature till which the process is spontaneous.
• Case 4: At ΔG = 0, T = ΔHΔS = minimum temperature above which the process is spontaneous.

57. Relation between ΔG° and K_eq

• At equilibrium: Q = K_eq and ΔG = 0.

58. Thermodynamic Partial Derivatives (Standard Results)

59. Quick Formula Revision