Thermodynamic Processes: Isothermal, Adiabatic, Isobaric & Isochoric

A thermodynamic process occurs when a system transitions from an initial state of equilibrium to a final state of equilibrium through a change in its state variables (P, V, T). The First Law of Thermodynamics (ΔQ = ΔU + ΔW) governs all such transformations.

1. Isothermal Process (T = Constant)

A process in which the temperature of the system remains strictly constant throughout (ΔT = 0).

• Equation of State: Boyle's Law: P V = constant ⇒ P1 V1 = P2 V2.
• Conditions: Walls of container must be perfectly conducting (diathermic), and the process must be carried out very slowly to allow continuous heat exchange with surroundings.
• Internal Energy Change: Since T is constant, for an ideal gas: ΔU = n Cv ΔT = 0.
• First Law Application: ΔQ = ΔW. All heat added is converted completely into work done.
• Work Done:
  W = ∫_V1^V₂ P dV = n R T ∫_V1^V₂ (dV / V) = n R T ln(V₂ / V₁) = 2.303 n R T log₁₀(V₂ / V₁)
  
  W = 2.303 n R T log₁₀(P₁ / P₂)
• Slope on P-V Diagram: Differentiating P V = const gives P dV + V dP = 0:
  (dP / dV)_iso = − P / V

2. Adiabatic Process (ΔQ = 0)

A process in which no heat enters or leaves the system across its boundaries.

• Conditions: Container walls must be perfectly insulating (adiabatic), and the process must occur extremely rapidly (e.g., bursting of a tire, propagation of sound waves in air) so heat has no time to transfer.
• Governing Poisson Relations:
  P V^γ = constant
  
  T V^γ−1 = constant
  
  P^1−γ T^γ = constant   (or P / T^γ/(γ−1) = const)
• Internal Energy Change: From First Law, ΔQ = 0 ⇒ ΔU = −ΔW.
  If gas expands adiabatically (W > 0), ΔU < 0 ⇒ gas cools down.
  If gas is compressed adiabatically (W < 0), ΔU > 0 ⇒ gas heats up.
• Work Done in Adiabatic Process:
  W = (P₁ V₁ − P₂ V₂) / (γ − 1) = n R (T₁ − T₂) / (γ − 1)
• Slope on P-V Diagram: Differentiating P V^γ = const gives:
  (dP / dV)_adi = − γ (P / V) = γ × (dP / dV)_iso
  Because γ > 1, the slope of an adiabatic curve is always γ times steeper than the slope of an isothermal curve passing through the same state point (P, V).

3. Comparison of Isothermal & Adiabatic Expansions and Compressions

4. Isobaric Process (P = Constant)

A process conducted at constant pressure (ΔP = 0).

• Equation: Charles's Law: V / T = constant ⇒ V1 / T1 = V2 / T2.
• Work Done: W = P (V2 − V1) = n R ΔT.
• Internal Energy Change: ΔU = n Cv ΔT.
• Heat Supplied: ΔQ = n Cp ΔT.
• Partitioning of Heat:
  Fraction of heat utilized to increase internal energy: ΔU / ΔQ = C_v / C_p = 1 / γ
  
  Fraction of heat utilized in external work: ΔW / ΔQ = 1 − (1 / γ) = (γ − 1) / γ

5. Isochoric (Isometric) Process (V = Constant)

A process occurring in a rigid container where volume does not change (ΔV = 0).

• Equation: Gay-Lussac's Law: P / T = constant ⇒ P1 / T1 = P2 / T2.
• Work Done: W = ∫ P dV = 0 (since dV = 0).
• First Law Application:
  ΔQ = ΔU = n C_v ΔT
  All heat supplied is stored as internal energy and increases temperature.
• Slope on P-V Diagram: Vertical line (slope = ∞).

6. Summary Comparison Table of All Four Processes

7. Polytropic Process (P Vⁿ = Constant)

Any generalized quasi-static process can be represented by P Vn = constant where n is the polytropic index:

• If n = 0 ⇒ P = const (Isobaric)
• If n = 1 ⇒ P V = const (Isothermal)
• If n = γ ⇒ P V^γ = const (Adiabatic)
• If n = ∞ ⇒ V = const (Isochoric)