Overview
This lecture covers adiabatic processes for an ideal gas, including their definition, equations, and differences from isothermal processes, with an application to engine compression.
Adiabatic Processes: Definition and Qualitative Features
- An adiabatic process occurs when no heat is exchanged with the environment (Q=0).
- In adiabatic compression, work done on the gas increases its temperature.
- In adiabatic expansion, the gas does work and its temperature decreases.
- Real-life example: quick compression in car engines approximates an adiabatic process.
Free Expansion and the Ideal Gas
- Free expansion is when gas expands into a vacuum in a thermally insulated system.
- In free expansion, both Q and work (W) are zero, so the internal energy is constant.
- For an ideal gas, internal energy only depends on temperature, so temperature remains unchanged during free expansion.
Mathematical Formulation of Adiabatic Processes
- For quasi-static (slow) adiabatic expansion: dW = p dV and dQ = 0.
- For ideal gases, dE_int = C_V n dT, so C_V n dT = –p dV.
- Combining with the ideal gas law yields: C_V (dP/P) + C_P (dV/V) = 0.
- Define γ (gamma) as the ratio of heat capacities: γ = C_P / C_V.
- Adiabatic condition: PV^γ = constant.
- Alternate forms: P^(1–γ) T^γ = constant, and T V^(γ–1) = constant.
Adiabatic vs. Isothermal Expansions
- On a p–V diagram, an adiabatic curve is steeper than an isothermal curve.
- Isothermal expansion keeps temperature (and PV) constant, while adiabatic does not.
Example: Adiabatic Compression in an Engine
- Compression modeled as quasi-static adiabatic process: PV^γ = constant.
- For a diatomic gas, γ ≈ 1.4.
- Final pressure calculated as P2 = P1 (V1/V2)^γ.
- Final temperature from ideal gas law: T2 = (P2V2/P1V1)T1.
- Work done is W = [P2V2 – P1V1]/(1–γ); negative sign means work is done on the gas.
Key Terms & Definitions
- Adiabatic Process — process with no heat transfer (Q = 0).
- Free Expansion — expansion into a vacuum with W = 0 and Q = 0.
- Quasi-static Process — a process that happens infinitely slowly, maintaining equilibrium.
- γ (gamma) — ratio of molar heat capacities, γ = C_P / C_V.
- Isothermal Process — process at constant temperature.
Action Items / Next Steps
- Practice problems involving adiabatic processes and engine compression.
- Review and memorize adiabatic process equations and γ values for monoatomic and diatomic gases.