Mechanical Engineering

Maxwell relations come from the Schwarz theorem applied to thermodynamic potentials (U, H, F, G). For example, from dG = -SdT + VdP: (∂S/∂P)_T = -(∂V/∂T)_P.

For Brayton cycle: η = 1 - 1/r_p^((γ-1)/γ). As pressure ratio r_p increases, efficiency increases. This is why modern turbojets use high pressure ratios.

For a reversible process: ΔS_universe = ΔS_system + ΔS_surroundings = 0. The system's entropy can increase or decrease, but total entropy doesn't change.

Clausius-Clapeyron: dp/dT = L/(T·Δv), where L is latent heat of vaporization and relates pressure-temperature relationship at phase equilibrium.

For Carnot refrigerator: COP_ref = T_C/(T_H - T_C) = 250/(300-250) = 250/50 = 5.

The throttling process is isenthalpic (constant enthalpy). The enthalpy remains the same before and after the expansion valve.

For a monatomic ideal gas with 3 translational degrees of freedom: Cv = (3/2)R. For diatomic: (5/2)R; for polyatomic: (7/2)R.

For an irreversible process in an isolated system, total entropy increases (ΔS_total > 0). Heat transfer between bodies at different temperatures is irreversible.

First Law: ΔE_system = Q - W, where Q is heat added to system and W is work done by system. This is the IUPAC convention.

When n = 0, PV⁰ = P = constant, which is an isobaric process. When n = 1 (isothermal), n = γ (adiabatic), n = ∞ (isochoric).