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Thermodynamics filled in class notes_Part_117

# Thermodynamics filled in class notes_Part_117 - 241 7.1...

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7.1. ISOTHERMAL, ISOCHORIC KINETICS 241 Near the first non-physical root at ρ e O = 0 . 003, a positive perturbation from equi- librium induces f < 0, which induces d ρ O /dt < 0, so ρ O returns to its equilibrium. Similarly, a negative perturbation from equilibrium induces d ρ O /dt> 0, so the system returns to equilibrium. This non-physical equilibrium point is stable. Note stability does not imply physicality! Perform the same exercise for the non-physical root at ρ e O = 0 . 000518944. We find this root is unstable . Perform the same exercise for the physical root at ρ e O = 0 . 000442414. We find this root is stable . In general if f crosses zero with a positive slope, the equilibrium is unstable. Otherwise, it is stable. Consider a formal Taylor series expansion of Eq. (7.85) in the neighborhood of an equi- librium point ρ 3 O : d dt ( ρ O ρ e O ) = f ( ρ e O ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright =0 + df d ρ O vextendsingle vextendsingle vextendsingle vextendsingle ρ O = ρ e O ( ρ O ρ e O ) + ... (7.86)

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