Pressure-Volume Diagrams

Discussion

math, math, math

Recall from the previous section…

∆U = Q + W

A system can be described by three thermodynamic variables — pressure, volume, and temperature. Well, maybe it's only two variables. With everything tied together by the ideal gas law, one variable can always be described as dependent on the other two.

Temperature is the slave of pressure and volume on a pressure-volume graph (PV graph).

Function of State

∆U = ³₂nR∆T

Function of Path: Work

W = ∫ F ⋅ ds = ∫ P dV

W = − area on PV graph

Function of Path: Heat

Q = ∆U + W = nc∆T

curves

• isobaric
  • constant pressure
  • "bar" comes from the greek word for heavy: βαρύς [varys]
  • examples: weighted piston, flexible container in Earth's atmosphere, hot air balloon
  • PV graph is a horizontal line

  W = −P∆V

  ⇒

  ∆U = Q − P∆V

  

• isochoric
  • constant volume
  • "chor" comes from the greek word for volume: χώρος [khoros]
  • examples: closed rigid container, constant volume thermometer
  • PV graph is a vertical line

  W = 0

  ⇒

  ∆U = Q

  

• isothermal
  • constant temperature
  • "therm" comes from the greek work for heat: θερμότητα [thermotita]
  • examples: "slow" processes, breathing out through a wide open mouth
  • PV graph is a rectangular hyperbola

  ∆U = 0

  ⇒

  Q = −W

  

• adiabatic
  • no heat exchange with the environment
  • adiabatic has a complex greek origin that means "not+through+go": α + ∆ια + βατός [a + dia + vatos]
  • examples: "fast" processes, forcing air out through pursed lips, bicycle tire pump
  • PV diagram is a "steep hyperbola"

  Q = 0

  ⇒

  ∆U = W

  

  PV^γ = constant

  γ =	cP	=	α + 1
  	cV		α

  3/2 + 1	=	5	monatomic
  3/2		3
  5/2 + 1	=	7	diatomic
  5/2		5

Superman saves the day with adiabatic cooling. The rapid expansion of the gas expelled from Superman's lungs cooled the overheating truck, thwarting the evil General Zod's attempt to explode its fuel tank. Thank you Superman.

… and the rest

liquids

solids