Pressure-Volume Diagrams

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

Math, math, math!

Recall from the previous section …

ΔU = Q + W
Q > 0		system absorbs heat
Q < 0		system releases heat
W > 0		work done on the system by the environment
W < 0		work done by the system on the environment

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.

		⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩	P =	nRT
				V
PV = nRT	⇒		V =	nRT
				V
			T =	PV
				nR

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

Function of State

ΔU =	3	nRΔT
	2

Function of Path: Work

W = ∫ F · ds = ∫ P dV = − area on PV graph

Function of Path: Heat

Q = ΔU + W = ncΔT
cP	specific heat at constant pressure
cV	specific heat at constant temperature

• 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 = −PΔV

  ⇒

  ΔU = Q − PΔV

  
  [magnify]

• 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

  ⇒

  ΔU = Q

  
  [magnify]

• 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

  ΔU = 0

  ⇒

  Q = −W

  
  [magnify]

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

  Q = 0

  ⇒

  ΔU = W

  
  [magnify]

  =

  ⎧ ⎪ ⎨ ⎪ ⎩

  3/2 + 1

  =

  5

  monatomic

  PVγ = constant

  ⇒

  γ =

  cP

  =

  α + 1

  3/2

  2

  cV

  α

  5/2 + 1

  =

  7

  diatomic

  5/2

  5

 

Superman illustrates adiabatic cooling brought about by the rapid expansion of a gas, thus preventing the evil General Zod from heating the truck's fuel tank to the point of explosion. Thank you Superman. You've saved us. [magnify]

liquids

solids

Summary

• bullet

Problems

practice

1. one mole ideal gas, rectangular cycle ABCDA on a PV diagram, find
   1. temperature at A, B, C, D
   2. ΔU, W and Q on AB, BC, CD, DA and total around one cycle
   3. engine or refrigerator?

   	A	B	C	D
   T
   	AB	BC	CD	DE	total
   W
   Q
   ΔU

   • Answer it.
2. Write something else.
   • Answer it.
3. Write something different.
   • Answer it.
4. Write something completely different.
   • Answer it.

numerical

1. Estimate the power of a relaxed human heart given an average blood pressure of 13 kPa, a volume change of 80 ml per beat, and a pulse rate of 72 beats per minute.
2. Estimate the power of a pair of relaxed human lungs from the graph below given a respiration rate of 20 breaths per minute.

   [magnify]

   
   [magnify]
3. One mole of an ideal gas runs through the zilch cycle, which consists of the following four processes …

   A→B	isothermal expansion at 600 K with 2.30 kJ of work done by the gas
   B→C	isochoric pressure drop
   C→D	isothermal expansion resumes at 300 K
   D→A	adiabatic compression returns the gas to its original state with 3.74 kJ of work done on the gas

   The cycle is rigged so that no net work is done over a complete cycle. The zilch cycle has no practical use. It's just an interesting thought problem. Given this information, complete the following table.

   process	ΔU(kJ)	Q(kJ)	W(kJ)
   A→B			−2.30
   B→C
   C→D
   D→A			+3.74
   ABCD			0

Resources

• miscellaneous
  • Respiratory Mechanics, Novametrix Medical Systems
  • Arrott, Anthony S. The zilch cycle: An application of the first law of thermodynamics. American Journal of Physics. Vol. 45, No. 7 (1977): 672-673.

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