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# Pressure-Volume Diagrams

## Discussion

### math, math, math

Recall from the previous section…

U = Q + W

 Q > 0 system absorbs heat from the environment Q < 0 system releases heat to the environment 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 P T = PV nR

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

Function of State

U = 32nRT

Function of Path: Work

W = ∫ F · ds = ∫ P dV

W = − area on PV graph

Function of Path: Heat

Q = ∆U + W = ncT

 cP = specific heat at constant pressure cV = specific heat at constant volume

### 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 • 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 PVγ = constant

 γ = cP = α + 1 cV α
 3/2 + 1 = 5 monatomic 3/2 3 5/2 + 1 = 7 diatomic 5/2 5

liquids

solids