Pressure-Volume Diagrams
math, math, math
Pressure-Volume Diagrams Recall from the previous section…
ΔU = 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
ΔU = | 3 | nRΔT |
2 |
Function of Path: Work
W = ∫ F · ds = ∫ P dV
W = − area on PV graph
Function of Path: Heat
Q = ΔU + W = ncΔT
c_{P} = | specific heat at constant pressure |
c_{V} = | 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 = −PΔV ⇒ ΔU = Q − PΔV - 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 - 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 - 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 PV^{γ} = constant
γ = c_{P} = α + 1 c_{V} α 3/2 + 1 = 5 monatomic 3/2 3 5/2 + 1 = 7 diatomic 5/2 5
… and the rest
liquids
solids