Adiabatic Free-Expansion - UW Oshkosh — University of Wisconsin

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Adiabatic Free-Expansion A B vacuum Q = 0 <— adiabatic thermal energy transfer to gas? work done by gas? W = 0 <— free 1st Law: ΔU = Q W = 0 eqn of state: U = 3 2 nRT so ΔU = 3 2 nRΔT and T A = T B Adiabatic Free-Expansion: Entropy Change ΔS gas = dQ rev T i f = ? process is irreversible — the gas will not spontaneously occupy one half of container —> entropy change must be positive P V A B to calculate change in entropy, connect A and B by a reversible, quasi-static isothermal “model” process ΔS gas: adiabatic freeexpansion = ΔS gas: quasi static isothermal expansion = nR ln V B V A > 0 reservoir Q V A V B quasi-static isothermal expansion System Entropy Change A B vacuum ΔS system = ΔS gas = nR ln V B V A > 0 irreversible ΔS system = ΔS gas + ΔS reservoir gas adiabatic free-expansion reversible ΔS system = +Q T + Q T = 0

Transcript of Adiabatic Free-Expansion - UW Oshkosh — University of Wisconsin

Page 1: Adiabatic Free-Expansion - UW Oshkosh — University of Wisconsin

Adiabatic Free-ExpansionA

B

vacuum

Q = 0 <— adiabatic

thermal energy transfer to gas?

work done by gas?

W = 0 <— free

1st Law: ΔU =Q −W = 0

eqn of state: U = 32nRT so ΔU = 3

2nRΔT

and TA = TB

Adiabatic Free-Expansion: Entropy Change

ΔSgas =dQrev

Ti

f

∫ = ?process is irreversible — the gas will not spontaneously occupy one half of container

—> entropy change must bepositive

P

V

A

B

to calculate change in entropy, connect A and B by a reversible, quasi-static isothermal “model” process

ΔSgas: adiabaticfree−expansion

= ΔSgas: quasi−staticisothermalexpansion

= nR lnVBVA

> 0

reservoir

gasQ

VA VB

quasi-static isothermal expansion

System Entropy ChangeA

B

vacuum ΔSsystem = ΔSgas = nR lnVBVA

> 0 irreversible

ΔSsystem = ΔSgas + ΔSreservoir

gas

adiabatic free-expansion

reversible

ΔSsystem = +QT

+ −QT

= 0

Page 2: Adiabatic Free-Expansion - UW Oshkosh — University of Wisconsin

Work Done by Gas

ΔSsystem > 0

A

B

vacuum

irreversible adiabatic free-expansion

ΔSsystem = 0

reservoir

Q

VA VB

reversible quasi-static isothermal expansion

gas

<— entropy change —>

Wa. f .e = 0 Wq.s.i.e. = nRT lnVBVA

<— work done —>

Note that energy of each system did not change, yet in one case the gas did work while in the other it did not

System ISystem II

Availability of Energy, or “Lost Work”• work could have been done by System I

—> the quality of the energy was lowered byfree-expansion/irreversible process

• in an irreversible process energy equal to becomes unavailable to do work, where is thetemperature of the coldest available reservoir—> this is known as “lost work”

TCΔSUniverseTC

Wlost = TCΔSUniverse

Entropy as a Measure of Quality

• example – heat engine:—> extracts thermal energy at high temperature—> exhausts thermal energy at low temperature

this energy is of lower quality because we need an even lower temperature reservoir to convert

some of it to work• given an amount of energy, it is high quality & lowentropy if it is localized, coherent, or at a high temp

QHQL