Equilibrium Processes in Gases

Slow (almost unchanging) with respect to the typical relaxation times of a system

Equilibrium processes are represented by lines in the state space

Processes in Ideal Gases

Constant volume (isochoric) processes:

No work done

TcnUQ v 0dV)V(PWf

0

V

V

1. law: WQU Q

V=const.

0

Equilibrium process: Quasi-static process which transforms the system through states of thermal equilibrium

f

0

V

V

dV)V(PW

Constant pressure (isobaric) processes

P=const.

1. law: WQU

WUQ )VV(PUU 0f0f

Q

m

P=const.)VV(PdVP 0f

V

V

f

0

Tcn P

Isothermal processes

Work done @ T=const. f

0

V

V

dV)V(PW f

0

V

V

dVV

nRT

T=const.

f

0

V

VV

dVnRT

0

f

V

VlnnRT

TcnU V

Adiabatic processes: No heat heat transfer in or out of the system Q=0

How to realize an adiabatic process

1

2

very good insulation

rapid changes in volume no time for heat to be exchanged

1. law: WQU

0

0WU

Note: although the constant volume specific heat appears, expression applies to any process that starts and ends in equilibrium

VV

V

0

0

dV)V(PW

V0

VP

In the limit V0

0VPTcn V

V

nRTP

0V

VnR

T

Tcn V

T

1

R

c

dT

dV

V

1 VV0

First order differential equation

or

:

T

1

R

c

dT

dV

V

1 VSolution of by integration T

dT

R

c

V

dV V

.constTlnR

cVln V with Rcc VP 1

c

RV

P

cc

V

1c

R

V

__.constTlnVln1

__1 .constTlnVln

__1 .constTVln .constTV 1

nR

PVT with

.constPV

different from constant above

What is the work done in an adiabatic equilibrium process

.constPV during entire adiabatic process

Starting at P0 and V0 00VPPV

V

VPP 00

f

0

V

V

dV)V(PW

V

VPP 00

f

0

V

V00

V

dVVP

10

1f

00

V

1

V

1

1

VP

Graphic summary of various processes on a PV diagram:

Constant volume(isochoric) process

Constant pressure(isobaric) process

Constant temperature(isothermal) process

V

.constP

Thigh

Tlow

Isotherms for reference

adiabatic processQ=0

V

.constP 14.1

Adiabat steeper than isotherm