Thermodynamics and Chemistry

APPENDIX G FORCES, ENERGY, AND WORK

G.8 CENTER-OF-MASSLOCALFRAME 501

The difference between the energy changes of the system in the cm frame and the lab
frame during a process is given, from Eqs.G.2.6andG.6.5, by

ÅUÅEsysDÅ

"

X

i

1

2 mi.v

0

i/

(^2) X
i
1
2 miv
2
i

CÅ

X

i



(^0) field
i

X

i

ifield

!

CÅ

X

i

iaccel

!

(G.8.6)

We will find new expressions for the three terms on the right side of this equation.
The first term is the difference between the total kinetic energy changes measured in
the cm frame and lab frame. We can derive an important relation, well known in classical
mechanics, for the kinetic energy in the lab frame:
X

i

1

2 miv

2

iD

X

i

1

2 mi.vcmCv

0

i/.vcmCv

0

i/

D^12 mv^2 cmC

X

i

1

2 mi

v^0 i

2

Cvcm

X

i

miv^0 i

!

(G.8.7)

The quantity^12 mvcm^2 is the bulk kinetic energy of the system in the lab frame—that is, the
translational energy of a body having the same mass as the system and moving with the
center of mass. The sum

P

imiv
0
iis zero (Eq.G.8.4). Therefore the first term on the right
side of Eq.G.8.6is

Å

"

X

i

1

2 mi.v

0

i/

(^2) X
i
1
2 miv
2
i

DÅ

1

2 mv

2

cm

(G.8.8)

Only by using a nonrotating local frame moving with the center of mass is it possible to
derive such a simple relation among these kinetic energy quantities.
The second term on the right side of Eq.G.8.6, with the help of Eqs.G.2.2,G.6.3, and
G.8.2becomes

Å

X

i



(^0) field
i

X

i

ifield

!

D

X

i

Z

Fifieldd.ri^0 ri/

D

Z^ X

i

Fifield

!

dRcm (G.8.9)

Suppose the only external field is gravitational:FifieldDFigravD migezwhereezis a
unit vector in the vertical (upward)Czdirection. In this case we obtain

Å

X

i



(^0) field
i

X

i

ifield

!

D

Z^ X

i

mi

!

gezdRcm

Dmg

Z

ezdRcmDmg

Z

dzcm

DmgÅzcm (G.8.10)