Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 109

The total internal energy isgiven by the integral of(S.4.21.3):

Rearranging(S.4.21.5), we have

Substituting(S.4.21.6) into (S.4.21.4), we obtain

The total gravitational energy may befound byintegrating (S.4.21.2):

Integrating by partsgives

The firstterm on the RHS of(S.4.21.9) is zero since at thelimits ofevalu-
ation either or so we have

We wish to change theintegralover into anintegraloverP. To do so,

first consider theforces on theslice ofatmosphere: