ABSTRACT
S E V N S E V N S E V N, , , , , , .( ) = ( ) + ( )1 1 1 1 2 2 2 2 (11.3) e entropies of systems 1 and 2 are added to give the entropy for the
combined system, as required for an extensive quantity. In equilibrium,
according to the second law, the entropy is a maximum. It follows that
d d dS S E
E S V
= =
∂ ∂
⎛
⎝
⎜
⎞
⎠
⎟
+ ∂ ∂
⎛
⎝
⎜
⎞
⎠
⎟
+ ∂ ∂
⎛
⎝
⎜
⎞
⎠
+ ∂ ∂
⎛
⎝
⎜
⎞
⎠
⎟
+ ∂ ∂
⎛
⎝
⎜
⎞
⎠
⎟
+ ∂ ∂
N
S E
E S V
V S N
d
d d 2
⎛
⎝
⎜
⎞
⎠
⎟
.d (11.4)
e constraints given in Equation 11.2 imply that dE1 = -dE2, dV1 = -dV2 , and dN1 = -dN2. Equation 11.4 may therefore be rewritten as
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎤
⎦
⎥
With the deˆnitions given in Section 3.12 on the basis of the general form of the fundamental relation, this equation becomes
1 1 0 1 2
2 1T T
E P T
P T
V T T
N−⎛ ⎝
⎜
⎞
⎠
⎟
+ −⎛ ⎝
⎜
⎞
⎠
⎟
− −
⎛
⎝
⎜
⎞
⎠
⎟
=d d dm m . (11.5)
Because dN1, dV1, and dE1 are independent and arbitrary in magnitude, all coeªcients in Equation 11.5 must be identically zero. e equilibrium conditions therefore become
T T P P1 2 1 2 1 2= = =, , .m m (11.6)
In equilibrium, the temperatures, pressures, and chemical potentials of the two subsystems are separately equal.
