582 
moreover that d.Q = 0 when the open system contains 
only a single constituent. 
The Second Law. Gibbs’ fundamental relation [2, 3, 5] 
dH = TdS + Vdp + Yiusdm, (4) 
J 
aG 
om; 
potential of constituent j, is valid whatever the changes 
dT, dp, dm; in state variables T, p, m; of the system; 
consequently it is applicable to open systems. By sub- 
stituting (2) into (4) it follows [7, 9, 14] that 
dS — sdm = dQ/T + dQ’/T, (5) 
where the uncompensated heat of Clausius dQ’ is de- 
fined by dQ’ = dQ’ + dQ’, in which 
dQ’ = —Lu,dam; > 0, 
J 
dQ’ = Le = uj)dm; z 0, 
J 
where p; = = h; — Ts; stands for the chemical 
(6) 
and p =h — Ts. The specific entropy of the system is 
denoted by s, and S = ms. The increase dS which the 
entropy S of an open system undergoes from time ¢ to 
time ¢ + dt (dt > 0) includes [7, 14]: 
1. The entropy change due to exchange of matter and 
energy during the time interval dt, 
dQ/T + d.Q’/T + sdm 2 0. 
2. The entropy production in the interior of the sys- 
tem during the same time interval, 
dQ'/T = 0. 
The second law states that the entropy production d;Q’/T 
due to the internal mass transformation is real, that is to 
say that the irreversibility of this transformation never 
entails destruction of entropy. By deduction from (8), 
(6), and (4), 
dQ = dQ’ = Tds. (7) 
Attention should be drawn to the fact that mass ex- 
changes d.m,; between the system and the surroundings 
must take place at the temperature 7’ and the pressure 
p of the system. This condition is usually satisfied in 
the case of a system which loses mass; when, on the 
other hand, mass enters the system the condition is no 
longer necessarily fulfilled. In the latter case it is well 
to make sure that this restriction is fully satisfied be- 
fore applying the two laws which we have just formu- 
lated. 
Fundamental Differences Between Open and Closed 
Systems. In the case of a closed system, the analytical 
expression dQ = dE + pdV for the first law consists 
of three terms for each of which there is an exact, well- 
known physical meaning. We have already seen what 
meaning is assumed by the term dQ in the case of an 
open system. It should be noted that in this more 
general case the other two terms no longer have physical 
meanings. For example, the term pdV is no longer “the 
mechanical work done by the system during time in- 
terval dt,” as a hasty generalization might lead one to 
believe [14]. Indeed, dV can be broken down into two 
DYNAMICS OF THE ATMOSPHERE 
additive terms, one of which d.V depends essentially 
on the arbitrary addition of mass from the outside. 
Furthermore, when the boundary of a system is opened 
to an exchange of matter with the environment (dm # 
0), the increase dX of any function of state X of the 
system is deprived of all physical sense. Mathemati- 
cally, one obviously has dX = mdx + xdm = >m;dx;+ 
4) 
>>2z,;dm;. However, since the specific state functions x 
J 
and «x; are determined only up to an arbitrary additive 
constant [14], the terms xdm and x;dm; of dX have 
indeterminate values. On the other hand the expression 
dX — «dm = mdz has a perfectly well-defined value, 
and consequently this is the case with expressions (2) 
and (5) for the two laws. Moreover, since the arbitrary 
constant which may be added to functions x and 2; is 
the same for all these functions [14], the differences 
x — x, and x; — x, (j, k = 1, 2, 3), have well-defined 
values, and this is the case with expressions (8) and 
(6) for d:Q, dQ, d:Q’, and d.Q’. 
In the case of a closed system, the second law can 
be written TdS — dQ = dQ’ = — Divsdm; > 0, where 
J 
dQ’ = 0 corresponds to reversible transformations and 
dQ’ > O corresponds to irreversible transformations. 
The physical meaning of the uncompensated heat of 
Clausius dQ’ appears in the simplest fashion when a 
closed isothermal cycle is considered. During such a 
change of state, the uncompensated heat of Clausius 
received by the system is equal to the excess of the 
heat effectively given up over that effectively received 
by the system. In the case of an open system, however, 
dQ’ is composed of two terms: the first depends on 
arbitrary addition or removal of mass and consequently 
may sometimes be positive and sometimes negative; 
the second depends on the internal physico-chemical 
transformation and is always positive. It is this latter 
term which actually generalizes, in the more realistic 
case of an open system, the classical concept of the un- 
compensated heat of Clausius. 
In order to make clearly evident the differences in 
the significances of dQ and dQ’, first for a closed and 
then for an open system, it is enough to set down the 
two laws for the case of a closed system consisting of 
two phases (a liquid and its vapor, for example) which 
undergoes only isobaric and isothermal transformations 
(vaporization and condensation at constant p and T), 
then to regard this system as being composed of two 
open systems (the gaseous phase and the liquid phase) 
and to see how the relations (2), (3), (5), and (6) reduce 
in this case [7]. 
Pseudoadiabatic Transformations. When the heat 
received by an open system does not alter the internal 
transformation of which the open system is the site, 
the system is said to undergo a pseudoadzabatic trans- 
formation [14]. In this case d;Q = 0, and as a result of 
(3) and (7), 
d:H = dH — Dv hj;d.m; = Vdp, 
) 
(8) 
d;S = dS — Dd sidem; = =e dams. 
) J 
