THERMODYNAMICS OF OPEN SYSTEMS 
Let us now consider a system containing mass mz, of 
dry air, mass m, of water vapor, and mass m, of water 
(aqueous cloud), and further let us suppose that the 
system receives water or gives it up (rain). In this case 
dm, = dima = dem =0, + dm=dim, 
d.m, = 0, AMy = dimMw + demMw, 
d;m, + dim» = 0, dm =dem=dem #0. (9) 
The equation for pseudoadiabatic transformations of 
this system can be deduced at once from (8); it follows 
that 
d;S = dS — syodem» = ie 
dm, = 0, (10) 
where A, = wy» — py is the affinity of vaporization [2]. 
If S in (10) is replaced by its value S = ma Sa (7, Pa) + 
My 8» (T, Pv) + Mw Sw (T), where p. and p, represent 
the partial pressures of dry air and water vapor, we 
obtain, after using the relation T (s, — sy) = L, + A» 
from [14], the differential equation of pseudoadiabatic 
transformations of moist air, 
ar 
if 
Here L, stands for the heat of vaporization and c, 
for the specific heat of water [14]. We suppose that the 
water vapor is at saturation value (A, = 0), and we 
assume that the water that is brought in evaporates 
as soon as it is introduced or that the water formed in 
the interior of the system leaves as soon as it is formed 
(m» = 0, dim» = —d.-m» = —d;m,). Under these con- 
ditions the pseudoadiabatic transformation of the air 
is said to be reversible and dry [9, 14]; following (11) 
d (« + a + hes = 0, 
where 7, = m,/m, represents the mixing ratio of the 
water vapor. This equation is immediately integrable; 
employing the theorem of the mean, we find [9, 14] 
Tip Lig 
d (m. Sa + a + (m, + mw) Cw 
(11) 
(12) 
(Cpa + TF Cw) In T — Ra ln pa + = const, (13) 
where 1; is a mean value of ry, Cpa is the specific heat of 
dry air at constant pressure, and FR, is the specific per- 
fect gas constant for dry air. 
The finite equation (13) of reversible, pseudoadia- 
batic transformations of air saturated with water vapor 
is one of the fundamental equations of atmospheric 
thermodynamics; we have just derived it rigorously. 
Polythermic Systems. A polythermic system is one 
in which all the phases do not have the same tempera- 
ture. Since, by hypothesis, the temperature is a quan- 
tity which is constant throughout the interior of each 
phase, we can treat each one of them as an open sys- 
tem whose state is defined by the temperature of the 
phase, by the pressure p which is assumed the same for 
533 
all phases, and by the masses of the constituents of the 
phase. The thermodynamics of polythermic systems, 
therefore, is only a special case of the thermodynamics 
of open systems [7, 14]. 
In order to establish our ideas, let us treat a closed 
polythermic system consisting of two phases: a gaseous 
phase (first phase) and a liquid phase (second phase). 
The gaseous phase comprises a mass m, of dry air and 
a mass m, of water vapor at temperature 7”; the liquid 
phase, a mass m, of water at temperature 7”; the 
two phases are assumed to be at atmospheric pressure 
p. In this case we have dim, = dim, = dimy» = 0, 
dem, = 0, dem» + dem» = 0, dm, = 0, dm, = dam, 
dm» = dmy,. Application of the first law (2) to each 
of the two phases [7, 14] leads to the relation 
[(@’Q)” + h’dm,] + [(d’Q)’ + h"dm.| = 0, (14) 
which shows that the heat (d’Q)” received by the first 
phase (gas) from the second phase (liquid) is not equal 
and opposite in sign to the heat (d’Q)’ given up by the 
first phase to the second phase. Furthermore, by a 
similar application [7, 14] of the second law (5) we 
obtain 
_ @Q* 
gp 
dS 
4 GQ, ( 1 
y TAN, 
T" pu Te (d Q) 
Pal) ee roi Al Dv) 
oF | pe ; qT! 
i le a 7 hal”) | dy, 
where (d’Q)* and (d’Q)* represent the heats added to 
the first and second phases by the surroundings of the 
system. It is clear that d’Q = (d’Q)* + (d’Q)” and 
d’Q = (d’Q)* + (d’Q)’, where d’Q and d’Q are the 
quantities of heat received by each of the two phases. 
Equation (15) shows that the increase dS in the entropy 
S of the system consisting of two phases includes: 
1. The change of entropy due to influx of heats 
(d’Q)* and (d’Q)* from the surroundings, and 
2. The production of entropy resulting from ex- 
changes of heat (d’Q)’ and matter dm, between the 
phases. The second law states that there definitely is 
production and never destruction of entropy in the 
interior of the system. 
Finally, the first law applied to the closed system 
consisting of the two phases provides the relation 
dQ = dH — Vap, (16) 
where dQ = (d’Q)* + (d’Q)*, and where the enthalpy 
H depends among other things on the temperatures 
T’ and T” of the first (gas) phase and the second 
(liquid) phase, respectively. 
The fundamental equations (14), (15), and (16) allow 
studies to be made of the “horizontal mixing” of two 
air masses of different temperature and of the evapora- 
tion of rain in the free atmosphere [14]. 
The Temperature of the Wet-Bulb Thermometer 
and the Equivalent Temperature. Let us consider a 
closed system consisting of a mass m, of dry air, a mass 
(15) 
