AN EQUATION OF STATE FOR SOIL NITROGEN BY HANS JENNY
I n this paper, soils are approached from the viewpoint of a general theory of state (allgemeine Zustandslehre) in which soil properties, soil processes, and soil-forming factors are united into a comprehensive system. At present, field soil investigations are still largely in a descriptive stage in which emphasis is laid on the accumulation of data. This is quite a natural and necessary phase in the development of a young science. Fortunately there are now available sufficient data, a t least along certain lines, to permit a search for functional relationships. If one is aware that soil formation is a chemical process in the broadest sense (including physiochemical and biochemical reactions), the possibility of combining soil characteristics and their conditioning parameters into a series of mathematical equations is a t once suggested. Such equations of state’ would completely describe the behavior of soil properties under any external conditions and, a t the same time, put field soil investigations upon an adequate scientific basis. The finding of such a “characteristic equation” for soil constituents is the object of this paper. Of the various soil properties that have been studied, the total nitrogen content, as determined 6y the Kjeldahl method, has probably attracted most attention. It has been found that the amount of total soil nitrogen varies greatly with soil texture, topography, nature of substrata, vegetation and climate. The last one of these factors is of special interest from the equation of state viewpoint, because it is a determining, external factor, not inherent in the parent material, and one which can be measured and expressed numerically. For mature soils, having similar texture, topography and vegetation, the following equation will thus hold: Soil nitrogen = f (climate) (1) From the standpoint of soil formation, the temperature, the precipitation and the evaporation are the most important climatic factors. Precipitation and evaporation are often combined into a quotient, called “moisture factor” or “humidity factor,” which allows one t o write equation (I) in the form: N = f(T, H) where Tu’ represents total nitrogen content of soil, T, air temperature and H, humidity factor. For the solution of equation ( 2 ) which contains one dependent (N) and two independent (T,H) variables, it is advisable to write it in the form of a differential equation: dN =
(g):T (g) f
dH
(3)
If the nature of the two partial differentialcoefficients (aN/dTa) and ( ~ N / ~ H ) T be known, one might be able to integrate equation (3) and obtain a solution of equation ( 2 ) .
’ G. S . Lewis and bl. Randall: “Thermodynamics”,
27 (1923).
( d S / d T ) H represents the first differential coefficient of the function
=f(T) which connects soil nitrogen with temperature in regions of constant humidity factors. This so-called nitrogen-temperature relation was previously studied for the semiarid, semihumid and humid regions of the Cnited States.' For loamy grass-land soils the relation is given by the a -kT (where a and k are constants) which was obtained equation [FIE= 1 S e from theoretical reasoning, based on the effect of temperature on the activity of microorganisms. This nitrogen-temperature relation, however, can also be described satisfactorily by the simpler empirical equation : [x]H
~
where K = average total nitrogen content of upland loamy grassland soils (surface 6-8 inch section) in per cent. T = mean annual centigrade temperature (oo-zzo, corresponding to the temperature range from Canada to the Gulf of Mexico.) C1 = constant, which includes H. kl = absolute constant (within experimental range). Taking logarithms, equation (4) becomes: 1 0 & [ N ] ~= lOg.C1 - klT (5) I t s differential coefficient has the form: (dS/dT)R = -klN (6) which is the required first partial derivative of the differential equation ( 3 ) . ( ~ N / B H ) Trepresents the first differential coefficient of the function [ N I T = f(H) which connects soil nitrogen with humidity factors in regions of constant temperature. This so-called nitrogen-humidity factor relation has also been investigated previously.2 For loamy grass-land soils in the temperate and subtropical regions of the United States, the relation was found t o be of the form: [NIT = Ai(I-e-k?LI) (7) where li = average total nitrogen content of upland loamy grass-land soils (surface 6-8 inch. section). in per cent. H =humidity factor, expressed as annual 1;.S. Quotient3 (0-400). h =constant, which includes T. kp =absolute constant (within experimental range) Taking logarithms, equation ( 7 ) becomes: log, [NIT = log, A log, ( I (8) 1 Its differential coefficient has the form:
+
soil Science, 27, 169-188 (1929). Soil Science, 29, (1930) (in print). A substitute for the true precipitation-evaporation ratio was used, namely, the socalled N. S. Quotient, which is obtained by dividing the precipitation by the absolute saturation deficit of the air. 2 3
AN EQU.4TION O F STATE FOR SOIL NITROGES
1055
which is the required second partial derivative of the differential equation ( 3 ) . Combining the partial differential equations ( 7 ) and (9) with the differential equation (3) results in:
After dividing the entire equation by ?i, the integration can be performed at sight. Assuming that k, and k2 are absolute constants, the integrated equation takes the form: log, S = - kl T
+ log,(I-e-ka)+c
(11)
or
h- = Ce-k'T( I -e-ka)
(12)
Since this equation is entirely an empirical one, the observational limits for T = oo - 2 2 ' and H = o - 400 should be constantly kept in mind. Equation ( I 2 ) gives the following information regarding the occurrence of soil nitrogen: I) If H = o also K = 0,or in other words, in desert regions the nitrogen content of the soil tends to be very low, no matter whether the deserts lie in northern or southern zones. 2) With an increasing humidity factor, soil nitrogen increases logarithmically. The rate of increase is greatest in northern regions (Canada) and smallest in southern regions (Texas). 3) With increasing temperature, soil nitrogen decreases exponentially. The rate of decrease is greatest in humid regions and smallest in arid regions. Southern regions have less nitrogen in the soil than northern regions, provided equal moisture districts are compared. As to the numerical magnitudes of the constants, the following experimental values were previously obtained:
TABLE I Values of the Constants Regions
Semiarid Semihumid Humid Temperate Subtropical
Humidity factor ( N . S. Q.)
kl
-
0.073
I25
250
- 380 - 420 280 300 -
0.095 0 .I O I
-
kr
Temperature
-
1 0 . 6~ 17.8'-
-
-
11.7'c
0.0034
20.0°C
0.0073
On account of a certain heterogeneity of the analytical material, some arbitrary selection in choosing the constants cannot be avoided. The following values satisfy the equation for a first approximation: kj = 0.08 ki = 0.005 C = 0.55
1056
c (
0
-E m I
HANS JENNY
AN EQUATION OF STATE FOR SOIL N I T R O G E S
Equation
(12) takes
IOj7
then the form:
h- =
.3 ,5e-0.08T(I -e--O.OOSH
1
(13)
.A comoarison between calculated and observed average soil nitrogen values is given in Table 11, and the corresponding nitrogen plane is shown in the three-dimensional graph of Fig. I . I
FIG.I T h e nitrogen content of loamy grassland soils in the United States as a function of annual temperature and annual humidity factor (N.S.Q.) The curves express the approximate trend of soil nitrogen of large areas.
Although this investigation is based on more than 1000 soil nitrogen values equation ( 1 3 ) must not be considered as final. From certain large areas no nitrogen analyses could be secured (e.g. Oklahoma, South Dakota) and in general the number of analyses from arid regions is too small to detertnine accurately the magnitude of the constant k2. One should also remember that equation ( I O ) was integrated on the assumption that kl and k2 are absolute constants. I t is quite possible that, in refining equation (1,3), a variation of the constants themselves may occur. Nevertheless, the agreement between calculated and observed data indicates that equation ( 1 3 ) is satisfactory for a first approximation, and, moreover, it shows that an equation of state viewpoint in field soil investigations can be applied with success. Department of Soils, IJrLiiIersity of Missouri. COliLrnbiQ, 210.
