IIN

A. M. P. TANSCentral Laboratory, Staatsmiinen in Limburg, Geleen, The Netherlands

A New Type of Nomogram

Aqueous Ammonium Sulfate SolutionsA single nomogram makes it possible to estimateseveral physical constants from iust two pointsTHE field of physical properties of solutions a large number of nomograms give properties such as density, vapor pressure, and viscosity as a function of temperature and concentration. However, a separate nomogram is made for every property. If all the properties were united in one nomogram, coupled to the same temperature and concentration scale, all other properties could be read at once, knowing only two points.

Sources of Physical Properties of Aqueous Ammonium Sulfate Solutions

The density, vapor pressure, and absolute viscosity of aqueous ammonium sulfate solutions are known as functions of temperature and concentration. Its solubility is also known. All these properties were included in the nomogram. As the kinematic viscosity is much used in technical calculations, this property has been calculated and is also shown in the nomogram. Density. For the density scale the data of Beattie (2) were used, as he has correlated the best literature values. When density is plotted against concentration, a family of isotherms is obtained which are quadratic functions in concentration. The coefficients of these equations in turn appeared to be quadratic functions of the temperature. The ultimate result was the following equation : d41 = 0.9999 6.29 X C10.33 X 10 -5 Cz - (0.17 17.98 C 0.2481 C2) 10-6 t - (4.388 0.1628 C 0.00214 C2)10-6 t2 (1) wherecis by weight and tis ' C. (Table I).

O n the nomogram, the density scale should not be used below 10' C. and below 5%. It should not be used for saturated solutions, because the nomogram could not be tested in this range. Vapor Pressure. I n constructing the p scale of the nomogram, values given by Edgar and Swan ( 5 ) ,Tammann ( 72), Adams and Merz (I), and Wexler and Hasegawa (73) were used. For construction of the scale it was necessary to know under what circumstances of temperature and concentration the solutions have a definite vapor pressure. The most obvious method in such cases is making graphs, from which a table can be compiled. This was also done for drawing u p the density scale. However, this method was too inaccurate for the vapor pressure scale. Therefore an equation was computed giving vapor pressure as a function of temperature and concentration. When log P was plotted against the concentration (in grams per 100 grams of water) a t constant temperature, a family of straight lines was obtained, having the same slope. This can be expressed as:(logP)~,oonJt, A =

Viscosity. For the dynamic viscosity scale the data of Pulvermacher (77), Kanitz (8), and Grunert (7) were used. The kinematic viscosity data were calculated from the dynamic viscosity and the density. The viscosity scales on the nomogram hold only for unsaturated solutions from 20' to 80' C. and up to a concentration of 38%, Solubility. The data given by Gmelin ( 6 ) , Nishizawa (70), Britton ( 3 ) , and Caven (4) were used to find point S on the nomogram. The saturation point, S, holds u p to 50%.Use of the Nomogram

The following examples illustrate use of the nomogram and demonstrate its possibilities. 1. The most general question one can ask is: What are the density, pressure, and viscosity of a 20% solution at 50' C. and what is its solubility? Connect point 50 on the t scale with point 20 on the C scale. The points of intersection of this line with the scales in question give the desired valuesd = 1.102(1.1035); p = 86 mm. of mercury (86.5); 7 = 0.84 cp. (0.83); V = 0.76 cs. (0.752) (values in brackets taken from the literature). The solubility is found by connecting point 50 on the t scale with the saturation point, S. Intersection with the C scale gives desired solubility: 46.2% (45.8) (Table 11). 2. What is the normal boiling point of 45y0 solution? Connect point 45 on the C scale with point 760 on the p scale. The intersection with the t scale gives the boiling point: 106.4' C. (106.6). 3. .At 60' C. a solution has a density of 1.0986; what are its concentration,

- 0,001190 C'

where C' is grams per 100 grams of water. Expressing A as a function of the temperature, the following equation was obtained :Logp = 8.0635t

- 0.001190 C'(2)

+

+

1727*6 m

+

Equation holds for unsaturated solutions from IO' to 110' C. and up to 100 grams per 100 grams of water--i.e., 500jo-with an average deviation below 1yo(Table I ) ,

Table 1.t,10%

' C.10 20 30 50 80 100

Calcd.1.0599 1.0575 1.0542 1.0466 1.0304 1.0166

Lit.1.0600 1.0574 1.0542 1.0460 1.0304 1.0185

--

20%

40%

Calcd.

Lit.1.1186 1.1154 1.1117 1.1033 1.0883 1.0772

Calcd.1.2317 1.2281 1.2241 1.2156 1.2009 1.1898

r

Lit.1.2314 1.2277 1.2237 1.2153 1.2011 1.1910

10 G./100 G. HzO Calcd. Lit.8.9 17.0 30.9 89.8 344.6 738.4 9.0 17.1 31.1 90.4 347.2 742.3

50 G./100 G. HzO Calcd. Lit.

80 G./100 G. H20

Calcd.

Lit.

1.1188 1.1156 1.1180 1.1036 1.0881 1.0756

8.0 15.3 27.7 80.5 308.9 661.8

8.0 15.3 27.7 80.6 309.7 662.1

74.2 284.5 609.5

73.9 283.9 607.0

VOL. 50, NO. 6

JUNE 1 9 5 8

971

vapor pressure, and viscosity? Is this solution saturated? Draw a line through point 60 on the t scale and through point 1.0986 of the d scale. O n the scales in question we find: C = 21% (20); p = 140 mm. of mercury (139); 7 = 0.73 cp. (0.72); y = 0.66 cs. (0.655). As the line does not pass through the saturation point, S, the solution is not saturated. I n practice question 3 is important because a thermometer and a hydrometer are always available, so that temperature and density can quickly be estimated and the other properties derived by means of the nomogram. 4. What is the normal boiling point of a solution saturated at 30' C.? Draw a line through point 30 on the t scale and through the saturation point, S; connect the point of intersection of this line with the C scale with point 760 on the p scale. The second line interTable II.

sects the t scale at 106.3' C., which is the desired boiling point (106.3). 5. What is the vapor pressure of a solution saturated a t 70" C.? Draw a line through point 70 on the t scale and through point 5'. O n the p scale the pressure is found to be 181 mm. of mercury (180.7). 6. What is the concentration of a solution if its vapor pressure amounts to 300 mm. of mercury and its absolute viscosity is 0.920 cp.? Connect point 300 on the p scale with point 0.92 on the 7 scale. The desired concentration found on the C scale is 37.7% (37.5). 7. What are the concentration and the normal boiling point of a solution saturated a t its boiling point? Draw a line through the point S and through point 760 on the p scale. O n the t scale we find the boiling point, 108.3' C. (108.7) (this is also the saturation temperature, because the line passes through point S ) , and on the C scale a

saturation concentration of 51.2% (51.8). Ordinary questions can also be answered. Thus it is possible to estimate the density, viscosity, or vapor pressure a t a given temperature and concentration, or to estimate the boiling point of a given solution at a given pressure.Construction of the Nomogram

Comparison of Values Read from Nomogram with Literature Data( p in mm. of Hg.

' c.1 0 20 30 40 50 60

t,

DensityNomogr.1 * 122 1.116 1.112 1.107 1.102 1.097 1.093 1.088 1.084 1.079

7080 90 100

Lit. 1.1186 1.1154 1.1117 1.1077 1.1035 1.099 1.093 1.088 1.083 1.077

7 in centipoises. y in centistokes) Concentration, 20y0 by Weight Pressure Abs. Viscos. Kin. Viscos. Nomogr. Lit. Komogr. Lit. Nomogr. Lit.

8.6 16.9 29.9 52.5 86 141 220 330 493 712

8.6 16.4 29.6 51.7 86.5 139.6 219.2 332 491.5 710.1

1.43 1.18 0.99 0.84 0.73 0.63 0.56

...

...

... ...

1.433 1.18 0.979 0.83 0.720 0.62 0.563

... ...

1.28 1.06 0.89 0.76 0.66 0.58 0.52

...

...

1.285 1.062 0,884 0.752 0.655 0.586 0.517

...

...8 . .

t

10Qj

E mE-1000E600

Figure 1. Physical properties of aqueous ammonium sulfate solutionsEnlarged copy available from author

The nomogram was constructed according to the method of Lo-Ho ( 9 ) . For this purpose graphs and tables were prepared showing the temperature and concentration at which a given physical property of different solutions has the same value. In constructing the nomogram a T and C scale were adopted. First, the P scale was constructed, because here the greatest difficulties could be expected. Equation 2 shows that at a constant temperature log p is a linear function of the concentration, which is expressed in grams per 100 grams of water. Therefore the C scale originally was linear, with grams of 100 grams of water as the unit. The same Cscale was used in constructing other scales. Scale C was transformed into per cent by weight units. so that the final scale is not linear. The whole method is rather a matter of trial and error, in which the T and C scales are chosen to give sufficient results for all properties. Much depends on the nature of the system in question. The simple relation among P, T , and C permits an easy and accurate construction of the P scale in the case of ammonium sulfate solutions. I n other systems a simple relation may not exist. The saturation point, S, could be introduced only because the solubility of ammonium sulfate in water is a nearly linear function of temperature; in most other cases this is not true.Liferature Cited (1) Adams, J. R . , Merz, A. R., IND. ENC. CHEM. 306 (1929). 21, (2) Beattie, J. A,, Internatl. Critical Tables, vol. 3, p. 60, McGrawHill, New York, 1928. (3) Britton, H. T. S., J . Chem. Soc. 121,982 (1922). (4) Caven, R. M., Mitchell, T. C., Ibid., 125,1428 (1924). ( 5 ) Edgar, G., Swan, W.O., J. Am. Chem. Soc. 44, 570 (1922). (6) "Gmelin's Handbuch der anorganischen Chemie," 8th ed., Verlag Chemie. Berlin. 1936. (7) Grunert, H., 2.' anorg. Chem. 145, 394 (1925). ( 8 ) Kanitz, P., J . phys. Chem. Stochiom. 22, 336 (1897). (9) T,o-Ho, J. Franklin Inst. 245, 227-44 . . (1 948). (IO) Nishizawa, K., J. Chem. Ind. Tokyo 23. 25. 1015 11920). (11) Pulvknacher, O . , 2. anorg. Chem. 113,144 (1920). (12) Tammann, G., Ann. Physik. 21, 523 (1885). (13) Wexler, A., Hasegawa, S., J. Research Natl. Bur. Standards 53.19 (1954). , . RECEIVED review June 12, 1957 for ACCEPTED November 25, 1957

d

C

30j20

10

0

413

Lt4

1,290 Density scale holds above IO'C, and 5 %

3

972

INDUSTRIAL AND ENGINEERING CHEMISTRY