By means of such equations, it is not only easy to calculate a table which is interpolated readily but also to obtain other special values not given in the table. Furthermore, values from the equation may be compared with observed data.

It is interesting to note that in most cases values calculated from these equations agree with Spengler's observed data better than they do with his graphic data; therefore, the use of such equations throughout the range of observed concentrations is justified. These equations are as follows:

100 purity

log10 At=2.6157X10-6X3-4.0185X10-X+4.2567X10-2X-1.1979,

90 purity

log10 At=3.1013X10-6X3-5.1209×10-4X2+4.9574X10-2X-1.2622,

80 purity

log10 At=2.9431X10-6X3-4.8211X10-4X2+4.7512×10-2X-1.1533,

70 purity

log10 At=3.3138×10-6X3-5.5257×10-X2+5.1177×10-2X-1.1305,

where At is the boiling-point elevation of the solution above that of pure water at a pressure of 1 standard atmosphere, and X is the concentration in percentage of solids.

Knowing the boiling-point elevation at 1 atmosphere (At160), the boiling-point elevation at another pressure (At) may be calculated by means of the following equation:

This is the same as eq. 127, the derivation of which is given on the following pages.

2. DEFINITION OF BOILING POINT

The boiling point of a liquid may be defined as the temperature at which its vapor pressure is equal to the pressure of the surrounding atmosphere [4]. If the liquid is considered as a solution of two substances, A the solvent and B the solute, then the total vapor pressure is equal to the sum of the partial vapor pressures resulting from A and B. If, however, the solute B is as nearly nonvolatile as sugar, the total vapor pressure is that arising from the solvent only and depends on the number of solvent molecules present in the vapor phase per unit volume.

3. RELATIONSHIP BETWEEN VAPOR PRESSURE AND
CONCENTRATION OF SOLUTE IN A LIQUID

According to the law of Raoult, a definite relationship exists between the number of molecules present in the vapor phase and the number of the same molecular species present in the liquid phase. The vapor pressure of a liquid, p, is proportional to 2, the mole fraction of the liquid which exists in the form of the same molecular species as the vapor, or

p=kro.

(111)

If we consider the pure solvent by itself, in which case the same molecular species exists in both the liquid and vapor phases, then the mole fraction, ro, becomes unity, and we have the proportionality factor, k, equal to the vapor pressure, Po, or

Substracting both sides of eq 113 from Po, we have

(112)

(113)

(114)

(115)

However, by definition, xo is the mole fraction of solvent present in the liquid, or

in which is the mole fraction of solute present in the solution, N refers to the number of moles of solute present which have the molecular weight of the solute, and No is the number of moles of solvent present which have the molecular weight of the solvent in the vapor

state.

4. RELATIONSHIP BETWEEN VAPOR-PRESSURE LOWERING AND BOILING-POINT ELEVATION

From the definition of the boiling point, it is readily seen that, should the vapor pressure of the boiling solvent be lowered by the addition of a solute it will cease boiling until equilibrium is again established and the vapor pressure is raised to the same value it had before the solute was added. An increase in vapor pressure will be accompanied by a proportionate increase in the temperature or

or

dpo Po-P
dto T-To

dpo
T-To-Po-P÷ dto

(118)

(119)

in which T-To is the elevation of the boiling point of the liquid which takes place in reestablishing equilibrium between the vapor pressure of the liquid and the surrounding atmosphere, and dpo/dto is the rate of change of vapor pressure of the solvent with change in temperature.

or

5. RELATIONSHIP BETWEEN BOILING-POINT ELEVATION AND CONCENTRATION OF SOLUTE

Substituting in eq 118 the value of Po-p from eq 117, we have

in which Bo is the boiling-point constant and represents the relation existing between the vapor pressure of the solvent and the rate of change in vapor pressure of the solvent with temperature. T-To or AT is the boiling-point elevation at an atmospheric pressure corresponding to To.

This equation is satisfactory for dilute solutions and is useful in determining the molecular weight of various substances. At high concentrations, however, the equation does not hold, because it does not correct for the mutual attraction between the solute and solvent molecules; therefore the relationship existing between the boilingpoint elevation and the concentration must be determined experimentally for each substance.

6. RELATIONSHIP BETWEEN BOILING-POINT ELEVATION AND ATMOSPHERIC PRESSURE

The boiling-point constant at any atmospheric pressure may be determined from the approximate Clausius-Clapeyron equation

in which R is the universal gas constant and AH, the molal heat of vaporization of the solvent at its boiling point, To.

If eq 122 is substituted for its value in eq 121, we have

which is the boiling-point elevation equation for concentration x. If this equation holds for any atmospheric pressure at which the boiling point of the solvent is To, then it is true for the pressure of 760 mm Hg. When we change the subscripts of the factors in eq 123 to correspond to these pressure conditions, we have

Inasmuch as AH760/AH, is the ratio of the molal heats of vaporization of the solvent under two pressure conditions, their values may be expressed in any unit, and we may rearrange eq 126 and write it as

where AT, is the boiling-point elevation at any pressure, p, at which the solvent boils at a temperature of T,, and L, is the latent heat of vaporization of the solvent at this temperature. AT760 is the boilingpoint elevation at a pressure of 760 mm Hg, in which case the solvent boils at a temperature of T760, which, for water, is equal to 373.16° K, and L60 is the latent heat of vaporization of the solvent at this temperature expressed in the same units as Lp. This equation, which is sometimes referred to as Tishchenko's equation [5], may be used to determine the boiling-point elevation at pressures other than that for which the values have been experimentally determined.

7. DERIVATION OF BOILING-POINT ELEVATION TABLE

As has been stated elsewhere, empirical equations were calculated from Spengler's observed data. These equations were developed in the following manner:

Inasmuch as no observations were made at a pressure of exactly 1 atmosphere, the first step was to adjust the values determined at the pressure nearest 1 atmosphere for each concentration and purity to the value it would have at exactly 1 atmosphere. This was done by means of eq 127 expressed in the form

in which L/L760 was determined for each observed temperature by the equation

1.1074-1024tX 10-6-53×10−9,

L730

760

(129)

where t is the temperature in degrees centigrade corresponding to the observed temperature, T,, which is expressed in degrees Kelvin, or

Values given by eq 129 in the range 60° to 130° C are in agreement with values of L, according to Osborne, Stimson, and Ginnings [6]. The use of eq 128 to adjust the observed values of the boiling-point elevation to the boiling-point elevation which the solution would have at 1 atmosphere resulted in a very small change from the observed values. In all cases this change was less than 0.10° C.

These adjusted values were fitted to four empirical equations, one for each purity reported, by the method of least squares. They have the form

log104T760-aX3+bX2+cX+d,

where AT760 is the boiling-point elevation of the solution at 1 atmosphere; X is the concentration of solids in solution, in percent (Brix); and a, b, c, and d are constants from the least-squares data. The equations on page 366 show the values of the constants for each of the four equations.

Boiling-point elevations at other pressures were calculated from the values found by these empirical equations by means of eq 127. The values so calculated deviate from the observed values by a lesser amount than they do from the graphic-method values.

8. REFERENCES

[1] H. Claassen, Z. Ver. deut. Zucker-Ind. 54, 1161 (1904).

[2] A. L. Holven, Ind. Eng. Chem. 28, 452 (1936).

[3] O. Spengler, S. Böttger, and E. Werner, Z. Wirtshaftsgruppe Zuckerind. 88, 521 (1938).

[4] H. S. Taylor, A Treatise on Physical Chemistry (D. Van Nostrand Co., Inc., New York, N. Y., 1931).

[5] I. A. Tishchenko, Soviet. Sakhar Nos. 11/12, 31-3 (1933); Chem. Zentr. 2, 352 (1934).

[6] N. S. Osborne, H. F. Stimson, and D. C. Ginnings, J. Research NBS 23, 261 (1939) RP1229.

XXIV. CANDY TESTS

1. INTRODUCTION
(a) GENERAL

The need of a standard or reference procedure as a starting point for a rational development of various types of candy tests has been felt for some time. The method outlined below for simple barleysugar tests, as well as the special equipment specified for carrying it out, has been developed at this Bureau, not only as a basic procedure but also as a standard procedure for the routine testing of commercial sucrose with respect to heat stability. It is founded upon an old procedure,20 which generally is attributed [2, 4, 5, 7, 9] to S. C. Hooker, whose directions, as transmitted to various laboratories under his supervision were stated in approximately the following words.

21

(b) HOOKER TEST

Half a pound (227 g) of sugar is placed in a copper casserole of the following dimensions, 416 inches diameter at the top, 24 inches diameter at the bottom, and height 216 inches (inside measurements). After the addition of 3 oz (89 cc)22 of distilled water, the casserole is placed over the naked flame of a burner. The flame should be regulated previously to such a size that the total time of heating required to bring the temperature to 350° F (177° C) is 21 to 25 minutes. been found that this condition will be fulfilled if 200 cc of water at room temperature is brought to a point of vigorous boiling in 41⁄2 to 5 minutes.)

(It has

20 The procedure was described in 1897 by Wiechmann [1], not as a control test but as a method of preparing "amorphous sugar" to be used in a study of allotropy in sucrose. The two sections of the description, separated in his paper, cover roughly every point of the Hooker procedure almost word for word. The paper includes analytical data on a dozen different candy plaques, one of which had been stored under a bell jar with calcium chloride desiccant while it developed a crystallizing area, of which there is presented a record of the rate of growth to a diameter of 51 mm and an excellent photograph at this stage.

21 Hooker's directions are quoted here, because no exact statement of them is known to be readily available elsewhere.

22 In certain copies of Hooker's directions, which probably were intended for use with wet-packed confectioners' sugars, the quantity of water was stated as "87 cc."