Freezing point,the temperature at which the solid and liquid phases of a substance are in equilibrium at a specified pressure (assumed atmospheric unless stated otherwise)^[1],is an important parameter in physical chemistry and pharmacology. The accurate measure­ment of the freezing point of chemical drug solutions is of critical importance,as it will be further used to determine other properties such as osmotic pressure,molecular weight,water activity,chemical purity,and construction of state diagram.

The best known methods of determining the freezing point include the cooling curve method^{[2, 3, 4]} and the differential scanning calorimetry (DSC)method^{[5, 6]},while each of them comes with considerable disadvan­tages. The cooling curvemethod detects the sudden temperature increase during cooling at the formation of the first crystals. This method is known to yield higher deviation at higher cooling rates^[4] or higher concentration^[3],the latter probably due to the increased viscosity and the subsequent inefficient stirring of the concentrated solution^[3],which is even more severe for macromolecules solutions.

The large subcooling may also result in inaccurate first registered freezing point,due to the excess formation of ice,which changes the liquid phase concentration. The DSC method detects the change in heat capacity over the transition temperature range and sets the end of melting as the measured freezing point. The measurement is known to suffer from the inaccuracy due to the “thermal lag” phenomenon,i.e. the limited thermal conduction ability of the frozen solutions and the thermal resistance between the sample pan and DSC furnace,thus the measurement from DSC may have a difference as high as several ℃ from those using other methods^[7]. Various experiment conditions such as cooling rate,warming rate,holding times - temperatures,annealing conditions and sample size,all affect the measurements of freezing point^[2]. Take the heating rate for example,it is generally believed that a slower heating rate will yield higher accuracies,and an accurate measurement using a slow heating rate may take at least 6 h^[6] per sample.

We propose a new isothermal method in this paper,which overcomes the aforementioned disadvantages of the known methods. It is generally believed that it’s difficult to measure a thermodynamic property accurately,when the system is undergoing a state change,while the measurement is much easier under an equilibrium state. Our method,namely air humidity cryoscopy (AHC),exploits the interrelationship between two colligative properties,the freezing point and the vapor pressure,as both of them depend on the number of solute molecules in diluted solutions^[8] and therefore once one is known,the other can be calculated.

The new AHC method is significantly cost-and- time efficient compared with the cooling curve and DSC method. Moreover,by avoiding the heating or cooling procedure and the crystallization process,our AHC method has the potential applications to concentrated and high viscous macromoleculessolutions. In the current study,we accomplish the first step of validation of the AHC method by measuring the freezing point of the NaCl solutions. NaCl solutions are prepared according to the CRC Handbook (2004) so that their freezing points vary from -1 ℃ to -17 ℃. We compare the measurements by AHC method,the cooling curve method and the DSC method and results indicate that AHC measurements are generally comparable to the cooling curve method and DSC method measurements,as well as the values given in the CRC Handbook (2004).

It is well known that the freezing point (T_f) and water activity (α_H2O) satisfy^[9] that

Where R is the universal gas constant,T_f is the freezing point of the solution and T_f is the freezing point of pure water. ∆_fusH is the fusion heat of ice in the solution,which is difficult to obtain and often approximately replaced by the fusion heat of ice in pure water ∆_fusH.

On the other hand,the vapor pressure of water (p_H2O) in equilibrium with the solution is proportional to the mole fraction of water (x_H2O) in ideal or diluted solution,known as Raoult’s Law

Where $p_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}^*$ is the vapor pressure of pure water,while in real solutions the water activity α_H2O replaces the mole fraction x_H2O as a correction for interactions among molecules in the liquid phase,known as the extended Raoult’s Law

Moreover,the relative air humidity (H_r) in equilib­rium with the solution is

Therefore we substitute the water activity (α_H2O) with H_r in Eq. (1)

Eq. (5) is the basis of the air humidity cryoscopy. By taking such an indirect approach,we avoid the heating or cooling procedure and the crystallization process.

NaCl (analytical reagent ≥ 99.5%) was purchased from Fuchen Chemical Reagents Factory,Tianjin,China. Distilled and degassed water was used in the preparation of all the aqueous solution.

A pocket humidity meter (Rotronic HydroPalm Hp22 with HygroClip HC2-S humidity sensor,Rotronic AG,Basserdorf,Switzerland),a cryoscope (FPD-4A,Nandawanhe Technology Co.,Ltd. Institute of Applied Physics,Nanjing University,Nanjing,China. The temperature range is -20 ℃ - 100 ℃ with the resolution 0.01 ℃),a differential scan­ning calorimeter (204F1,Netzsch,Germany. The temperature range is -170 ℃ - 500 ℃ with the repro­ducibility ≤ 0.1 ℃),an environmental chamber (SDH401,Chongqing experiment equipment factory,Chongqing,China. The accuracy of temperature is ≤ 2 ℃,precision and reproducibility are ≤ 0.5 ℃) and an electronic balance with 0.1 mg precision (FA2004,Liangping Co.,Ltd.,Shanghai,China.) were used in our study. For the humidity meter,the H_r range is 0 - 1 with resolution ± 0.008; the temperature range is -50 ℃ - 100 ℃ with the accuracy ± 0.1 ℃.

Measuring freezing point by proposed air humidity cryoscopyhe AHC equipment,a humidityprobe sealed at the headspace of a conical flask,is shown in Figure 1. A total of 50 mL of sample solution was prepared at each concentration. In each trial,about 8 mL of sample solution was placed in the conical flask,which was put in an environmental chamber maintained at 25 ℃. After 2 h incubation (although the respond time of the humidity sensor is only about 1s,as long as 1 h will be needed to reach the gas-liquid equilibrium) the relative air humidity in the conical flask was recorded and the freezing point of the sample solution was calculated using Eq. (5).

图 1

Figure 1

Figure 1 Assembly used in the Hr determination by air humidity cryoscopy

Measuring freezing point by cooling curve methodbout 20 mL of sample solution was prepared at each concentration. Obtaining the freezing point using computer aided cooling curve thermal analysis technique. Temperature of the solidifying composite was recorded at intervals of 1s through a cryoscope. Finally,the freezing points were obtained from the cooling curve.

In the DSC device,two Hastelloy cells were enclosed in the same temperature-controlled furnace. Both cells are sealed during the measurement with 5-10 mg sample in the sample cell,while the reference cell is empty. The DSC was equipped with a thermostat for cooling and dry nitrogen was used as the purge gas. The calibration of the DSC was made using pure distilled water. The furnace temperature (T) was ramped down from 288 K to 233 K at 10 K×min^-1 and then kept at 233 K for 5 min to freeze the liquid completely. Then the temperature was programmed from 233 K to 288 K at 3 K×min^-1. Finally,the freezing points were obtained from the DSC curve.

Freezing points obtained by the proposed air humidity cryoscopy,the cooling curve method,the DSC method and the literature are all listed in Table 1 and shown in Figure 2. Results indicate that the freezing points obtained by our new cryoscopy are comparable to those by the other two methods and the literature.

表 1 (Table 1)

Table 1 Freezing point of NaCl solutions determined by air humidity cryoscopy (AHC),cooling curve method,differential scanning calorimetry (DSC) method and reported by literature. n = 3,x± s w /g·g-1 Freezing point/℃ AHC Cooling curve method DSC CRC Handbook 0.02 -1.19 0.021 1 -1.353 ± 0.082 -1.347 ± 0.008 -0.873 ± 0.350 0.04 -2.41 0.040 4 -2.483 ± 0.045 -2.539 ± 0.025 -2.682 ± 0.005 0.06 -3.7 0.060 3 -3.966 ± 0.009 -3.919 ± 0.003 -3.906 ± 0.006 0.079 6 -5.118 ± 0.000 -5.045 ± 0.005 -4.890 ± 0.013 0.08 -5.08 0.10 -6.56 0.100 3 -7.031 ± 0.009 -7.057 ± 0.015 -6.645 ± 0.003 0.12 -8.18 0.120 1 -8.453 ± 0.000 -8.428 ± 0.005 -7.833 ± 0.001 0.14 -9.94 0.140 3 -10.045 ± 0.046 -10.136 ± 0.006 -9.742 ± 0.001 0.16 -11.89 0.160 3 -12.174 ± 0.005 -12.190 ± 0.007 -13.421 ± 0.008 0.18 -14.04 0.180 1 -14.503 ± 0.001 -14.528 ± 0.006 - 0.20 -16.46 0.200 1 -17.072 ± 0.000 -17.038 ± 0.014 -

Table 1 Freezing point of NaCl solutions determined by air humidity cryoscopy (AHC),cooling curve method,differential scanning calorimetry (DSC) method and reported by literature. n = 3,x± s

图 2

Figure 2

Figure 2 Comparison of freezing point data of NaCl solutions obtained by air humidity cryoscopy (□),cooling curve method (△),DSC method (○) and reported by literature (×)

In this work we proposed the air humidity cryoscopy,a simple and cost-and-time efficient isothermal measurement of the freezing point,and validated this method in measuring the freezing point of NaCl solu­tions,which is not by itself a challenging task. However,more importantly,by its isothermal nature the AHC method has promising potential applications in concen­trated and high viscous macromolecules solutions.