Calorimetric study of the entropy relation in the NaCl–KCl system

Highlights

• •The heat capacity of three (Na, K)Cl samples was measured from 5 to 300 K.
• •From these data, the vibrational entropy at T = 298.15 K was determined.
• •The calorimetrically determined solvus was compared to the directly determined solvus.

Abstract

The heat capacity of one Na-rich and two K-rich samples of the NaCl–KCl (halite–sylvite) crystalline solution was investigated between 5 and 300 K. It deviated positively from ideal behaviour with a maximum at 40 K. The thereby produced excess entropy at 298.15 K was described by a symmetric Margules mixing model yielding $W_m^S$ = 8.73 J/mol/K. Using enthalpy of mixing data from the literature and our data on the entropy, the solvus was calculated for a pressure of 10⁵ Pa and compared with the directly determined solvus. The difference between them can be attributed to the effect of Na–K short range ordering (clustering).

1. Introduction

The heat capacity (C_p,m) of crystalline solutions often deviates from that of a mechanical mixture at low temperatures (e.g., 100 K) [1–4]. This behaviour generates excess vibrational entropies at higher temperatures that stabilise (if positive) or destabilise (if negative) the crystalline solution. The position of the miscibility gap (solvus), depends thus strongly on these properties (e.g., [5]).

Investigating the physical nature of the excess vibrational entropy, first principles studies proposed a so-called “bond stiffness versus bond length” interpretation [6,7]. When comparing the MgO–CaO with the NaCl–KCl crystalline solution systems, which both have a similar size mismatch but different excess vibrational entropies, it became clear that bond stiffness relations play an important role in producing vibrational excess behaviour [7]. For the NaCl–KCl system, this first principles study found a significant softening of Na–Cl bonds with increasing K content producing positive excess vibrational entropies. In two recent studies [8,9], the excess vibrational entropy of several silicate solid solutions and binary alloys was described by a simple relationship, which is based on considerations that the elastically stiffer end member forces the softer one to fit to its size. Using this relationship, the maximum extent of the molar excess vibrational entropy (${\mathrm{\Delta }}_{\max }{S}_m^{\text{exc}}$) was described by the differences of the end member volumes (ΔV_i,m) and the end member bulk moduli (ΔK_i), i.e.,

$${\mathrm{\Delta }}_{\max }{S}_m^{\text{exc}}/(\text{J}·{\text{mol}}^{-1}·{\text{K}}^{-1})=(\mathrm{\Delta }{V}_{i\text{,}m}/(\text{J}·{\text{mol}}^{-1}·{\text{Pa}}^{-1})+a\mathrm{\Delta }{K}_i/\text{Pa})b\text{,}$$ (1)

where a and b are fit parameters. ΔV_i,m is defined to be positive whereas ΔK_i has a positive or negative value. Its sign depends on which end member (larger or smaller) is elastically stiffer (i.e., ΔV_i,m = V_1,m − V_2,m and ΔK_i = K₁ − K₂, where end member 1 has a larger volume than 2). According to these investigations, strongly negative ΔK_i values correspond to negative excess vibrational entropies, whereas positive ΔK_i values correspond to positive ones. In spite of the simplicity of this relationship, it was successfully applied to several silicate solid solutions and binary alloys [8,9].

The thermodynamics of the NaCl–KCl (halite–sylvite) binary has been thoroughly investigated (e.g., [7,10–17]). Although many studies found indirect evidence for an excess entropy in this system (e.g., [11,14]), the direct experimental proof on the existence of an excess vibrational entropy is missing, because there are no low temperature heat capacities of samples with crystalline solution composition. The excess entropy derived from phase equilibrium experiments ($G_m^{\text{mix}}$) and enthalpy of mixing data ($H_m^{\text{mix}}$) contains two different entropic contributions, i.e., the excess vibrational entropy and the excess configurational entropy coming from short range ordering (clustering). It is to be expected that these contributions have different temperature dependencies. The excess configurational entropy due to short range ordering or clustering decreases with increasing temperature. An unlike behaviour is to be expected for the excess vibrational entropy, which is generated at low temperatures and typically does not change at higher temperatures. It is, therefore, necessary to separate the entropic contributions in order to understand the thermodynamic processes of this crystalline solution system more precisely. For this purpose, three samples of the NaCl–KCl binary have been investigated by low temperature heat capacity measurements in this study to obtain the vibrational entropy.

2. Experimental methods

2.1. Relaxation calorimetry (PPMS)

Low temperature heat capacities from 5 to 300 K were measured on sample powders using a commercially available relaxation calorimeter (heat capacity option of the PPMS by Quantum Design®). The samples were put in Al cups made out of an Al foil and pressed to discs with 5 mm in diameter and 0.5 mm thickness (for details of the relaxation technique, see e.g., [18,19] and references therein).

2.2. Differential scanning calorimetry (DSC)

The heat capacity between 273 and 300 K was also measured using a Perkin Elmer Diamond DSC®. The evaluation of the DSC heat flow data was performed as described in [20,21].

2.3. Evaluation of the raw C_p,m data

To calculate the entropy, the measured heat capacities were integrated numerically using an interpolation function of Mathematica® (interpolation order 2). The entropy at 298.15 K determined by the PPMS has relative uncertainties of 1% to 2% [18], when investigating powder samples. To improve the precision of the entropy, the low temperature heat capacities have been corrected by the DSC data in several previous studies (e.g., [3,20,22]). This correction, however, was not necessary in case of the (Na, K)Cl samples. Here, perfect agreement between the PPMS and DSC data exists at around ambient temperatures, where they overlap. This is most likely because compact sample discs could be produced during preparation enabling a high thermal coupling between individual grains, which was not the case with other materials (e.g., corundum, silicates). The uncertainties in the entropy were, therefore, corrected to lower values using a relationship between the uncertainty and the thermal coupling given by [4].

2.4. Calculation of the solvus

In equilibrium, the two coexisting phases (ph1 and ph2, separated by the miscibility gap) have the same chemical potential for both components (${\mu }_{\text{NaCl}}^{\text{ph}1}$ = ${\mu }_{\text{NaCl}}^{\text{ph}2}$ and ${\mu }_{\text{KCl}}^{\text{ph}1}$ = ${\mu }_{\text{KCl}}^{\text{ph}2}$). Using experimentally determined mixing parameters and solving the two equations simultaneously, the common tangent of the Gibbs free energy of mixing function is found. The calculations for this paper used a Mathematica® routine, which searched the compositions of the coexisting phases numerically. In order to investigate the uncertainties of the solvus, a Monte Carlo method was used. It generated 10⁴ new sets of mixing parameters, which were normally distributed around the experimentally determined parameters with their corresponding standard deviations. Using these parameter sets, 10⁴ new solvi were calculated from which the standard deviation of the solvus at a given mole fraction could be obtained.

2.5. Samples of the NaCl−KCl system

Weighted mixtures of NaCl and KCl (high purity reagent with 99.9%, Merck®) were melted at 1073 K and then crystallised at 800 K. The (Na, K)Cl crystalline solution is stable above ∼750 K and decomposes very fast with cooling, especially if small amounts of H₂O are present (air humidity). It was not possible to cool it down in air without becoming partly decomposed. However, using the DSC (operating under a flow of pure Ar gas), the crystalline solution of Na- and K-rich samples could be cooled down metastably to room temperature. The successful cooling procedure was verified by the DSC signal in repeated temperature scans (the mixing procedure of a decomposed sample at temperatures of ∼750 K is endothermic producing a large calorimetric peak). The PPMS samples (wrapped in Al foil and pressed to a compact disc) were also homogenised in the DSC. The crystalline solution state was verified again by the DSC signal after the PPMS run. X-ray diffractometry, performed on samples homogenised in the DSC but not sealed in Al foils, documented partly decomposed samples after minutes. The results from X-ray diffractometry and DSC were consistent, i.e., samples, which were not wrapped in an Al foil and pressed to a disc, could not be removed from the DSC without an immediate start of decomposition. The preparation of samples with more intermediate composition was not possible because the cooling in the DSC was not fast enough (1.7 K ⋅ s⁻¹). The correct composition of the samples was checked by microprobe analyses using a defocused beam (20 μm). It agreed with the expected composition within one standard deviation (table 1) and no impurities were detected.

TABLE 1.

Microprobe analyses in wt% and mole fractions of KCl of the investigated samples synthesised from pure NaCl and KCl (see chapter 2.5 for more details). The standard deviation is given in parentheses and refers to the last digit.

Sample	Na	K	Cl	XKCl
Na90K10Cl	34.3 (9)	6.7 (4)	59.0 (1)	0.10 (1)
Na20K80Cl	6.5 (4)	43.8 (5)	49.7 (1)	0.80 (1)
Na10K90Cl	3.1 (4)	48.4 (5)	48.6 (1)	0.90 (1)

3. Results and discussion

3.1. Calorimetric results

The measured molar heat capacities of the investigated samples are listed in table 2. The excess heat capacity, defined as the deviation from C_p,m of a mechanical mixture, is plotted in figure 1 for the Na₂₀K₈₀Cl sample (C_p,m of the end members were taken from [23,24]). It shows a positive deviation with a maximum of 0.9 J ⋅ mol⁻¹ ⋅ K⁻¹ at 40 K. The calculated vibrational entropies at 298.15 K are listed in table 3 and plotted as a function of composition in figure 2. Their deviations from a mechanical mixture were described by a symmetric Margules mixing model defined as

$$S_m^{\text{exc}}/(\text{J}·{\text{mol}}^{-1}·{\text{K}}^{-1})=(1-X_{\text{KCl}})·X_{\text{KCl}}·W_{\text{m}}^S/(\text{J}·{\text{mol}}^{-1}·{\text{K}}^{-1})\text{,}$$ (2)

yielding $W_m^S$ = 8.73 J ⋅ mol⁻¹ ⋅ K⁻¹, from which the maximum extent of the excess vibrational entropy can be calculated, i.e., ${\mathrm{\Delta }}_{\max }{S}_m^{\text{exc}}$ = 2.2 J ⋅ mol⁻¹ ⋅ K⁻¹. Using this value, equation (1) can be tested on the NaCl−KCl system. Applying volumes and bulk moduli for halite and sylvite from the literature [26,27], a difference in the end member volumes of ΔV_i,m = 1.05 ⋅ 10⁻⁵ J ⋅ mol⁻¹ ⋅ Pa⁻¹ and a difference in the end member bulk moduli of ΔK_i = −6.75 GPa is obtained. The fit parameters of equation (1) are a = 1.089 ⋅ 10⁻¹⁶ J ⋅ mol⁻¹ ⋅ Pa⁻² and b = 2.505 ⋅ 10⁵ Pa ⋅ K⁻¹ (the values from [9] were converted to SI units), and results in ${\mathrm{\Delta }}_{\max }{S}_m^{\text{exc}}$ = 2.4 J ⋅ mol⁻¹ ⋅ K⁻¹, which is in good agreement with the calorimetrically determined value (2.2 J ⋅ mol⁻¹ ⋅ K⁻¹).

TABLE 2.

Measured molar PPMS heat capacities (C_p,m) of the NaCl–KCl samples. The uncertainties in T and C_p,m are given as follows: σ_T = ±T/K ⋅ (0.0004 − 3.6 ⋅ 10⁻⁶ ⋅ T/K + 1.2 ⋅ 10⁻⁸ ⋅ (T/K)²); σ_Cp,m = ±C_p,m/(J ⋅ mol⁻¹ ⋅ K⁻¹) ⋅ (0.004 − 2.7 ⋅ 10⁻⁵ ⋅ T/K + 9.7 ⋅ 10⁻⁸ ⋅ (T/K)²).

Na10K90Cl		Na20K80Cl		Na90K10Cl
T/K	Cp,m/(J ⋅ mol−1 ⋅ K−1)	T/K	Cp,m/(J ⋅ mol−1 ⋅ K−1)	T/K	Cp,m/(J ⋅ mol−1 ⋅ K−1)
5.095	0.0387	5.086	0.0361	5.041	0.0176
5.458	0.0485	5.454	0.0452	5.409	0.0219
5.879	0.0601	5.856	0.0564	5.793	0.027
6.294	0.0738	6.274	0.07	6.208	0.0335
6.752	0.0936	6.728	0.0878	6.664	0.0417
7.188	0.1141	7.176	0.1084	7.14	0.0518
7.702	0.1425	7.691	0.1356	7.653	0.0641
8.252	0.1787	8.238	0.1702	8.202	0.0801
8.84	0.2244	8.829	0.215	8.791	0.1001
9.475	0.2828	9.461	0.2712	9.422	0.1252
10.15	0.357	10.14	0.3435	10.1	0.157
10.89	0.4512	10.87	0.4354	10.83	0.198
11.67	0.5685	11.65	0.5511	11.61	0.2499
12.5	0.7153	12.48	0.6946	12.45	0.3169
13.4	0.8969	13.38	0.8732	13.34	0.4021
14.36	1.1196	14.34	1.0918	14.23	0.51
15.39	1.3953	15.37	1.3623	15.33	0.6479
16.49	1.7261	16.47	1.6913	16.43	0.8219
17.67	2.1251	17.65	2.0874	17.61	1.0395
18.94	2.5991	18.91	2.55	18.87	1.31
20.29	3.1559	20.26	3.1048	20.24	1.6491
21.74	3.8101	21.71	3.7558	21.7	2.0608
23.29	4.5784	23.26	4.5137	23.25	2.5635
24.96	5.4629	24.93	5.3961	24.92	3.165
26.74	6.5085	26.72	6.4288	26.71	3.8591
28.66	7.6739	28.63	7.596	28.64	4.7159
30.71	8.8238	30.69	8.7684	30.7	5.6711
32.91	10.287	32.89	10.163	32.9	6.785
35.28	11.821	35.25	11.673	35.27	8.0388
37.81	13.477	37.78	13.296	37.81	9.4367
40.52	15.247	40.5	15.059	40.53	10.981
43.43	17.109	43.41	16.885	43.45	12.635
46.57	19.041	46.53	18.806	46.58	14.398
49.91	21.012	49.87	20.732	49.93	16.28
53.5	23.001	53.47	22.726	53.53	18.229
57.34	25.022	57.32	24.731	57.38	20.247
61.47	26.984	61.44	26.705	61.51	22.264
65.89	28.949	65.86	28.669	65.94	24.311
70.63	30.883	70.6	30.604	70.69	26.324
75.7	32.61	75.67	32.387	75.77	28.265
81.13	34.397	81.11	34.171	81.21	30.128
86.95	36.068	86.93	35.847	87.05	32.292
93.14	37.493	93.15	37.318	93.3	34.009
99.83	39.053	99.85	38.835	100.02	35.792
107.02	40.326	107.03	40.148	107.21	37.382
114.72	41.626	114.72	41.454	114.92	38.916
122.97	42.707	122.97	42.568	123.18	40.265
131.81	43.696	131.82	43.531	132.04	41.449
141.3	44.687	141.3	44.511	141.54	42.607
151.46	45.557	151.45	45.392	151.71	43.668
162.37	46.296	162.35	46.183	162.61	44.814
174.06	47.074	174.02	46.929	174.32	45.519
186.58	47.688	186.55	47.504	186.88	46.208
199.99	48.315	199.96	48.186	200.33	47.112
214.36	48.855	214.35	48.716	214.74	47.779
229.76	49.264	229.74	49.194	230.17	48.365
246.27	49.829	246.25	49.804	246.73	49.073
263.93	50.305	263.91	50.181	264.44	49.551
282.87	50.877	282.86	50.682	283.42	50.221
303.18	51.268	303.12	51.207	303.77	50.613

FIGURE 1.

Molar excess heat capacity of mixing ($C_{p\text{,}m}^{\text{exc}}$) plotted against temperature (T) for the Na₂₀K₈₀Cl sample. Error bars represent one standard deviation.

TABLE 3.

Molar vibrational entropy at T = 298.15 K ($S_m^{298.15-0}$) with its standard deviation for the NaCl–KCl binary.

Sample	$S_m^{298.15-0}$/(J ⋅ mol−1 ⋅ K−1)
NaCl	72.115a
Na90K10Cl	73.96 ± 0.30
Na20K80Cl	81.85 ± 0.33
Na10K90Cl	82.30 ± 0.33
KCl	82.554a

^aJANAF-tables [25].

FIGURE 2.

Molar vibrational entropy ($S_m^{\text{vib}}$) at T = 298.15 K plotted against composition. End member data (open circles) are taken from the JANAF tables [25]. The data from this study are marked by closed symbols (error bars represent one standard deviation). Solid line represents a fit using a Margules mixing model with $W_m^S$ = 8.73 J ⋅ mol⁻¹ ⋅ K⁻¹, the broken line a mechanical mixture.

The measured excess vibrational entropy is also in good agreement with computer simulation studies [15,17] for which the first study used the double defect method. It resulted in excess vibrational entropies, which depend indirectly on the temperature, generated due to the temperature dependence of the cation distribution. At ∼623 K, the calculated excess vibrational entropies of [15] agree well with the calorimetrically determined value.

3.2. Solvus in the NaCl–KCl binary

The solvus of the NaCl–KCl binary at a pressure of 10⁵ Pa was calculated using our new mixing parameter on the entropy and enthalpic mixing parameters determined from literature data [10] (see table 4). The configurational entropy was calculated according to the one site mixing model, i.e., with a disordered Na–K distribution. This solvus is compared to the directly determined one in figure 3. The calorimetrically based solvus is more symmetric and has a by ∼100 K lower critical temperature than the directly determined solvus. The difference between them can be minimised, if the configurational entropy is lowered from that of a disordered Na–K distribution. Negative deviation from a random distribution can be produced by Na–K short range ordering or clustering. On the other hand, Schottky defects, which were found in this system [10,11], would increase the configurational entropy. Both, ordering/clustering and Schottky defects may be present in (Na, K)Cl crystals. In the vicinity of the solvus, however, Na–K short range ordering or clustering is more effective compared to Schottky defects because a slight net reduction of the configurational entropy as shown in figure 4 is necessary to achieve good agreement between calorimetric and directly determined solvus. The reduction of the configurational entropy was described by an asymmetric Margules mixing model with $W_{\text{NaK,}m}^S$ = 0 and $W_{\text{KNa,}m}^S$ = −3 J ⋅ mol⁻¹ ⋅ K⁻¹ and has a maximum extent of 0.5 J ⋅ mol⁻¹ ⋅ K⁻¹ in the Na-rich region, which agrees well with the results at 673 K of a computer simulation study [15].

TABLE 4.

Margules mixing parameters and equations used to calculate the activities of the NaCl and KCl components.

	NaK	KNa	Reference
$W_m^H$/(J ⋅ mol−1)	17060 ± 1131	18073 ± 1196	Data from [10]
$W_m^S$/(J ⋅ mol−1 ⋅ K−1)	8.73 ± 0.07	8.73 ± 0.07	This study
$W_m^V$ ⋅ 105/(J ⋅ Pa−1)	0.063 ± 0.01	0.054 ± 0.01	[16]
Asymmetric Margules model	$\mathrm{\Delta }{\mathrm{\Phi }}_m^{\text{mix}}$ = (1 − XK)2XK$W_{\text{NaK,}m}^{\mathrm{\Phi }}$ + (1 − XK)$X_K^2{W}_{\text{KNa,}m}^{\mathrm{\Phi }}$(Φ = H, S, and V)
Mixing parameter	$W_m^G$/(J ⋅ mol−1) = $W_m^H$/(J ⋅ mol−1) − T/K $W_m^S$/(J ⋅ mol−1 ⋅ K−1) + P/Pa $W_m^V$/(J ⋅ Pa−1)
Activity coefficient NaCl	RT ln γNaCl = XKCl2 ($W_{\text{NaK,}m}^G$ + 2($W_{\text{KNa,}m}^G$ − $W_{\text{NaK,}m}^G$) (1 − XKCl))
Activity coefficient KCl	RT ln γKCl = (1 − XKCl)2($W_{\text{KNa,}m}^G$ + 2($W_{\text{NaK,}m}^G$ − $W_{\text{KNa,}m}^G$)XKCl)
Ideal activity	$a_i^{\mathit{id}}$ = Xi	i = NaCl, KCl
Activity	ai = $a_i^{\mathit{id}}$γi	i = NaCl, KCl

FIGURE 3.

Solvus of the NaCl–KCl binary. Solid line: Calculated solvus using calorimetrically determined mixing parameters from table 4. Error bars represent one standard deviation at three compositions. Broken line: same parameters plus reduction of the configurational entropy as shown in figure 4. Symbols represent the results from studies, where the solvus was directly determined (triangles[10], diamonds[12]and circles[13]).

FIGURE 4.

Molar configurational entropy ($S_m^{\text{cfg}}$) calculated by a one site mixing model (solid line) and reduced by Na–K clustering or short range ordering in the Na-rich region (broken line). The reduction was described by an asymmetric Margules mixing model with $W_{\text{NaK,}m}^S$ = 0 and $W_{\text{KNa,}m}^S$ = −3 J ⋅ mol⁻¹ ⋅ K⁻¹.

4. Conclusions

The calorimetric data for the NaCl–KCl binary are consistent with the directly determined solvus, if the configurational entropy is slightly reduced in the Na-rich region from that of a fully disordered Na–K distribution. In the NaAlSi₃O₈–KAlSi₃O₈ system, Na-rich clusters were in fact detected by ²³Na NMR investigations in samples homogenised above the solvus, and they were attributed to early stages of exsolutions [28]. Such behaviour is also consistent with results made in binary alloys in which preferences for like neighbours were found above the solubility temperature by X-ray scattering and electrical resistivity measurements (e.g., [29,30]).