Heat Capacity and Standard Thermodynamic Functions of NaTi2(PO4

Nov 18, 2009 - Thermodynamic investigation of Rb2FeTi(PO4)3 phosphate of langbeinite structure. V. I. Pet'kov , E. A. Asabina , A. V. Markin , A. A. A...
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J. Chem. Eng. Data 2010, 55, 856–863

Heat Capacity and Standard Thermodynamic Functions of NaTi2(PO4)3 and NaHf2(PO4)3 Vladimir I. Pet’kov, Elena A. Asabina, Alexey V. Markin, and Natalia N. Smirnova Lobachevsky State University of Nizhni Novgorod, Gagarina Pr. 23, Nizhni Novgorod 603950, Russia

Heat capacity measurements of the crystalline phosphates NaTi2(PO4)3 and NaHf2(PO4)3 were performed between (6 and 650) K. Their thermodynamic functions, molar heat capacities C0p,m, enthalpy [Ho(T) Ho(0)], entropy So(T), and Gibbs free energy Go(T) - Ho(0), over the range from T f 0 to 650 K, were calculated, and the fractal dimension Dfr was evaluated. Standard entropies of formation at T ) 298.15 K were estimated to be [(1174 ( 2) and (1207 ( 2)] J · K-1 · mol-1 for NaTi2(PO4)3 and NaHf2(PO4)3, respectively.

Introduction Crystalline phosphates of the kosnarite structure type are of interest because of their considerable thermal and chemical stability, resistance to radiation damage, and other valuable physical and chemical properties that lead to industrial application possibilities as constructional and functional ceramics.1,2 The mineral kosnarite, KZr2(PO4)3, is isostructural with the large class of solid ionic conductors typified by the material NASICON, Na1+xZr2SixP3-xO12,3 and by NZP, NaZr2(PO4)3.4 The NZP structure is characterized by the high isomorphic capacity of cations of various sizes and oxidation states. The basis of their structure is a three-dimensional framework {[L2(PO4)3]p-}3∞ (L is an octahedrally coordinated cation) built up of corner-shared LO6 octahedra and PO4 tetrahedra.2,4 Two kinds of cavities (1:3 multiplicity) with different oxygen environments, referred to as M1 and M2, are situated within this framework, which is why cations from 0 to 4 per formula unit of the NZP may be included in the structural cavities. In the complex phosphates of sodium and IVB group metals of the general composition NaL2(PO4)3 (L ) Ti, Zr, Hf), the crystallographic site M1 is occupied by Na+ ions, and M2 sites remain vacant.4,5 Knowledge of the NZP compound thermodynamic functions over a wide temperature range is necessary as they are fundamental characteristics and needed for the different chemical process calculations in which these substances participate. In earlier work6 the temperature dependence of the NaZr2(PO4)3 heat capacity was measured over the range from (7 to 640) K, and the standard (po ) 0.1 MPa) thermochemical parameters of its formation at T ) 298.15 K were reported. The purpose of the present investigation is to measure the heat capacities by adiabatic calorimetry and differential scanning calorimetry over the temperature range of T ) (6 to 650) K of the NaTi2(PO4)3 and NaHf2(PO4)3 compounds, to calculate the 0 , Ho(T) - Ho(0), So(T), standard thermodynamic functions (Cp,m and Go(T) - Ho(0)) of the phosphates over the range from T f 0 K to T ) 650 K), to determine the characteristic temperatures and fractal dimensions Dfr, to establish the structure topology, to calculate the standard entropies of formation at T ) 298.15 K of NaTi2(PO4)3(cr) and NaHf2(PO4)3(cr), and to compare the * Corresponding author. E-mail: [email protected].

thermal behavior and thermodynamic characteristics of the series of isostructural compounds NaL2(PO4)3 containing IVB group metals where L ) Ti, Zr, and Hf.

Experimental Section Synthesis and Characterization of Samples. The compounds NaTi2(PO4)3 and NaHf2(PO4)3 were synthesized by solid-state reactions starting from NaCl, TiO2, HfO(NO3)2 · 2H2O, and NH4H2PO4. A fine mixture of the stoichiometric amounts of the reactants was dried at 473 K for (10 to 16) h and then thermally processed in unconfirmed air access at (873 and 1073) K for at least 24 h at each stage. Thermal stages were alternated with careful grinding. All chemicals used were provided by REACHEM,7 and their purity was not less than 99.5 % (except for HfO(NO3)2 · 2H2O, purity > 98 %). The purity of the starting HfO(NO3)2 · 2H2O is Table 1. Results of Element Analyses (Mass Fraction w) on the Calorimetric Samplesa sample

Na

NaTi2P3O12 1 2 3 4 5 average

Ti or Hf

P

Electron Microprobe Analysis 5.65 23.80 23.06 5.64 23.77 22.98 5.68 23.70 23.10 5.72 23.76 23.05 5.66 23.79 23.00 5.67 23.76 23.04 Chemical Analysis 5.73 23.80

47.49 47.61 47.52 47.47 47.55 47.53

22.95

47.52

Electron Microprobe Analysis 3.43 53.81 13.90 3.41 53.70 13.98 3.49 53.73 13.86 3.44 53.76 13.95 3.47 53.79 13.92 3.45 53.66 13.92

28.86 29.51 28.92 28.85 28.82 28.87

NaHf2P3O12 1 2 3 4 5 average

O

w/%

w/%

Chemical Analysis 3.49 53.59

14.01

28.91

Actual composition of NaTi2P3O12: w(Na) ) 5.69 %, w(Ti) ) 23.73 %, w(P) ) 23.02 %, w(O) ) 47.56 % and of NaHf2P3O12: w(Na) ) 3.46 %, w(Hf) ) 53.69 %, w(P) ) 13.98 %, w(O) ) 28.88 %. a

10.1021/je900494g  2010 American Chemical Society Published on Web 11/18/2009

Journal of Chemical & Engineering Data, Vol. 55, No. 2, 2010 857 Table 2. Experimental Molar Heat Capacities of Crystalline NaTi2(PO4)3 (M ) 403.704 g · mol-1, po ) 0.1 MPa) T

0 Cp,m

T

0 Cp,m

T

0 Cp,m

K

J · K-1 · mol-1

K

J · K-1 · mol-1

K

J · K-1 · mol-1

2.23 2.58 2.95 3.41 3.81 4.300 5.405 7.590 10.91 13.83 16.86 19.97 24.94 30.14

45.01 48.56 52.13 55.71 59.31 62.92 66.54 70.17 74.16 77.80 81.43 85.54 88.90

35.31 40.52 45.97 51.23 56.47 61.82 67.41 72.93 79.24 85.01 90.18 96.45 101.4

135.4 138.8 141.7 145.1 147.8 150.4 153.4 156.4 159.9 163.7 166.6 170.1 173.3 260.9 263.4 265.2 267.3 269.0 270.7 272.9 274.5 276.0 277.4 279.1 281.0 282.5 284.1 285.8 287.4 289.2 291.0 292.4

147.97 150.93 153.90 156.86 159.83 162.79 165.76 168.73 171.70 174.67 177.64 180.63 184.36 298.62 301.14 303.97 306.76 309.55 312.32 315.80 317.84 320.60 323.34 326.07 328.79 331.51 334.51 336.39 339.08 341.75 344.41

176.7 179.9 183.1 185.9 188.8 192.1 195.2 198.1 200.7 204.2 206.3 209.5 213.0 293.7 294.9 296.4 297.5 299.2 300.8 302.4 303.4 305.0 306.7 308.6 310.0 311.9 313.5 313.9 315.3 316.7 318.0

324 325 326 326 327 328 329 330 331 331 332 356 357 358 359 359 360 361 362 363 363 364 365 366 367

374.6 376.3 378.1 379.9 381.6 383.6 385.5 387.5 389.4 391.3 393.2 480.2 482.0 483.8 485.6 487.4 489.2 491.0 492.9 494.7 496.5 498.3 500.1 501.9 503.7

333 334 335 336 337 338 338 339 340 341 342 380 381 381 382 383 383 384 385 385 386 387 387 388 389

Series 1 7.17 7.83 8.49 9.15 9.81 10.26 10.88 11.50 12.15 12.83 13.63 14.41 15.16 15.89

0.135 0.169 0.228 0.295 0.373 0.445 0.540 0.652 0.782 0.951 1.15 1.37 1.63 1.93

16.61 17.33 18.03 18.73 19.43 20.10 21.67 23.96 26.98 29.46 31.81 34.18 37.80 41.49 Series 2

81.75 84.30 87.01 89.06 91.43 93.81 96.18 98.56 100.93 103.42 105.81 108.32 110.84 187.43 190.39 193.37 196.34 199.32 202.30 205.27 208.36 211.41 214.37 217.34 220.31 223.26 226.95 229.39 232.40 235.35 238.29 241.23

90.81 94.70 98.52 102.0 105.3 108.7 112.3 115.3 118.7 121.6 124.9 128.4 131.9 215.8 218.8 221.7 224.2 226.4 229.0 231.8 233.7 236.2 239.0 240.9 243.3 245.7 248.8 250.8 252.6 254.5 256.7 259.1

113.35 115.73 118.11 120.48 122.86 125.24 127.62 129.99 133.00 136.12 139.08 142.04 145.01 244.17 247.10 250.03 252.96 255.68 258.59 261.50 264.40 267.30 270.20 273.08 275.95 278.82 281.68 284.53 287.37 290.20 293.03 295.85

331.0 333.8 336.4 339.0 341.5 344.0 346.3 348.5 350.5 352.5 354.4 395.0 396.9 398.7 400.5 402.3 404.0 405.8 407.6 409.3 411.1 412.8 414.6 416.4 418.2

312 313 315 316 317 318 319 320 321 322 323 343 344 345 346 346 347 348 349 350 350 351 352 353 354

356.2 357.8 359.4 361.0 362.4 363.9 365.3 367.5 369.3 371.0 372.8 423.5 425.3 427.1 429.0 430.8 432.6 434.5 436.3 438.2 440.0 441.9 443.7 445.6 447.5

Series 3

858

Journal of Chemical & Engineering Data, Vol. 55, No. 2, 2010

Table 2 Continued T

0 Cp,m

T

0 Cp,m

T

0 Cp,m

K

J · K-1 · mol-1

K

J · K-1 · mol-1

K

J · K-1 · mol-1

419.9 395.0 396.9 398.7 400.5 402.3 404.0 405.8 407.6 409.3 411.1 412.8 414.6 416.4 418.2 419.9 421.7 536.5 538.3 540.1 541.9 543.8 545.6 547.4 549.2 551.1 552.9 554.7 556.6 558.4 560.2 562.0 563.9 565.7 567.5 569.3 571.1 572.9

354 343 344 345 346 346 347 348 349 350 350 351 352 353 354 354 355 400 401 401 402 402 403 403 404 405 405 406 406 407 407 408 408 409 409 410 410 411

450.3 451.2 453.0 454.9 456.7 458.6 460.4 462.2 464.0 465.8 467.6 469.4 471.2 473.0 474.8 476.6 478.4 574.8 576.2 578.1 579.9 581.8 583.6 585.4 587.2 589.0 590.8 592.6 594.4 596.2 598.0 599.8 601.6 603.3 605.1 606.9 608.7

368 368 369 370 370 371 372 373 373 374 375 376 376 377 378 379 379 411 412 412 413 413 413 414 414 415 415 416 416 417 417 417 418 418 419 419 419

505.6 507.4 509.2 511.0 512.9 514.7 516.5 518.3 520.2 522.0 523.8 525.6 527.4 529.2 531.0 532.8 534.6 610.5 612.3 614.1 615.9 617.8 619.6 621.4 623.2 625.1 626.9 628.7 630.6 632.4 634.2 636.0 637.8 639.6 641.4 643.3 645.1

389 390 391 391 392 393 393 394 395 395 396 396 397 398 398 399 399 420 420 420 421 421 421 422 422 423 423 423 424 424 424 424 425 425 425 426 426

explained by the uncertainty in the H2O content in this chemical. That is why the hafnium concentration in the solution taken for synthesis was confirmed gravimetrically with cupferron, following a documented procedure.8 The samples obtained were colorless polycrystalline powders. Confirmation of the desired compounds was obtained on a Shimadzu XRD-6000 powder X-ray diffractometer over the 2θ range of (10 to 60)°. A Cu anode (30 mA and 30 kV) with filtered monochromatic KR radiation (λ ) 1.54178 Å) was used in the determination. The X-ray patterns of the samples contained only reflections of NaTi2(PO4)39 and NaHf2(PO4)310 phosphates, respectively. The symmetry of these crystals was rhombohedral (space group R3jc). The unit cell parameters for the synthesized phosphates were derived from the least-squares refinement of powder X-ray diffraction data: a ) 8.479(4) Å, c ) 21.77(2) Å, and V ) 1355 Å3 for NaTi2(PO4)3, and a ) 8.780(5) Å, c ) 22.62(3) Å, and V ) 1510 Å3 for NaHf2(PO4)3. These agreed favorably with those reported elsewhere.9,10 The IR spectra of the samples were recorded on an IR Fourier spectrometer FSM-1201 over the range of (1400 to 400) cm-1. They agreed with data presented elsewhere11 and showed no evidence of condensed phosphate groups. The homogeneity and chemical composition of the samples were checked by electron microprobe analysis on a CamScan MV-2300 device with a Link Inca Energy 200C energy dispersion detector operated at 20.0 kV. The results showed the homogeneity of the calorimetric samples. Microprobe analysis (Table 1) confirmed the stoichiometry of the samples

to be close to the theoretical composition of NaTi2P3O12 and NaHf2P3O12. There were no substantial impurities (within 0.5 %) of other elements in the samples. The chemical compositions of the samples were also confirmed by chemical analysis. Known masses of the samples were dissolved in the HF aqueous solutions. The sodium mass contents were determined by an atomic absorption method on a Perkin-Elmer device. The titanium and hafnium mass contents were determined gravimetrically with cupferron, following a demonstrated procedure.8 The phosphorus mass contents were determined colorimetrically according to the method employing solutions of ammonium vanadate and ammonium molybdate.12 Owing to the presence of fluoride, colorimetric determinations were performed with perspex cells on a SF-46 spectrophotometer. Results of analyses (Table 1) proved that the stoichiometries of the obtained samples were close to ideal. Adiabatic Calorimetry. A precision automatic adiabatic calorimeter (BCT-3) was used to measure heat capacities over the temperature range of 6 e (T/K) e 350. The calorimeter was established in the Termis joint-stock company at the AllRussian Metrology Research Institute, Moscow region, Russia. The principle and structure of the adiabatic calorimeter are described in detail elsewhere.13,14 Briefly, all measurements were monitored by a complex-measuring system consisting of a computer, analog-to-digital and digital-to-analog converters, and a switch. The calorimetric ampule is a thin-walled cylindrical vessel made from titanium of 1.5 · 10-6 m3. Its mass is 2.0400 g. A miniature iron-rhodium resistance thermometer (a nominal

Journal of Chemical & Engineering Data, Vol. 55, No. 2, 2010 859 Table 3. Experimental Molar Heat Capacities of Crystalline NaHf2(PO4)3 (M ) 664.884 g · mol-1, po ) 0.1 MPa) T

0 Cp,m

T

0 Cp,m

T

0 Cp,m

K

J · K-1 · mol-1

K

J · K-1 · mol-1

K

J · K-1 · mol-1

13.01 16.78 20.66 24.40 28.09 31.91 35.37 39.15 42.57 45.88 49.04 52.27 55.41

58.15 60.61 63.14 65.61 68.03 70.45 72.85 75.29 77.74 80.18 82.61 85.04 87.48

58.57 61.70 65.07 68.54 71.80 75.29 78.58 81.81 85.10 88.32 91.50 94.48 97.50

Series 1 6.03 7.27 8.51 9.75 10.99 12.23 13.48 14.72 15.96 17.16 18.43 19.71 22.56

0.190 0.320 0.510 0.740 1.17 1.66 2.21 2.85 3.71 4.63 5.78 6.99 9.760

25.33 27.97 30.58 33.18 35.76 38.28 40.80 43.31 45.79 48.27 50.75 53.22 55.69 Series 2

83.57 86.77 89.82 92.85 95.88 98.89 101.90 104.89 107.89 110.89 113.87 116.85

92.59 96.60 100.4 104.1 107.8 111.2 114.5 118.1 121.4 124.5 127.7 130.9

119.52 122.49 125.47 128.44 131.41 134.37 137.32 140.28 143.23 146.18 149.12 152.06

184.11 188.28 192.33 196.37 200.40 204.42 208.45 212.47 216.49 220.52 224.30 228.56 232.58 236.30 240.59

194.5 198.2 201.6 205.1 208.7 212.1 215.7 219.2 222.9 226.1 229.8 233.1 236.8 240.0 244.0

244.40 248.80 252.59 256.58 260.58 264.57 268.56 272.56 276.55 280.54 284.80 288.49 292.70 295.90 298.12

322.9 324.9 326.9 328.9 331.0 333.0 335.0 337.0 339.0 341.1 343.1 345.1 347.1 349.2 403.7 405.5 407.2 409.0 410.8 412.6 414.4 416.2 418.0 419.9 421.7 423.6 425.4 427.3 429.2

311 312 314 316 317 319 320 322 323 325 326 327 329 330 364 364 365 366 367 368 369 370 371 371 372 373 374 375 376

351.2 353.2 355.2 357.2 359.2 361.3 363.3 365.3 367.3 369.3 371.3 373.2 375.1 377.0 459.8 461.7 463.5 465.4 467.2 469.1 470.9 472.8 474.6 476.5 478.3 480.2 482.0 483.9 485.7

133.7 136.6 139.5 142.4 145.4 148.5 151.7 154.7 157.5 160.3 162.8 165.7

155.00 157.93 160.87 163.85 166.80 169.71 172.63 175.70 178.48 181.39 184.31 187.22

168.1 170.7 173.3 176.1 178.7 181.3 183.8 186.7 189.2 192.0 194.8 197.1

247.4 251.0 254.5 257.9 261.2 264.5 267.7 271.3 274.9 278.3 281.6 284.6 288.0 290.5 292.2

300.95 302.62 305.60 309.02 312.30 315.30 318.72 321.84 324.93 327.94 330.90 333.40 336.63 339.39

294.4 295.6 298.0 300.6 303.2 305.5 308.1 310.4 313.2 314.6 317.0 318.9 321.4 323.1

331 332 334 335 336 338 339 340 342 343 345 346 347 348 390 391 392 392 393 394 395 395 396 397 398 399 399 400 401

378.9 380.7 382.6 384.4 386.1 387.9 389.7 391.4 393.2 394.8 396.7 398.4 399.7 401.9 516.7 518.5 520.2 522.0 523.8 525.5 527.3 529.1 530.9 532.7 534.6 536.4 538.2 540.1 541.9

349 350 351 353 354 355 356 357 358 359 360 361 361 363 413 413 414 415 416 416 416 417 418 419 420 420 420 421 422

Series 3

Series 4

860

Journal of Chemical & Engineering Data, Vol. 55, No. 2, 2010

Table 3 Continued T

0 Cp,m

T

0 Cp,m

T

0 Cp,m

K

J · K-1 · mol-1

K

J · K-1 · mol-1

K

J · K-1 · mol-1

431.0 432.9 434.7 436.5 438.6 440.1 441.9 443.8 445.5 447.4 449.1 450.9 452.7 454.5 456.2 458.0 572.8 574.6 576.4 578.2 580.0 581.9 583.7 585.5 587.3 589.1 590.9 592.7 594.5 596.3 598.0

377 377 378 379 380 381 382 382 383 384 385 386 387 388 388 389 432 432 433 433 434 434 435 435 435 436 436 437 438 438 438

487.6 489.4 491.3 493.2 495.0 496.9 498.7 500.6 502.4 504.2 506.0 507.8 509.6 511.4 513.1 514.9 599.8 601.6 603.4 605.1 606.9 608.7 610.5 612.3 614.1 615.9 617.7 619.5 621.3 623.1

401 403 403 404 405 406 406 407 408 408 409 410 410 411 412 412 439 439 440 440 440 441 441 442 442 443 443 443 444 444

543.7 545.5 547.3 549.1 550.9 552.7 554.5 556.3 558.1 559.9 561.8 563.6 565.5 567.3 569.1 571.0 624.9 626.7 628.5 630.3 632.1 633.9 635.7 637.5 639.3 641.2 643.0 644.9 646.8 648.6

423 423 424 424 425 425 426 427 427 427 428 429 429 430 431 431 444 445 445 445 446 446 446 447 447 447 447 447 448 448

Figure 1. Temperature dependence of the heat capacity of the crystalline substances: NaHf2(PO4)3 (1) and NaTi2(PO4)3 (2).

resistance of 100 Ω) was applied to measure the temperature of the sample. The thermometer was calibrated on the basis of IST-90 by the Russian Metrology Research Institute, Moscow region, Russia. The difference in temperature between the ampule and an adiabatic shield was controlled by a four-junction copper-iron-chromel thermocouple. The sensitivity of the thermometric circuit was 1 · 10-3 K and of the analog-to-digital converter 0.1 µV. The energy introduced into the sample cell and the equillibrium temperature of the cell after the energy input were automatically recorded and processed online by a computer. The speed of the computer-measuring system was 10 measurements per second.

To verify the accuracy of the calorimeter, the heat capacities of the standard reference materials (K-2 benzoic acid and R-Al2O3)15,16 prepared at the Institute of Metrology of the State Standard Committee of the Russian Federation were measured over the temperature range of 6 e (T/K) e 350. The sample masses were (0.7682 and 1.5000) g, respectively. It was established that the apparatus and the measurement technique enable the determination of the heat capacities of substances with an error not exceeding ( 2 % over the temperature range of 6 e (T/K) e 15, ( 0.5 % between T ) (15 and 40) K and ( 0.2 % over the temperature range 40 e (T/K) e 350 and the measuring of the phase transition temperatures within about ( 0.01 K and the enthalpies of transitions with the error of ( 0.2 %. Heat capacity measurements were continuously and automatically carried out by means of the standard method of intermittently heating the sample and alternately measuring the temperature. The heating rate and temperature increments were controlled at 0.01 K · s-1 and (0.5 to 2) K. The heating duration was ∼ 10 min, and the temperature drift rates of the sample cell measured in an equilibrium period were always kept within 0.01 K · s-1 during the acquisition of all heat capacity results. Liquid helium and nitrogen were used as coolants. The ampule with the substance was filled with dry helium as a heat exchange gas to the pressure of 4 kPa at room temperature. The sample masses used for calorimetric measurements were (1.013 and 1.250) g for NaTi2(PO4)3 and NaHf2(PO4)3, respectively. The molar masses of the objects under study were calculated from the IUPAC table of atomic weights.17 The experimental values 0 of Cp,m (136 and 119 points for NaTi2(PO4)3 and NaHf2(PO4)3, respectively) were obtained in the experiments. Differential Scanning Calorimetry. An automatic thermoanalytical complex, namely, a differential scanning calorimeter operating on the principle of a triple thermal bridge (ADCTTB),

Journal of Chemical & Engineering Data, Vol. 55, No. 2, 2010 861 Table 4. Smoothed Molar Heat Capacities and Thermodynamic Functions of Crystalline NaTi2(PO4)3 (M ) 403.704 g · mol-1, po ) 0.1 MPa) T K 5 10 15 20 25 30 35 40 45 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 298.15 300 310 320 330 340 350 360 370 380 390 400 420 440 460 480 500 520 540 560 580 600 620 640 650

[Ho(T) - Ho(0)]

0 Cp,m -1

-1

J · K · mol

0.0502 0.400 1.58 4.221 8.649 14.55 21.09 28.02 35.27 42.74 57.51 72.69 88.18 103.1 117.2 130.8 144.5 156.3 167.8 178.8 189.2 199.3 209.0 218.2 227.0 235.4 243.3 250.8 258.1 265.3 271.5 277.6 283.5 289.0 293.2 294.4 299.9 305.3 310.4 315.6 320.9 325.9 330.9 336 341 345 355 363 372 380 387 395 401 407 413 417 422 425 427

-1

kJ · mol

0.000100 0.00100 0.00540 0.01920 0.05070 0.1081 0.1971 0.3197 0.4778 0.6727 1.174 1.825 2.629 3.587 4.689 5.929 7.305 8.809 10.43 12.16 14.00 15.95 17.99 20.13 22.35 24.66 27.06 29.53 32.07 34.69 37.37 40.12 42.93 45.79 48.16 48.70 51.68 54.70 57.78 60.91 64.10 67.33 70.61 73.9 77.3 80.8 87.8 94.9 102 110 117 125 133 141 150 158 166 175 179

-[Go(T) - Ho(0)]

So(T) -1

-1

J · K · mol

0.0167 0.133 0.477 1.247 2.630 4.712 7.438 10.70 14.42 18.52 27.62 37.62 48.34 59.60 71.20 83.01 94.97 107.0 119.0 131.0 142.8 154.6 166.3 177.8 189.3 200.5 211.7 222.7 233.5 244.2 254.7 265.0 275.3 285.3 293.4 295.2 304.9 314.5 324.0 333.4 342.6 351.7 360.7 370 378 387 404 421 437 453 469 484 499 514 528 542 556 570 576

-1

kJ · mol

0.0000210 0.000326 0.00172 0.005749 0.01504 0.03326 0.06328 0.1084 0.1711 0.2534 0.4832 0.8088 1.238 1.777 2.431 3.202 4.092 5.102 6.232 7.482 8.851 10.34 11.94 13.66 15.50 17.45 19.51 21.68 23.96 26.35 28.85 31.44 34.15 36.95 39.31 39.85 42.85 45.95 49.14 52.43 55.81 59.28 62.84 66.5 70.2 74.1 82.0 90.2 98.8 108 117 126 136 146 157 168 179 190 196

was used to measure the heat capacities over the temperature range of 330 e (T/K) e 650.18 The design of the device and the measurement procedure of the heat capacities were demonstrated in detail.18 The reliability of the calorimeter operation was checked by measuring the heat capacity of a standard sample of synthetic corundum and the thermodynamic characteristics of fusion of indium, tin, and lead. As a result of the calibrations, it was found that the calorimeter and the measurement technique make it possible to obtain the heat capacity values with a maximum error of ( 2 % and the transformation temperatures within 0.5 K. Since the heat capacity of the compound examined was also measured over the temperature range of 330 e (T/K) e 350 in the adiabatic calorimeter with

Table 5. Smoothed Molar Heat Capacities and Thermodynamic Functions of Crystalline NaHf2(PO4)3 (M ) 664.884 g · mol-1, po ) 0.1 MPa) T K 5 10 15 20 25 30 35 40 45 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 298.15 300 310 320 330 340 350 360 370 380 390 400 410 420 440 460 480 500 520 540 560 580 600 620 640 650

[Ho(T) - Ho(0)]

0 Cp,m -1

-1

J · K · mol 0.106 0.840 3.05 7.250 12.59 19.78 27.03 34.33 41.47 48.16 60.92 74.59 88.04 100.6 112.6 123.6 134.2 144.0 154.4 163.8 172.7 181.5 190.8 199.7 208.4 217.2 225.9 234.6 243.5 252.2 260.7 269.1 277.8 285.9 292.3 293.6 301.4 309.1 316.5 323.6 330.5 337.1 343.5 350 356 362 367 371 381 390 398 407 414 421 428 434 439 443 447 448

-1

kJ · mol

0.000100 0.00210 0.0109 0.03580 0.08620 0.1670 0.2840 0.4373 0.6270 0.8511 1.397 2.074 2.888 3.831 4.898 6.081 7.368 8.762 10.25 11.85 13.53 15.30 17.16 19.11 21.15 23.28 25.50 27.80 30.19 32.67 35.23 37.88 40.62 43.44 45.79 46.33 49.31 52.36 55.49 58.69 61.96 65.30 68.70 72.2 75.7 79.3 82.9 86.6 94.1 102 110 118 126 134 143 151 160 169 178 182

-[Go(T) - Ho(0)]

So(T) -1

-1

J · K · mol

0.0352 0.280 0.963 2.371 4.592 7.514 11.11 15.20 19.66 24.37 34.28 44.69 55.54 66.64 77.87 89.13 100.3 111.5 122.5 133.5 144.4 155.1 165.8 176.3 186.8 197.1 207.4 217.7 227.9 238.0 248.0 258.0 268.0 277.9 285.9 287.7 297.4 307.1 316.8 326.3 335.8 345.2 354.5 364 373 382 391 400 417 435 451 468 484 500 515 530 545 559 574 580

kJ · mol-1 0.0000440 0.000704 0.00354 0.01157 0.02862 0.05841 0.1048 0.1706 0.2575 0.3675 0.6604 1.055 1.556 2.167 2.889 3.724 4.672 5.731 6.902 8.181 9.572 11.07 12.67 14.38 16.20 18.12 20.14 22.27 24.50 26.82 29.25 31.78 34.41 37.14 39.44 39.97 42.90 45.92 49.04 52.25 55.56 58.97 62.47 66.1 69.7 73.5 77.4 81.3 89.5 98.0 107 116 126 135 146 156 167 178 189 195

an error of ( 0.2 % and the conditions of measurements in the dynamic device were chosen such that within the above 0 temperature interval the Cp,m values measured with the use of both calorimeters coincided, it was assumed that at T g 350 K the heat capacities were determined with an error of ( (0.5 to 2.0) %. The results for the heat capacity of the objects under study were obtained over the range of 330 e (T/K) e 650 at the average rate of heating of the calorimeter and the substance of 1.5 K · s-1. The sample masses used for the calorimetric measurements were (0.9720 and 0.4704) g for NaTi2(PO4)3 and NaHf2(PO4)3, respectively.

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Journal of Chemical & Engineering Data, Vol. 55, No. 2, 2010

Table 6. Parameters of the Debye Heat Capacity Functions for NaTi2(PO4)3 and NaHf2(PO4)3 substance

ΘD/K

δ/%

temperature interval/K

NaTi2(PO4)3 NaHf2(PO4)3

169.1 132.0

1.1 1.0

7 to 12 6 to 11

Table 7. Absolute Entropies of the Simple Substances Necessary to Calculate the Standard Entropy of Formation of NaTi2(PO4)3 and NaHf2(PO4)3; T ) 298.15 K, p ) 0.1 MPa substance

physical statea

K 0 ∆298.15 Sm/J · K-1 · mol-1 0

Na Ti Hf P O2

cr cr cr cr, white g

51.30 ( 0.02 30.72 ( 0.10 43.555 ( 0.209 41.09 ( 0.25 205.152 ( 0.005

Results and Discussion All experimental results of the molar heat capacities of NaTi2(PO4)3 and NaHf2(PO4)3 over the range from T ) (6 to 650) K are shown in Tables 2 and 3 and plotted in Figure 1. The heat capacities of the samples were between (30 and 70) % of the overall heat capacity of the calorimetric ampules with the substances under temperature change from (6 to 650) K. The heat capacities of the samples rise gradually with increasing temperature, and no phase changes or thermal decompositions occurred. 0 in the above temperature The experimental points of Cp,m interval were fitted by means of the least-squares method using polynomial dependences, and the smoothed heat capacity values are shown in Tables 4 and 5. The root-mean-square deviations of the experimental points from the smoothed C0p,m ) f(T) curves were within ( 0.6 % in the range T ) (6 to 80) K, ( 0.3 % from T ) (80 to 200) K, ( 0.05 % between T ) (200 to 350) K, and ( 0.5 % over the range T ) (350 to 650) K, for both samples under study. As Figure 1 illustrates, the phosphates do not undergo phase transitions in the studied temperature interval. It is seen that the temperature dependences of the molar heat capacities of NaTi2(PO4)3 and NaHf2(PO4)3 cross: at 100 e (T/K) e 300 the 0 0 (NaTi2(PO4)3) > Cp,m (NaHf2(PO4)3), and at T g 300 K the Cp,m 0 0 Cp,m(NaTi2(PO4)3) < Cp,m(NaHf2(PO4)3). It may be explained by the influence of sample’s density on its heat capacity. The larger ionic radius and bigger molar mass of the framework-forming cation Hf4+ (compared with Ti4+) differently influence the density of the hafnium sample (compared with titanium sample). At high temperatures the heat capacities of both samples are close to the value theoretically predicted from the Dulong and 0 (predicted) ) 3RN ) 3 · 8.314 · 18 ) 449 Pti rule: CV,m -1 -1 J · K · mol (where N is number of atoms in the formula unit). The values of the fractal dimension Dfr for NaTi2(PO4)3 and NaHf2(PO4)3 were determined from the experimental results of their heat capacities. In the fractal theory of the heat capacity19-21 0 of solids, Dfr is the exponent at T in the CV,m ) f(T) function. The significance of the Dfr value for solids gives information on the topology type of their structure. The relation “heat capacity versus T” is proportional to T1 in the lower temperature range for chain-structured bodies, T2 for solids with layer structures, and T3 in the case of three-dimensional structures. For sufficient accuracy, it can be taken that isobaric and isochoric heat capacities are equal at temperatures below 50 K. From the ln C0p,m versus lnT plot, it is found that at temperatures between (20 and 45) K Dfr ) 3 for NaTi2(PO4)3 and NaHf2(PO4)3, within uncertainties of (1.1 and 1.0) %, respectively. This means that crystalline NaTi2(PO4)3 and NaHf2(PO4)3 have three-dimensional structures. At lower temperatures, to calculate the thermodynamic functions (Tables 4 and 5), the heat capacities of the studied 0 ∼ T 3: phosphates were described by the Debye function, Cp,m 0 Cp,m ) nD(ΘD /T)

where D denotes the Debye heat capacity function and n and ΘD are specially selected parameters. Their values for NaTi2(PO4)3 and NaHf2(PO4)3 and uncertainties of their deter-

a

cr, crystalline; g, gaseous.

mination are presented in Table 6. While calculating the thermodynamic functions, it was assumed that, at T < (6 or 7) 0 K, this equation reproduces the Cp,m values of NaTi2(PO4)3 and NaHf2(PO4)3 almost with the same accuracies. The thermodynamic functions of crystalline NaTi2(PO4)3 and 0 ) f(T) curves in NaHf2(PO4)3 were calculated from the Cp,m the range from (0 to 650) K. The enthalpies [Ho(T) - Ho(0)] and entropies So(T) were calculated by numerical integration of the curves with respect to T and ln T, respectively. The Gibbs functions [Go(T) - Ho(0)] were calculated from [Ho(T) - Ho(0)] and So(T) values at respective temperatures.22 The uncertainties for the function values are ( 2 % at T < 15 K, ( 0.5 % from (15 to 40) K, ( 0.3 % from T ) (40 to 350) K, and ( 1.5 % in the temperature range between T ) (350 and 650) K. Using the value of the absolute entropies (Tables 4 and 5) and the elemental substances23,24 (Table 7), the standard entropies of NaTi2(PO4)3 and NaHf2(PO4)3 formation at T ) 298.15 K were estimated to be [(1174 ( 2) and (1207 ( 2)] J · K-1 · mol-1, respectively. The values obtained fit to the following equations:

Na(cr) + 2Ti(cr) + 3P(cr, white) + 6O2(g) ) NaTi2(PO4)3(cr) and

Na(cr) + 2Hf(cr) + 3P(cr, white) + 6O2(g) ) NaHf2(PO4)3(cr) Conclusions The general aim of these investigations was to report the results of a thermodynamic study of the NaTi2(PO4)3 and NaHf2(PO4)3 compounds belonging to the kosnarite structure type. The heat capacities of these phosphates were measured by adiabatic calorimetry and differential scanning calorimetry in the temperature range from [(6 or 7) to 650] K. The standard and thermodynamic functions for NaTi2(PO4)3(cr) 0 , [Ho(T) - Ho(0)], So(T), and [Go(T) NaHf2(PO4)3(cr), Cp,m Ho(0)], over the range from T f 0 to T ) 650 K and the standard entropies of formation at T ) 298.15 K were derived from these experimental results. The low-temperature (T e 50 K) dependences of the heat capacity were analyzed on the basis of the heat capacity theory of Debye and the multifractal variant, and as a result, their three-dimensional structure was established. The thermodynamic characteristics of the studied compounds were compared. Supporting Information Available: Tables with the experimental molar heat capacities of crystalline NaHf2(PO4)3 and NaTi2(PO4)3. This material is available free of charge via the Internet at http://pubs.acs.org.

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