Thermodynamics for proton dissociations from aqueous l -histidine at temperatures from 278.15 K to 393.15 K and at the pressure 0.35 MPa: apparent molar volumes and apparent molar heat capacities of the protonated cationic, neutral zwitterionic, and deprotonated anionic forms

J. Chem. Thermodynamics 2001, 33, 1419–1440 doi:10.1006/jcht.2001.0841 Available online at http://www.idealibrary.com on

Thermodynamics for proton dissociations from aqueous L-histidine at temperatures from 278.15 K to 393.15 K and at the pressure 0.35 MPa: apparent molar volumes and apparent molar heat capacities of the protonated cationic, neutral zwitterionic, and deprotonated anionic forms J. J. Jardine, T. G. Call, B. A. Patterson, M. L. Origlia-Luster, and E. M. Woolleya Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602-5700, U.S.A.

Apparent molar volumes Vφ and apparent molar heat capacities C p,φ were determined for individual solutions of aqueous L-histidine, of aqueous L-histidine with equimolal HCl, and of aqueous L-histidine with equimolal NaOH at molalities m = (0.015 to 0.66) mol · kg−1 , at temperatures T = (278.15 to 393.15) K, and at the pressure p = 0.35 MPa. Apparent molar volumes were generated from density measurements obtained with a vibrating-tube densimeter. Apparent molar heat capacities were generated from heat capacity measurements with a twin fixed-cell, power-compensation, differential-output, temperature-scanning calorimeter. These results were then fitted by regression to empirical equations to describe the (Vφ ,m,T ) and (C p,φ , m, T ) surfaces for each of the three systems. These regression equations were then used to calculate the changes in partial molar volume 1r Vm and partial molar heat capacity 1r C p,m as functions of m and T for both the first and second proton dissociation reactions for protonated aqueous L-histidine. The changes in enthalpy 1r Hm and entropy 1r Sm and the acid dissociation molality quotient Q a were then obtained as functions of m and T for each proton dissociation reaction by integration, using our (1r C p,m , m, T ) results and literature values for 1r Hm and Q a . Our results illustrate the unique thermodynamic properties of the cationic, neutral zwitterionic, and c 2001 Academic Press anionic forms of L-histidine in aqueous solution. KEYWORDS: apparent molar volume; apparent molar heat capacity; L-histidine; hydrochloric acid; sodium hydroxide; zwitterion; ionization; acidity; proton dissociation

a To whom correspondence should be addressed (E-mail: earl [email protected]).

0021–9614/01/101419 + 22 $35.00/0

c 2001 Academic Press

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1. Introduction The physical and chemical characteristics of virtually all biologically relevant molecules depend strongly upon the pH of their aqueous environment. Enzymes optimally catalyse biological reactions only within a narrow pH range. Beyond this range, catalytic activity is lost and the enzymes are partially denatured under increasingly basic or acidic conditions. On the molecular level, changes in pH determine the proportions of protonated and deprotonated residues within proteins, nucleic acids, and other biomolecules. Changes in the equilibrium distribution among species with changes in temperature T , pressure p, and molality m can be determined by thermodynamic calculations using free energy and enthalpy information at a reference T , p, and m, combined with the change in partial molar volume 1r Vm and partial molar heat capacity 1r C p,m as functions of T , p, and m for proton dissociation reactions of isolated amino acids in aqueous solution. Values for 1r Ym (where Y = V or C p ) for proton dissociation of protonated L -histidine(aq), H2 · His+ (aq), reaction (1), can be estimated from apparent molar properties Yφ for solutes in process (2) by using equation (3): H2 · His+ (aq) = H · His(aq) + H+ (aq), H2 · His+ Cl− (aq) = H · His(aq) + H+ Cl− (aq), 1r Ym,1 = Yφ {H+ Cl− (aq)} + Yφ {H · His(aq)} − Yφ {H2 · His+ Cl− (aq)}.

(1) (2) (3)

Similarly, the change in partial molar property 1r Ym for the proton dissociation reaction of zwitterionic L-histidine, H · His(aq), reaction (4), can be estimated from apparent molar properties Yφ for reactants and products for process (5) by using equation (6): H · His(aq) = His− (aq) + H+ (aq), H · His(aq) + Na+ Cl− (aq) = Na+ His− (aq) + H+ Cl− (aq), 1r Ym,2 = Yφ {Na+ His− (aq)} + Yφ {H+ Cl− (aq)} − Yφ {H · His(aq)} − Yφ {Na+ Cl− (aq)}.

(4) (5) (6)

2. Experimental Anhydrous L-histidine {H · His(c)} (molar mass = 155.16 g · mol−1 , Fluka Chemie AG, Neu-Ulm, Switzerland, analysis number 350 371/1 30 397, stated purity >0.995 mass fraction, 0.005 mass fraction assumed to be water) was used to prepare H · His(aq) solutions in small preweighed glass bottles. All solutions were prepared by mass dilution with distilled, deionized, autoclaved, and degassed water. The L-histidine + NaOH solutions {Na+ His− (aq)} were prepared from {H · His(c)} and a stock solution of NaOH(aq) that was previously standardized by titration against primary standard potassium hydrogen phthalate (1) and further diluted with water to yield a series of solutions with a 1 : 1 molar ratio of L-histidine and NaOH. Similarly, the L-histidine + HCl {H2 · His+ Cl− (aq)} solutions were prepared from H · His(c) and a solution of HCl(aq) that was previously standardized by titration against the above NaOH(aq) solution (1) and further diluted

Thermodynamics of protonated, neutral, and deprotonated L-histidine

1421

to yield a series of solutions with a 1:1 molar ratio of L-histidine and HCl. All mass measurements were corrected from apparent mass to mass, and the formation of water was accounted for in the calculation of the Na+ His− (aq) solution molalities. Solution densities were obtained using a vibrating-tube densimeter (model DMA 512, Anton PAAR, Austria) that was described in previous work. (1) The temperature of the densimeter experiments ranged from T = (278.15 to 368.15) K at 5 K or 10 K intervals, while the cell pressure p remained constant at (0.350 ± 0.002) MPa. The period of oscillation τ of the vibrating tube as well as T and p of the experiments were recorded at intervals of about 40 s as described in detail previously. (1) Solution densities were calculated by using equation (7): ρs = ρw + {kρ · (τs2 − τw2 )}.

(7)

In equation (7), ρs and ρw are the densities of the solution of interest and of pure water respectively, and τs and τw are the periods of oscillation of the vibrating tube when it contains the experimental solution and pure water respectively. The T - and p-dependent calibration constant kρ was obtained by measuring (at the same T and p) τw for water and τs for 1.0 mol · kg−1 NaCl(aq), both of which have well-known densities. (2,3) Measurements were made on water every 4 or 5 days to account for any drift in τw over time. The temperature of the vibrating tube was monitored with a platinum resistance thermometer as described previously. (1) This method yields ρs values with a maximum standard deviation of 20 µg · cm−3 at each rounded T for all solutions investigated. A twin fixed-cell, power-compensation, differential-output, temperature-scanning calorimeter (model 6100 NanoDSC, Calorimetry Sciences Corporation, Spanish Fork, UT, U.S.A.) was used to measure the volumic heat capacity of each solution. The calibration process for this calorimeter is described in previous work. (4,5) The calorimetric cell pressure p was held constant at (0.350 ± 0.015 ) MPa, while the scan rate r was set to ±16.6667 mK · s−1 for the heating and cooling modes at 273.15 < T /K < 398.15. Each experiment was run in an alternating cycle of five heating and five cooling scans, and average calorimetric output values were then used to calculate the solution heat capacities. The massic heat capacities for the solutions at each T were obtained by using equation (8): c p,s = {kc · (1Ps − 1Pw )/(r · ρs )} + (c p,w · ρw /ρs ).

(8)

In equation (8), c p,s and c p,w are the massic heat capacities of the experimental solution and water respectively, while 1Ps and 1Pw are the differences in power applied to the heaters on the two cells to maintain the same temperature when the probe cell contains solution and water respectively. The T - and p-dependent calibration constant kc was determined from measurements on water and on 1.0 mol · kg−1 NaCl(aq) which have well-known massic heat capacities. (2,3) Any drift in 1Pw with time was determined by measuring water in the probe cell every 4 or 5 days, and the calculation of c p,s accounted for this drift. Values of Vφ and C p,φ were determined from results of the density and heat capacity experiments by using equations (9) and (10): Vφ = (M2 /ρs ) − {1000 · (ρs − ρw )/(ρs · ρw · m)}, C p,φ = (M2 · c p,s ) + {1000 · (c p,s − c p,w )/m}.

(9) (10)

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J. J. Jardine et al.

TABLE 1. Apparent molar volumes Vφ for aqueous L-histidine {H · His(aq)} at p = 0.35 MPa.a The ± uncertainties are r.m.s. deviations from multiple measurements taken during a 1200 s interval while the period of vibration was stable m mol · kg−1

Vφ

Vφ

Vφ

Vφ

Vφ

cm3 · mol−1

cm3 · mol−1

cm3 · mol−1

cm3 · mol−1

cm3 · mol−1

T = 278.15 K

T = 288.15 K

T = 298.15 K

T = 308.15 K

T = 318.15 K

0.01569

95.05 ± 0.21

99.77 ± 0.27

97.95 ± 0.33

100.05 ± 0.27

103.00 ± 0.64

0.03134

96.45 ± 0.10

99.28 ± 0.14

100.06 ± 0.18

101.01 ± 0.20

101.72 ± 0.27

0.06238

96.16 ± 0.04

98.29 ± 0.07

99.29 ± 0.09

100.37 ± 0.11

101.19 ± 0.06

0.12004

96.07 ± 0.03

97.75 ± 0.02

99.02 ± 0.02

100.11 ± 0.03

100.99 ± 0.03

0.25029

96.21 ± 0.01

98.07 ± 0.02

99.31 ± 0.03

100.35 ± 0.02

101.22 ± 0.03

T = 328.15 K

T = 338.15 K

T = 348.15 K

T = 358.15 K

T = 368.15 K

0.01569

103.06 ± 0.29

104.65 ± 0.15

102.94 ± 0.35

104.84 ± 0.30

104.70 ± 0.23

0.03134

102.27 ± 0.13

102.73 ± 0.17

102.45 ± 0.10

103.42 ± 0.12

103.85 ± 0.27

0.06238

102.10 ± 0.10

102.49 ± 0.08

102.58 ± 0.06

103.26 ± 0.09

104.00 ± 0.07

0.12004

101.76 ± 0.03

102.08 ± 0.03

102.73 ± 0.04

103.23 ± 0.04

103.54 ± 0.03

0.25029

101.89 ± 0.03

102.63 ± 0.03

102.98 ± 0.02

103.44 ± 0.03

103.87 ± 0.02

a Average experimental values of ρ can be obtained with equation (9) and with ρ given in s w reference 7.

In equations (9) and (10), M2 is the molar mass of the solute. Estimated uncertainties in Vφ and C p,φ that result from uncertainties in solution compositions are 60.3 cm3 · mol−1 and 62.2 J · K−1 · mol−1 respectively at all T and m of this investigation.

3. Results and discussion Values of Vφ for H · His(aq), H2 · His+ Cl− (aq), and Na+ His− (aq) are given in tables 1 to 3 respectively. Figures 1 to 3 show the results for these three L-histidine systems. Also shown in figures 1 to 3 are the (Vφ , m, T ) surfaces obtained by regression using the empirical equations (11) and (12) and the regression parameters in table 4: Vφ = ν0 + ν1 · T + ν2 /T, Vφ = A V · m

1/2

+ ν0 + ν1 · T + ν2 /T + ν3 · m.

(11) (12)

Equation (11) was used to regress the results for H · His(aq) in table 1, and equation (12) was used to regress the results for both H2 · His+ Cl− (aq) and Na+ His− (aq) in tables 2 and 3 respectively. The regressions used weighting factors inversely proportional to the uncertainties given in tables 1 to 3. The term A V in equation (12) is the Debye–H¨uckel coefficient for apparent molar volumes. Values of A V used in this investigation are based on the equations of Bradley and Pitzer (6) and are given by Ford et al. (7) Our Vφ results for H · His(aq) are compared with literature values at 278.15 < T /K < 328.15 and 0.1 MPa in figure 1. (8–13) The deviations 1 of our Vφ from these range from

Thermodynamics of protonated, neutral, and deprotonated L-histidine

1423

TABLE 2. Apparent molar volumes Vφ for aqueous L-histidine + HCl {H2 His+ Cl− (aq)} at p = 0.35 MPa.a The ± uncertainties are r.m.s. deviations from multiple measurements taken during a 1200 s interval while the period of vibration was stable m mol · kg−1

Vφ

Vφ

Vφ

Vφ

Vφ

cm3 · mol−1

cm3 · mol−1

cm3 · mol−1

cm3 · mol−1

cm3 · mol−1

T = 278.15 K

T = 288.15 K

T = 298.15 K

T = 308.15 K

T = 318.15 K

0.03000

115.10 ± 0.11

115.04 ± 0.18

116.75 ± 0.35

117.53 ± 0.28

118.83 ± 0.37

0.06002

110.30 ± 0.04

112.39 ± 0.14

113.66 ± 0.16

115.76 ± 0.10

116.94 ± 0.17

0.12004

112.52 ± 0.04

114.36 ± 0.06

115.73 ± 0.09

117.00 ± 0.08

117.51 ± 0.07

0.25007

112.38 ± 0.02

114.29 ± 0.03

115.82 ± 0.04

117.05 ± 0.04

117.65 ± 0.04

0.49629

112.93 ± 0.005

115.02 ± 0.02

116.50 ± 0.013

117.68 ± 0.02

118.43 ± 0.02

T = 328.15 K

T = 338.15 K

T = 348.15 K

T = 358.15 K

T = 368.15 K

0.03000

117.83 ± 0.33

118.30 ± 0.13

117.60 ± 0.23

117.98 ± 0.15

116.87 ± 0.26

0.06002

116.89 ± 0.11

116.86 ± 0.07

117.40 ± 0.07

117.75 ± 0.08

116.93 ± 0.07

0.12004

117.65 ± 0.06

117.96 ± 0.03

118.06 ± 0.05

117.63 ± 0.05

117.49 ± 0.05

0.25007

118.08 ± 0.03

118.49 ± 0.014

118.52 ± 0.02

118.60 ± 0.03

118.30 ± 0.015

0.49629

118.80 ± 0.02

119.23 ± 0.009

119.36 ± 0.007

119.47 ± 0.013

119.28 ± 0.012

a Average experimental values of ρ can be obtained with equation (9) and with ρ given in reference 7. s w

TABLE 3. Apparent molar volumes Vφ for aqueous L-histidine + NaOH {Na+ His− (aq)} at p = 0.35 MPa.a The ± uncertainties are r.m.s. deviations from multiple measurements taken during a 1200 s interval while the period of vibration was stable m mol · kg−1

Vφ

Vφ

Vφ

Vφ

Vφ

cm3 · mol−1

cm3 · mol−1

cm3 · mol−1

cm3 · mol−1

cm3 · mol−1 T = 318.15 K

T = 278.15 K

T = 288.15 K

T = 298.15 K

T = 308.15 K

0.03002

94.71 ± 0.10

96.69 ± 0.09

97.64 ± 0.06

98.49 ± 0.11

99.38 ± 0.09

0.04996

92.82 ± 0.06

95.56 ± 0.04

97.65 ± 0.06

99.20 ± 0.04

100.27 ± 0.07

0.08491

93.18 ± 0.03

95.84 ± 0.03

97.87 ± 0.03

99.29 ± 0.03

100.37 ± 0.03

0.17435

93.42 ± 0.014

96.37 ± 0.016

98.30 ± 0.019

99.79 ± 0.016

100.94 ± 0.015

0.34750

94.68 ± 0.013

97.19 ± 0.010

99.02 ± 0.012

100.51 ± 0.03

101.66 ± 0.008

0.66092

96.73 ± 0.004

99.07 ± 0.003

100.89 ± 0.004

102.31 ± 0.004

103.44 ± 0.006

T = 328.15 K

T = 338.15 K

T = 348.15 K

T = 358.15 K

T = 368.15 K

0.03002

99.54 ± 0.11

99.92 ± 0.07

100.28 ± 0.17

100.31 ± 0.06

100.18 ± 0.09

0.04996

100.46 ± 0.06

101.38 ± 0.08

100.98 ± 0.06

101.50 ± 0.07

101.33 ± 0.07

0.08491

100.77 ± 0.022

101.19 ± 0.03

101.54 ± 0.03

101.76 ± 0.02

101.67 ± 0.03

0.17435

101.73 ± 0.013

102.04 ± 0.02

102.62 ± 0.02

102.82 ± 0.02

102.99 ± 0.02

0.34750

102.41 ± 0.006

103.20 ± 0.005

103.64 ± 0.011

104.09 ± 0.008

104.31 ± 0.009

0.66092

104.26 ± 0.006

104.96 ± 0.006

105.48 ± 0.006

105.89 ± 0.005

106.20 ± 0.005

a Average experimental values of ρ can be obtained with equation (9) and with ρ given in reference 7. s w

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J. J. Jardine et al.

104

102 100 98 96

m

0.25 0.20 0.15 0.10

. kg ol

m

/(

0.05

−1

)

0.00

280

300

320

340

360

380

T/K

FIGURE 1. Apparent molar volumes Vφ for L-histidine(aq) {H · His(aq)} at p = 0.35 MPa plotted against temperature T and molality m. , Experimental values from table 1; surface generated from regression parameters given in table 4 with equation (11). Literature values at p = 0.1 MPa: O, reference 8; , reference 9; N, reference 14; ♦, reference 10; H, reference 11; , reference 12; , reference 13.

◦

•

TABLE 4. Regression parameters of equations (11) and (12) for the apparent molar volumes Vφ of aqueous L-histidine {H · His(aq)}, L-histidine + HCl {H2 · His+ Cl− (aq)}, and Lhistidine + NaOH {Na+ His− (aq)}. The ± values are chosen to reproduce the generated Vφ values to within ±0.01 cm3 · mol−1 at m 6 0.66 mol · kg−1 and at 278.15 6 T /K 6 393.15

Parameter v0 /(cm3 · mol−1 ) 100 · v1 /(cm3 · mol−1 · K−1 ) v2 /(cm3 · K · mol−1 ) v3 /(kg · cm3 · mol−2 ) 1a /(cm3 · mol−1 )

H · His(aq) equation (11)

H2 · His+ Cl− (aq) equation (12)

Na+ His− (aq) equation (12)

238.167 ± 0.006 −17.326 ± 0.001 −26044 ± 1

385.286 ± 0.006 −39.414 ± 0.001 −45568 ± 1

0.27

0.32

364.903 ± 0.005 −37.287 ± 0.001 −46855 ± 1 4.038 ± 0.007 0.23

a Standard deviations of the regression.

−0.1 to +0.9 cm3 · mol−1 , except for one value at 298.15 K which is lower than all others by about 4 cm3 · mol−1 . (14) Our Vφ results for H2 · His+ Cl− (aq) are compared with one value from the literature (8) at 298.15 K and 0.1 MPa in figure 2. The deviation 1 of our Vφ from this value from the literature is 1.9 cm3 · mol−1 . Figure 4 shows the (1r Vm,1 , m, T ) surface for proton dissociation of H2 · His+ (aq)

Thermodynamics of protonated, neutral, and deprotonated L-histidine

1425

120

118 116 114 112 0.5 0.4

m

0.3

. kg ol

m

/(

0.2 0.1

−1

)

0.0

280

300

320

340

360

380

T/K

FIGURE 2. Apparent molar volumes Vφ for L-histidine + HCl(aq) {H2 · His+ Cl− (aq)} at p = 0.35 MPa plotted against temperature T and molality m. , Experimental values from table 2; surface generated from regression parameters given in table 4 with equation (12). Literature value at p = 0.1 MPa: O, reference 8.

◦

106 104 102 100 98 96 94 92 0.6 0.5 m 0.4

/(

m

0.3

ol . kg

0.2 −1

)

320

0.1 0.0

280

300

340

360

380

T/K

FIGURE 3. Apparent molar volumes Vφ for L-histidine + NaOH(aq) {Na+ His− (aq)} at p = 0.35 MPa plotted against temperature T and molality m. , Experimental values from table 3; surface generated from regression parameters given in table 4 with equation (12).

◦

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J. J. Jardine et al.

3.0

2.5

2.0

1.5 1.0 0.4

m

0.3

. kg ol

m

/(

0.2 0.1

−1

)

0.0

280

300

320

340

360

380

T/K

FIGURE 4. Change in volume 1r Vm,1 for proton dissociation reaction (1) for protonated L -histidine(aq) {H2 · His+ (aq)} at p = 0.35 MPa generated from regression parameters given in table 4 and from regression equations for H+ Cl− (aq) in reference 16.

200 160 120 80 40 0.5 0.4

m

0.3

/(

0.1

−1

. kg ol

m

0.2

)

0.0

280

300

320

340

360

380

T/K

FIGURE 5. Change in volume 1r Vm,2 for proton dissociation reaction (4) for L-histidine(aq) {H · His(aq)} at p = 0.35 MPa generated from regression parameters given in table 4, from equations for Na+ Cl− (aq) in reference 3, and from regression equations for H+ Cl− (aq) in reference 16.

Thermodynamics of protonated, neutral, and deprotonated L-histidine

1427

475 450 425 400 375 350 0.25 0.20 m / ( 0.15 m 0.10 ol

.k g

−1

)

320

0.05

300 0.00

280

340

360

380

T/K

FIGURE 6. Apparent molar heat capacities C p,φ for L-histidine(aq) {H · His(aq)} plotted against temperature and molality. , Experimental values from table 5; surface generated from regression parameters given in table 8 with equation (13).

◦

described by reaction (1), and figure 5 shows the (1r Vm,2 , m, T ) surface for proton dissociation of H · His(aq) described by reaction (4). The surface (Vφ , m, T ) for HCl(aq) used in these calculations was generated from the equations of Sharygin and Wood, (15) and that for NaCl(aq) was generated from the equations of Archer. (3) The differences between 1r Vm,1 for the isoelectric process of reaction (1) and 1r Vm,2 for the ionic process of reaction (4) are readily apparent. We estimate the total uncertainty in 1r Vm,1 and 1r Vm,2 as 1 6 ±1.1 cm3 · mol−1 . Values of C p,φ for H · His(aq), H2 · His+ Cl− (aq), and Na+ His− (aq) are given in tables 5 to 7 respectively. Figures 6 to 8 show the results for these three L-histidine systems. We are not aware of values from the literature for these solutions. Also shown in figures 6 to 8 are the (C p,φ , m, T ) surfaces obtained by regression using the empirical equations (13) and (14) and the regression parameters given in table 8: C p,φ = c0 + c1 · T + c2 /{(T /K) − 270} + c3 /{1000 − (T /K)} + c4 · m,

(13)

C p,φ = AC · m + c0 + c1 · T + c2 /{(T /K) − 270} + c4 · m + c5 · ln(T /K) + c6 /T.

(14)

1/2

The term AC in equation (14) is the Debye–H¨uckel coefficient for apparent molar heat capacities. Values of AC used in this investigation are based on the equations of Bradley and Pitzer (6) and are given by Ford et al. (7) The regressions used weighting factors inversely proportional to the uncertainties given in tables 1 to 3. Figure 9 shows the (1r C p,m,1 , m, T ) surface for proton dissociation of H2 · His+ (aq)

1428

J. J. Jardine et al.

250

200 150 100 50 0.5 0.4

m

0.3

−1

. kg ol

m /(

0.2 0.1

)

0.0

280

300

320

340

360

380

T/K

FIGURE 7. Apparent molar heat capacities C p,φ for L-histidine(aq) + HCl(aq) {H2 · His+ Cl− (aq)} plotted against temperature T and molality m. , Experimental values from table 6; surface generated from regression parameters given in table 8 with equation (14).

◦

300 250 200 150 100 0.6 0.5

m

/ ( 0.40.3 m ol . 0.2 kg −1 0.1 )

340 0.0

300 280

320

360

380

T/K

FIGURE 8. Apparent molar heat capacities C p,φ for L-histidine(aq) + NaOH(aq) {Na+ His− (aq)} plotted against temperature T and molality m. , Experimental values from table 7; surface generated from regression parameters given in table 8 with equation (14).

◦

Thermodynamics of protonated, neutral, and deprotonated L-histidine

1429

TABLE 5. Apparent molar heat capacities C p,φ for aqueous L-histidine {H · His(aq)} at p = 0.35 MPa.a The ± uncertainties are r.m.s. deviations for the averages obtained from a minimum of eight scans C p,φ C p,φ C p,φ C p,φ C p,φ C p,φ m mol · kg−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 T = 278.15 K T = 283.15 K T = 288.15 K T = 293.15 K T = 298.15 K T = 303.15 K 0.00784

341 ± 17

362 ± 18

378 ± 19

390 ± 19

401 ± 19

410 ± 18

0.06238

341.1 ± 2.9

362.4 ± 4.1

379.5 ± 4.3

393.4 ± 4.4

404.7 ± 4.6

414.3 ± 4.7

0.12004

336.0 ± 1.9

358.7 ± 2.9

376.4 ± 3.1

390.4 ± 3.0

401.7 ± 2.9

411.3 ± 2.9

0.25029

362.5 ± 1.2

381.2 ± 1.6

396.3 ± 1.6

408.6 ± 1.6

418.7 ± 1.6

427.2 ± 1.7

T = 308.15 K T = 313.15 K T = 318.15 K T = 323.15 K T = 328.15 K T = 333.15 K 0.00784

417 ± 21

424 ± 19

429 ± 20

436 ± 20

440 ± 20

445 ± 20

0.06238

422.3 ± 4.7

429.2 ± 4.8

435.0 ± 5.0

440.4 ± 4.9

444.9 ± 4.9

449.1 ± 4.9

0.12004

419.7 ± 2.8

426.7 ± 2.4

432.5 ± 2.4

437.7 ± 2.5

442.5 ± 2.6

446.8 ± 2.5

0.25029

434.4 ± 1.7

440.7 ± 1.7

446.0 ± 1.8

450.7 ± 1.8

454.8 ± 1.7

458.5 ± 1.7

T = 338.15 K T = 343.15 K T = 348.15 K T = 353.15 K T = 358.15 K T = 363.15 K 0.00784

449 ± 21

453 ± 22

457 ± 23

459 ± 23

461 ± 23

463 ± 25

0.06238

452.9 ± 5.1

456.2 ± 5.4

459.2 ± 5.7

462.1 ± 5.9

464.4 ± 6.0

466.5 ± 6.3

0.12004

450.6 ± 2.6

453.8 ± 2.7

456.7 ± 2.9

459.6 ± 3.0

462.1 ± 2.9

464.3 ± 2.9

0.25029

461.9 ± 1.8

464.8 ± 1.9

467.4 ± 1.9

469.8 ± 2.0

471.9 ± 2.0

473.6 ± 2.0

T = 368.15 K T = 373.15 K T = 378.15 K T = 383.15 K T = 388.15 K T = 393.15 K 0.00784

465 ± 27

468 ± 29

468 ± 31

470 ± 34

470 ± 39

468 ± 47

0.06238

468.4 ± 6.6

470.1 ± 7.0

471.4 ± 7.5

472.7 ± 8.1

473.6 ± 9.1

474.1 ± 10

0.12004

466.0 ± 2.9

467.7 ± 3.1

468.8 ± 3.2

470.0 ± 2.9

470.8 ± 2.9

471.5 ± 2.9

0.25029

475.3 ± 2.2

476.7 ± 2.3

477.6 ± 2.4

478.6 ± 2.5

479.3 ± 2.8

479.8 ± 3.5

a Average experimental values of c p,s can be obtained with equation (10) and with c p,w given in reference 7.

described by reaction (1), and figure 10 shows the (1r C p,m,2 , m, T ) surface for proton dissociation of H · His(aq) described by reaction (4). The surfaces (C p,φ , m, T ) for HCl(aq) and NaCl(aq) used for these calculations were obtained from Patterson et al. (16) and from the equations of Archer (3) respectively. The differences between (1r C p,m,1 , m, T ) for the isoelectric process of reaction (1) and (1r C p,m,2 , m, T ) for the ionic process of reaction (4) are readily apparent in figures 9 and 10. There are several sources of the proton dissociation constants for aqueous L-histidine at p = 0.1 MPa in the literature. (17–22) We use the values of the acid dissociation molality quotients Q a,1 for reaction (1) and Q a,2 for reaction (4) at T = 298.15 K, m = 0.1 mol · kg−1 and m = 0.5 mol · kg−1 , and p = 0.1 MPa from Martell and Smith (17) to obtain the thermodynamic equilibrium constants K a , as pK a,1 = (6.02±0.03) and pK a,2 = (9.06 ± 0.03). We also use pQ a,1 = pK a,1 at T = 298.15 K for m 6 0.5 mol · kg−1

1430

J. J. Jardine et al.

TABLE 6. Apparent molar heat capacities C p,φ for aqueous L-histidine + HCl {H2 His+ Cl− (aq)} at p = 0.35 MPa.a The ± uncertainties are r.m.s. deviations for the averages obtained from a minimum of eight scans C p,φ C p,φ C p,φ C p,φ C p,φ C p,φ m mol · kg−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 T = 278.15 K T = 283.15 K T = 288.15 K T = 293.15 K T = 298.15 K T = 303.15 K 0.01497

21.0 ± 7.2

55.2 ± 9.6

81.8 ± 9.7

104 ± 10

122 ± 12

137 ± 12

0.03000

−12.7 ± 3.3

22.7 ± 5.2

51.1 ± 5.1

75.0 ± 5.4

94.2 ± 5.8

110.4 ± 5.8

0.06002

25.6 ± 2.9

58.1 ± 5.1

84.7 ± 5.5

106.7 ± 5.5

124.3 ± 5.6

139.3 ± 5.6

0.12004

28.5 ± 1.8

60.4 ± 3.3

86.2 ± 3.4

107.5 ± 3.2

124.7 ± 3.1

139.3 ± 3.2

0.25007

29.8 ± 0.7

60.2 ± 1.4

84.9 ± 1.6

105.4 ± 1.8

122.0 ± 1.9

136.4 ± 1.9

0.49629

67.3 ± 0.2

93.6 ± 0.7

115.0 ± 1.0

133.1 ± 1.3

147.8 ± 1.4

160.6 ± 1.4

T = 308.15 K T = 313.15 K T = 318.15 K T = 323.15 K T = 328.15 K T = 333.15 K 0.01497

149 ± 13

160 ± 12

168 ± 13

168 ± 13

182 ± 13

187 ± 13

0.03000

123.7 ± 5.4

135.2 ± 5.5

144.5 ± 5.9

144.5 ± 5.9

159.1 ± 5.7

165.1 ± 5.4

0.06002

151.8 ± 5.2

162.5 ± 5.0

171.1 ± 5.0

171.1 ± 4.8

185.0 ± 4.6

190.7 ± 4.3

0.12004

151.6 ± 3.3

162.1 ± 3.3

170.5 ± 3.3

170.5 ± 3.3

184.4 ± 3.2

190.3 ± 3.2

0.25007

148.3 ± 1.9

158.8 ± 2.0

167.4 ± 2.1

167.4 ± 2.1

181.6 ± 2.0

187.5 ± 2.0

0.49629

171.6 ± 1.8

181.2 ± 1.7

189.1 ± 1.9

189.1 ± 1.8

202.4 ± 1.7

208.1 ± 1.8

T = 338.15 K T = 343.15 K T = 348.15 K T = 353.15 K T = 358.15 K T = 363.15 K 0.01497

192 ± 13

195 ± 14

200 ± 15

202 ± 16

205 ± 16

206 ± 17

0.03000

170.1 ± 5.1

174.5 ± 5.3

178.4 ± 5.6

181.8 ± 5.8

184.6 ± 5.8

186.4 ± 6.4

0.06002

195.7 ± 4.1

199.7 ± 4.1

203.2 ± 4.3

206.5 ± 4.4

209.1 ± 4.2

211.1 ± 4.5

0.12004

195.5 ± 3.2

199.6 ± 3.2

203.3 ± 3.3

206.7 ± 3.3

209.6 ± 3.4

212.0 ± 3.5

0.25007

193.0 ± 2.0

197.5 ± 2.1

201.5 ± 2.1

205.3 ± 2.0

208.5 ± 2.0

211.3 ± 2.1

0.49629

213.3 ± 1.8

217.6 ± 1.9

221.6 ± 1.8

225.2 ± 1.7

228.5 ± 1.8

231.3 ± 1.8

T = 368.15 K T = 373.15 K T = 378.15 K T = 383.15 K T = 388.15 K T = 393.15 K 0.01497

208 ± 19

209 ± 22

208 ± 27

209 ± 32

208 ± 38

208 ± 47

0.03000

188.2 ± 7.7

189.9 ± 9.2

190 ± 11

191 ± 13

190 ± 17

190 ± 22

0.06002

212.8 ± 5.2

214.2 ± 5.9

214.7 ± 6.9

215.6 ± 8.1

216 ± 10

216 ± 13

0.12004

214.0 ± 3.7

215.8 ± 4.1

216.8 ± 4.6

218.1 ± 5.3

218.9 ± 6.4

219.6 ± 7.8

0.25007

213.8 ± 2.2

216.0 ± 2.1

217.5 ± 2.2

219.1 ± 2.5

220.4 ± 3.4

221.5 ± 4.7

0.49629

233.9 ± 1.8

236.2 ± 1.7

237.8 ± 1.6

239.7 ± 1.6

241.2 ± 1.6

242.4 ± 1.7

a Average experimental values of c p,s can be obtained with equation (10) and with c p,w given in reference 7.

since reaction (1) is isoelectric, and pQ a,2 = pK a,2 + (1.022 kg1/2 · mol−1/2 ) · m1/2 for reaction (4) based on Debye–H¨uckel behavior at T = 298.15 K, p = 0.1 MPa, and m 6 0.66 mol · kg−1 . We have used experimental results from the literature (17–22) at m 6 0.5 mol · kg−1 for pK a,1 (or pQ a,1 ) at 273.15 6 T /K 6 313.15 to estimate the standard enthalpy change

Thermodynamics of protonated, neutral, and deprotonated L-histidine

1431

TABLE 7. Apparent molar heat capacities C p,φ for aqueous L-histidine + NaOH {Na+ His− (aq)} at p = 0.35 MPa.a The ± uncertainties are r.m.s. deviations for the averages obtained from a minimum of eight scans C p,φ C p,φ C p,φ C p,φ C p,φ C p,φ m mol · kg−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 J · K−1 · mol−1 T = 278.15 K T = 283.15 K T = 288.15 K T = 293.15 K T = 298.15 K T = 303.15 K 0.00749

91 ± 13

135 ± 13

166 ± 13

190 ± 13

210 ± 14

225 ± 15

0.08491

115.1 ± 2.5

149.8 ± 2.6

177.3 ± 2.1

198.9 ± 1.9

216.5 ± 1.7

230.7 ± 1.5

0.17435

114.8 ± 2.8

149.0 ± 3.0

176.0 ± 2.9

197.6 ± 2.8

215.0 ± 2.5

229.4 ± 2.5

0.34750

123.8 ± 2.1

155.8 ± 2.0

181.1 ± 2.0

201.7 ± 2.0

218.4 ± 1.9

232.3 ± 1.9

0.66092

168.3 ± 1.3

194.4 ± 1.3

215.4 ± 1.4

232.7 ± 1.5

246.8 ± 1.4

258.8 ± 1.4

T = 308.15 K T = 313.15 K T = 318.15 K T = 323.15 K T = 328.15 K T = 333.15 K 0.00749

239 ± 16

250 ± 14

257 ± 14

263 ± 14

269 ± 15

274 ± 15

0.08491

242.3 ± 1.4

252.2 ± 1.6

260.0 ± 1.6

266.6 ± 1.5

272.2 ± 1.6

276.8 ± 1.6

0.17435

241.3 ± 2.4

251.3 ± 2.3

259.4 ± 2.2

266.3 ± 2.1

272.1 ± 2.1

277.0 ± 2.1

0.34750

243.9 ± 1.8

253.8 ± 1.8

261.8 ± 1.8

268.8 ± 1.8

274.7 ± 1.7

279.7 ± 1.7

0.66092

268.9 ± 1.5

277.6 ± 1.5

284.8 ± 1.6

291.0 ± 1.5

296.4 ± 1.5

301.0 ± 1.6

T = 338.15 K T = 343.15 K T = 348.15 K T = 353.15 K T = 358.15 K T = 363.15 K 0.00749

277 ± 14

279 ± 14

281 ± 15

281 ± 15

283 ± 15

282 ± 16

0.08491

280.6 ± 1.6

283.4 ± 1.5

285.7 ± 1.6

287.4 ± 1.6

288.5 ± 1.7

289.0 ± 1.9

0.17435

281.1 ± 2.1

284.3 ± 2.1

286.8 ± 2.2

288.8 ± 2.2

290.3 ± 2.2

291.1 ± 2.2

0.34750

284.0 ± 1.7

287.4 ± 1.8

290.2 ± 1.8

292.6 ± 1.9

294.3 ± 1.8

295.5 ± 1.8

0.66092

305.0 ± 1.6

308.2 ± 1.7

311.0 ± 1.7

313.2 ± 1.8

315.1 ± 1.8

316.4 ± 1.9

T = 368.15 K T = 373.15 K T = 378.15 K T = 383.15 K T = 388.15 K T = 393.15 K 0.00749

280 ± 15

279 ± 17

277 ± 20

278 ± 23

278 ± 27

278 ± 27

0.08491

289.1 ± 2.0

288.6 ± 2.0

287.5 ± 2.2

286.1 ± 2.3

284.1 ± 2.3

281.8 ± 2.5

0.17435

291.5 ± 2.2

291.4 ± 2.2

290.7 ± 2.2

289.8 ± 2.2

288.4 ± 2.3

286.8 ± 2.5

0.34750

296.3 ± 1.9

296.7 ± 1.9

296.4 ± 1.8

295.9 ± 1.8

294.9 ± 1.8

293.8 ± 2.0

0.66092

317.3 ± 1.9

317.9 ± 1.9

317.9 ± 1.9

317.8 ± 2.0

317.3 ± 2.1

316.6 ± 2.3

a Average experimental values of c p,s can be obtained with equation (10) and with c p,w given in reference 7.

◦ for reaction (1) (by linear regression against 1/T ) 1r Hm,1 = (29.0 ± 2.9) kJ · mol−1 ◦ at T = 298.15 K for at T = 298.15 K and p = 0.1 MPa. We use 1r Hm,1 = 1r Hm,1 −1 m 6 0.5 mol · kg since reaction (1) is isoelectric. We have also used all the experimental results from the literature (17,18,20,21) at m 6 0.5 mol · kg−1 for pK a,2 (or pQ a,2 ) at 273.15 6 T /K 6 313.15 to determine the standard enthalpy change for reaction (4) (by ◦ = (38.9±2.3) kJ · mol−1 at T = 298.15 K. We use linear regression against 1/T ) 1r Hm,2

1432

J. J. Jardine et al.

∆ rCp,m,1 / (J . K −1. mol −1)

190 180 170 160 150 140 0.4 0.3

m . kg ol / (m

0.2 340

0.1

−1 )

0.0

280

300

320

360

380

T/K

FIGURE 9. Change in heat capacity 1r C p,m,1 plotted against temperature T and molality m for proton dissociation reaction (1) for protonated L-histidine(aq) {H2 · His+ (aq)} at p = 0.35 MPa generated from regression parameters given in table 8 and from regression equations for H+ Cl− (aq) in reference 16.

TABLE 8. Regression parameters of equations (13) and (14) for the apparent molar heat capacities C p,φ of aqueous L-histidine {H · His(aq)}, L-histidine + HCl{H2 · His+ Cl− (aq)}, and L-histidine + NaOH {Na+ His− (aq)}. The ± values are chosen to reproduce the generated C p,φ values to within ±0.1 J · K−1 · mol−1 at m 6 0.66 mol · kg−1 and at 278.15 6 T /K 6 393.15

Parameter c0 /(J · K−1 · mol−1 ) c1 /(J · K−2 · mol−1 ) c2 /(J · K−1 · mol−1 ) 10−1 · c3 /(J · K−1 · mol−1 ) c4 /(J · kg · K−1 · mol−2 ) c5 /(J · K−1 · mol−1 ) 10−1 · c6 /(J · mol−1 ) 1a /(J · K−1 · mol−1 )

H · His(aq)

H2 · His+ Cl− (aq)

Na+ His− (aq)

equation (13)

equation (14)

equation (14)

1732.53 ± 0.04

26184.66 ± 0.05

3032.03 ± 0.04

5.3430 ± 0.0001

9.6899 ± 0.0001

−413.1 ± 0.3

−633.1 ± 0.3

−204442 ± 2 72.6 ± 0.1

a Standard deviations of the regression.

−399712 ± 2 33.25 ± 0.09

12.93 ± 0.07

−3766.698 ± 0.008 −137888 ± 1 5

12

9

Thermodynamics of protonated, neutral, and deprotonated L-histidine

1433

TABLE 9. Best standard thermodynamic values for the acid dissociation constant and changes in partial molar volume, heat capacity, enthalpy, and entropy for proton dissociation from aqueous protonated L-histidine {H2 · His+ Cl− (aq)}, reaction (1), at the pressure p = 0.35 MPa T K

◦ 1r Vm,1

1r C ◦p,m,1

◦ 1r Hm,1

cm3 · mol−1

J · K−1 · mol−1

kJ · mol−1

278.15

2.7

151.6

25.63

pK a,1 6.355

◦ 1r Sm,1

J · K−1 · mol−1 −29.52

283.15

2.8

168.8

26.43

6.270

−26.70

288.15

2.8

174.7

27.29

6.186

−23.72

293.15

2.9

176.3

28.17

6.102

−20.73

298.15

3.0

176.0

29.05

6.018

−17.78

303.15

3.0

174.6

29.92

5.934

−14.89

308.15

3.0

172.5

30.79

5.850

−12.08

313.15

3.0

170.2

31.65

5.767

−9.34

318.15

3.0

167.8

32.49

5.684

−6.69

323.15

3.0

165.3

33.33

5.602

−4.12

328.15

3.0

163.0

34.15

5.520

−1.62

333.15

2.9

160.8

34.96

5.439

0.80

338.15

2.9

158.8

35.76

5.358

3.16

343.15

2.8

157.0

36.55

5.278

5.46

348.15

3.0

155.4

37.33

5.198

7.69

353.15

3.0

153.9

38.10

5.119

9.88

358.15

3.0

152.6

38.87

5.041

12.01

363.15

3.0

151.3

39.63

4.963

14.10

368.15

3.0

150.0

40.38

4.886

16.14

373.15

3.0

148.7

41.13

4.810

18.13

378.15

3.0

147.2

41.87

4.734

20.08

383.15

2.9

145.5

42.60

4.659

21.99

388.15

2.9

143.5

43.32

4.584

23.84

393.15

2.8

141.2

44.03

4.511

25.65

1a

1.1

9

0.03

0.00005

0.06

a Maximum uncertainties from propagation of experimental uncertainties in our 1 V ◦ and r m,1 1r C ◦p,m,1 values.

◦ + (1.987 kJ · kg1/2 · mol−3/2 ) · m 1/2 for reaction (4) at T = 298.15 K, 1r Hm,2 = 1r Hm,2 p = 0.1 MPa, and m 6 0.66 mol · kg−1 . The enthalpy changes (1r Hm,1 , m, T ) and (1r Hm,2 , m, T ) for reactions (1) and (4), respectively, were calculated by integrating our (1r C p,m,1 , m, T ) and (1r C p,m,2 , m, T ) surfaces over temperature and over molality. The integration constants we used are 1r Hm,1 , and 1r Hm,2 at T = 298.15 K given above at p = 0.1 MPa. The effect of

1434

J. J. Jardine et al.

∆ rCp,m,2 / (J . K −1. mol −1)

− 230 − 240 − 250 − 260 − 270 0.4

m

0.3

/ (m

0.2 −1 )

. kg ol

0.1

320 0.0

300 280

360

340

380

T/K

FIGURE 10. Change in heat capacity 1r C p,m,2 plotted against temperature T and molality m for proton dissociation reaction (4) for L-histidine(aq) {H · His(aq)} at p = 0.35 MPa generated from regression parameters given in table 8, from equations for Na+ Cl− (aq) in reference 3, and from regression equations for H+ Cl− (aq) in reference 16.

∆ r Hm,1 / (J . K −1. mol −1)

45 40 35 30 25 0.4

m

0.3

−1 )

. kg ol

/ (m

0.2 0.1 0.0

280

300

320

340

360

380

T/K

FIGURE 11. Change in enthalpy of reaction 1r Hm,1 plotted against temperature T and molality m for proton dissociation reaction (1) for protonated L-histidine(aq) {H2 · His+ (aq)} at p = 0.35 MPa. See text for a description of the calculations.

Thermodynamics of protonated, neutral, and deprotonated L-histidine

1435

TABLE 10. Best standard thermodynamic values for the thermodynamic acid dissociation constant and changes in partial molar volume, heat capacity, enthalpy, and entropy for proton dissociation from aqueous zwitterionic L -histidine {H · His(aq)}, reaction (4), at the pressure p = 0.35 MPa T K

◦ 1r Vm,2

1r C ◦p,m,2

◦ 1r Hm,2

cm3 · mol−1

J · K−1 · mol−1

kJ · mol−1

278.15

18.7

−229.5

43.59

9.572

−26.52

283.15

56.8

−228.2

42.45

9.432

−30.65

288.15

78.2

−229.8

41.31

9.300

−34.70

293.15

93.4

−231.5

40.15

9.176

−38.71

298.15

105.5

−233.0

38.99

9.060

−42.68

303.15

115.6

−234.1

37.82

8.951

−46.60

308.15

124.3

−235.0

36.65

8.848

−50.47

313.15

131.9

−235.8

35.47

8.752

−54.28

318.15

138.5

−236.6

34.29

8.662

−58.05

323.15

144.3

−237.3

33.11

8.578

−61.77

328.15

149.3

−238.0

31.92

8.499

−65.45

333.15

153.5

−238.8

30.73

8.426

−69.07

338.15

157.0

−239.7

29.53

8.357

−72.66

343.15

159.7

−240.7

28.33

8.293

−76.20

348.15

161.6

−241.8

27.12

8.233

−79.71

353.15

162.8

−243.1

25.91

8.177

−83.18

358.15

163.3

−244.6

24.69

8.126

−86.62

363.15

163.0

−246.3

23.47

8.078

−90.04

368.15

161.9

−248.4

22.23

8.034

−93.43

373.15

160.0

−250.7

20.98

7.994

−96.81

378.15

157.2

−253.4

19.72

7.956

−100.17

383.15

153.7

−256.5

18.45

7.922

−103.53

388.15

149.3

−260.1

17.15

7.892

−106.89

393.15

144.1

−264.2

15.84

7.864

−110.25

1a

1.0

8

0.02

pK a,2

0.00004

◦ 1r Sm,2

J · mol−1

0.05

a Maximum uncertainties from propagation of experimental uncertainties in our ◦ and 1 C ◦ 1r Vm,2 r p,m,2 values.

changing pressure to p = 0.35 MPa changes these reference values only insignificantly compared with their estimated uncertainties noted above. The calculated (1r Hm,1 , m, T ) and (1r Hm,2 , m, T ) surfaces are shown in figures 11 and 12. We estimate total uncertainties in these two surfaces as 11 6 ±3.6 kJ · mol−1 for 1r Hm,1 and 12 6 ±3.1 kJ · mol−1 for 1r Hm,2 , including uncertainties that result from integration of (1C p,m , m, T ). The two (1r Hm , m, T ) surfaces were then integrated using the values of pQ a,1 and pQ a,2

1436

J. J. Jardine et al.

∆ r Hm,2 / (J . K −1. mol −1)

60 50 40 30 20 0.5 0.4

m

0.2

−1 )

. kg ol

/ (m

0.3 0.1 0.0

280

300

320

340

360

380

T/K

FIGURE 12. Change in enthalpy of reaction 1r Hm,2 plotted against temperature T and molality m for proton dissociation reaction (4) for L-histidine(aq) {H · His(aq)} at p = 0.35 MPa. See text for a description of the calculations.

given above at T = 298.15 K and p = 0.1 MPa. Again, the effect of changing pressure to p = 0.35 MPa changes these reference values only insignificantly compared with their estimated uncertainties. The (pQ a,1 , m, T ) and (pQ a,2 , m, T ) surfaces are shown in figures 13 and 14. We estimate total uncertainties 1 6 ±0.15 for both of these surfaces, including uncertainties propagated from the two integrations. The greatest contributing factor to this uncertainty is the uncertainty in the 1r Hm reference values at T = 298.15 K. The (1r Sm , m, T ) surfaces shown in figures 15 and 16 were calculated from equation (15) using the (1r Hm , m, T ) and (pQ a , m, T ) surfaces obtained above for each proton dissociation, and by using R = 8.31451 J · K−1 · mol−1 : 1r Sm = 1r Hm /T + R · ln Q a .

(15)

◦ , and calculated values of 1r Vm◦ , 1r C ◦p,m , 1r Hm◦ , 1r Sm 10. The values of both pK a,1 and pK a,2 were regressed

Summaries of our pK a are given in tables 9 and by using equation (16) with Tr = 333.15 K, and the resulting parameters an are given in table 11: pK a =

3 X {an · (T − Tr )n }.

(16)

n=0

4. Conclusions We have measured apparent molar heat capacities C p,φ for individual solutions of aqueous L -histidine, of aqueous L -histidine with equimolal HCl, and of aqueous L -histidine with

Thermodynamics of protonated, neutral, and deprotonated L-histidine

1437

6.4

p Qa 1

6.0 5.6 5.2 4.8

0.5

0.1 380

m

360

m

0.2

340

T/K

ol . kg

0.3 320

/(

300

−1

280

)

0.4

0.0

FIGURE 13. Plot of pQ a,1 against temperature T and molality m for reaction (1) for protonated L -histidine(aq) {H2 · His+ (aq)} at p = 0.35 MPa. See text for a description of the calculations.

9.6

p Qa 2

9.2 8.8 8.4 8.0 7.6

0.5

0.1

360 380

0.0

m

340

T/K

/(

320

ol . kg −

0.3 0.2

m

300

1

7.2

)

0.4

FIGURE 14. Plot of pQ a,2 against temperature T and molality m for reaction (4) for L-histidine(aq) {H · His(aq)} at p = 0.35 MPa. See text for a description of the calculations.

1438

J. J. Jardine et al.

30

∆ r Sm,1 / (J . K −1. mol −1)

20 10 0 − 10 − 20 − 30

m

0.4 0.3

/ (m

0.1

−1 )

. kg ol

0.2

0.0

280

300

320

340

360

380

T/K

FIGURE 15. Change in entropy 1r Sm,1 plotted against temperature T and molality m for reaction (1) for protonated L-histidine(aq) {H2 · His+ (aq)} at p = 0.35 MPa. See text for a description of the calculations.

∆ r Sm,2 / (J . K −1. mol −1)

20 0 − 20 − 40 − 60 − 80

m

− 100 0.5 0.4 0.3

−1 )

. kg ol

/ (m

0.2 0.1 0.0

260

280

300

320

340

360

380

400

T/K

FIGURE 16. Change in entropy 1r Sm,2 plotted against temperature T and molality m for reaction (4) for L-histidine(aq) {H · His(aq)} at p = 0.35 MPa. See text for a description of the calculations.

Thermodynamics of protonated, neutral, and deprotonated L-histidine

1439

TABLE 11. Regression parameters for equation (16) using Tr = 333.15 K for pK a,1 and pK a,2 from tables 9 and 10 for proton ionizations from protonated L-histidine, reactions (1) and (4). The ± values for each parameter are chosen to reproduce the generated pK a,1 and pK a,2 values to within ±0.001 at 278.15 6 T /K 6 393.15

Coefficient

Reaction (1)

Reaction (4)

pK a,1

pK a,2

a0

5.4389 ± 0.0006

8.4243 ± 0.0005

102 · a1 /(K−1 ) 105 · a2 /(K−2 ) 107 · a3 /(K−3 ) 1a

−1.613 ± 0.001

−1.425 ± 0.001

1.04 ± 0.03

10.112 ± 0.02 −3.321 ± 0.02

0.0012

0.0012

a Standard deviations of the regressions.

equimolal NaOH at molalities m = (0.015 to 0.66) mol · kg−1 , at temperatures T = (278.15 to 393.15) K, and at the pressure p = 0.35 MPa with greater precision than has been reported previously. Standard deviations 1 of our regressions of (C p,φ , m, T ) are all 69 J · mol−1 · K−1 . Apparent molar volumes Vφ were also measured for the three systems, and standard deviations 1 of our regressions of (Vφ , m, T ) are all 6 ±0.4 cm3 · mol−1 . The regression equations were then used to calculate (1r C p,m , m, T ) and (1r Vm , m, T ) surfaces for the first and second proton dissociation reactions for protonated aqueous L -histidine with estimated maximum uncertainties 11 = ±6.7 J · mol−1 · K−1 for (1r C p,m,1 , m, T ) and 12 = ±9.6 J · mol−1 · K−1 for (1r C p,m,2 , m, T ), and 11 6 ±1.1 cm3 · mol−1 for 1r Vm,1 and 12 6 ±1.0 cm3 · mol−1 for 1r Vm,2 . Reference values of pQ a,1 , pQ a,2 , 1r Hm,1 , and 1r Hm,2 at T = 298.15 K from the literature were used as integration constants to obtain the surfaces (1r Hm , m, T ), (pQ a , m, T ), and (1r Sm , m, T ) for both proton dissociation reactions. This integration process tends to minimize uncertainties in the resulting enthalpies and free energies, rather than amplifying the uncertainties as is the case when performing differentiation of free energies to obtain enthalpies and heat capacities, or when performing differentiation of enthalpies to obtain heat capacities.

Note added in proof We note that Vφ values for H · His(aq) reported in this paper differ by only −0.2 6 1/(cm3 · mol−1 ) 6 0.76 from recently reported results at 278.15 6 (T /K) 6 308.15, at 0.025 6 m/(mol · kg−1 ) 6 0.2, and at p = 0.010 MPa (Chen, J.-L.; Li, Z.-F.; Wang, B.-H.; Zhang, Y.-M. J. Chem. Thermodynamics 2000, 32, 805–819; doi:10.1006/jcht.2000.0658). REFERENCES 1. Ballerat-Busserolles, K.; Ford, T. D.; Call, T. G.; Woolley, E. M. J. Chem. Thermodynamics 1999, 31, 741–762; doi:10.1006/jcht.1999.0484.

1440 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

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