- Email: [email protected]
Titration Calorimetry With Dilute Titrant☆
Eq. (D-1) to (D-4)
APPENDIX D
Titration Calorimetry With Dilute Titrant Isothermal titration calorimetry is designed for a liquid sample of low viscosity. Thus, the method is not applicable for a solid sample. If the titrating sample is a viscous liquid, the titration becomes problematic in quantitative delivery. In such cases, a dilute solution of the sample, instead of pure liquid, can be titrated. In this manner, a solid sample could also be used. The conversion of the raw data to the excess partial molar enthalpy is straightforward (Westh et al., 1998; Koga, 2003b). Consider a small amount of an aqueous solution of B of the fixed mole ðt Þ ðtÞ fraction x(t) titrated into the titrand mixture consistB with r ¼ xB = 1 xB ing of nB of B and nW of W. Thus, the process is described as ½nB of B, nW of W + ½δnB of B, δnW of W ðtÞ ! ½ðnB + δnB Þ of B, ðnW + δnW Þ of W ,
(D-1)
where superscript (t) stands for the titrant whose composition is fixed at x(t) B. The total enthalpy difference for the above process (Eq. D-1) is determined as δq. Hence, δq ¼ H ðnB + δnB , nW + δnW Þ H ðnB , nW Þ H ðtÞ ðδnB , δnW Þ,
(D-2)
with ðt Þ
ðtÞ
H ðtÞ ðδnB ,δnW Þ ¼ δnB HB + δnW HW : It follows then that δq H ðnB + δnB ,nW + δnW Þ H ðnB , nW + δnW Þ ¼ h δnB δnB ðt Þ H ðnB ,nW + δnW Þ H ðnB , nW Þ H ðtÞ + HB W : r rδnW
(D-3)
(D-4)
We used the relationship r ¼ δnB =δnW in deriving Eq. (D-4). If δnB and δnW are sufficiently small, the right-hand side of Eq. (D-4) becomes
403
404 ðtÞ
h ¼ HB +
HW H ðt Þ HB W , r r
(D-5)
where HB and HW are the partial molar enthalpy of B and W, respectively. The last two terms on the right of Eq. (D-5) are associated with the titrant and constant. A small increment of Eq. (D-5), δh, can be written, using the Gibbs-Duhem relation as δHW xB δh ¼ δHB + : (D-6) ¼ δHB 1 r rxW Since we take the symmetric reference states, Eq. (D-6) is rewritten by using the excess partial molar enthalpies, HBE ¼ HB HB∗ , etc., as E δHW xB E E δh ¼ δHB + , (D-7) ¼ δHB 1 r rxW where HB∗ is the enthalpy of pure B, which is taken as zero. The same is true for W. Therefore, δh : δHBE ¼ xB 1 rxW It follows then HBE ¼
X δHBE + constant:
(D-8)
(D-9)
The integration constant on the right of Eq. (D-9) can be determined if a value of HEB is available at one point. For ethylene glycol (EG)-H2O, the values of HEm are available in the literature and they show a minimum. Its minimum value was used as HEEG (Koga, 2003b). Recall Eq. (II-75) E @Hm E HB ¼ ð1 xB Þ + HmE : @xB E @Hm E ¼ 0, hence HEG ¼ HmE at the minimum. At the minimum, @xB The B-B enthalpic interaction, HEB-B, is then readily calculated using HEB data obtained by Eq. (D-9) as E δHB E HBB ¼ ð1 xB Þ : (D-10) δxB
APPENDIX D
Titration Calorimetry With Dilute Titrant Isothermal titration calorimetry is designed for a liquid sample of low viscosity. Thus, the method is not applicable for a solid sample. If the titrating sample is a viscous liquid, the titration becomes problematic in quantitative delivery. In such cases, a dilute solution of the sample, instead of pure liquid, can be titrated. In this manner, a solid sample could also be used. The conversion of the raw data to the excess partial molar enthalpy is straightforward (Westh et al., 1998; Koga, 2003b). Consider a small amount of an aqueous solution of B of the fixed mole ðt Þ ðtÞ fraction x(t) titrated into the titrand mixture consistB with r ¼ xB = 1 xB ing of nB of B and nW of W. Thus, the process is described as ½nB of B, nW of W + ½δnB of B, δnW of W ðtÞ ! ½ðnB + δnB Þ of B, ðnW + δnW Þ of W ,
(D-1)
where superscript (t) stands for the titrant whose composition is fixed at x(t) B. The total enthalpy difference for the above process (Eq. D-1) is determined as δq. Hence, δq ¼ H ðnB + δnB , nW + δnW Þ H ðnB , nW Þ H ðtÞ ðδnB , δnW Þ,
(D-2)
with ðt Þ
ðtÞ
H ðtÞ ðδnB ,δnW Þ ¼ δnB HB + δnW HW : It follows then that δq H ðnB + δnB ,nW + δnW Þ H ðnB , nW + δnW Þ ¼ h δnB δnB ðt Þ H ðnB ,nW + δnW Þ H ðnB , nW Þ H ðtÞ + HB W : r rδnW
(D-3)
(D-4)
We used the relationship r ¼ δnB =δnW in deriving Eq. (D-4). If δnB and δnW are sufficiently small, the right-hand side of Eq. (D-4) becomes
403
404 ðtÞ
h ¼ HB +
HW H ðt Þ HB W , r r
(D-5)
where HB and HW are the partial molar enthalpy of B and W, respectively. The last two terms on the right of Eq. (D-5) are associated with the titrant and constant. A small increment of Eq. (D-5), δh, can be written, using the Gibbs-Duhem relation as δHW xB δh ¼ δHB + : (D-6) ¼ δHB 1 r rxW Since we take the symmetric reference states, Eq. (D-6) is rewritten by using the excess partial molar enthalpies, HBE ¼ HB HB∗ , etc., as E δHW xB E E δh ¼ δHB + , (D-7) ¼ δHB 1 r rxW where HB∗ is the enthalpy of pure B, which is taken as zero. The same is true for W. Therefore, δh : δHBE ¼ xB 1 rxW It follows then HBE ¼
X δHBE + constant:
(D-8)
(D-9)
The integration constant on the right of Eq. (D-9) can be determined if a value of HEB is available at one point. For ethylene glycol (EG)-H2O, the values of HEm are available in the literature and they show a minimum. Its minimum value was used as HEEG (Koga, 2003b). Recall Eq. (II-75) E @Hm E HB ¼ ð1 xB Þ + HmE : @xB E @Hm E ¼ 0, hence HEG ¼ HmE at the minimum. At the minimum, @xB The B-B enthalpic interaction, HEB-B, is then readily calculated using HEB data obtained by Eq. (D-9) as E δHB E HBB ¼ ð1 xB Þ : (D-10) δxB