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Question 8.E.A: Using activity coefficients correctly, find the pH of 1.0 × ...

Using activity coefficients correctly, find the pH of 1.0 × 10−210^{−2} M NaOH.

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pH = −log⁡AH+−\log \mathcal{A}_{H^{+}} . But AH+AOH−=Kw\mathcal{A}_{H^{+}} \mathcal{A}_{OH^{−}} = K_{w} ⇒ AH+=Kw/AOH−\mathcal{A}_{H^{+}} = K_{w}/\mathcal{A}_{OH^{−}}.

For 1.0 × 10−210^{−2} M NaOH, [OH−]=1.0×10−2[OH^{−}] = 1.0 × 10^{−2} M and γOH−\gamma _{OH^{−}} = 0.900

(Table 7-1, with μ = 0.010 M).

TABLE 7-1      Activity coefficients for aqueous solutions at 25°C
 

 

Ion

 Ion

size

(α  , pm)

Ionic strength (μ , M)

0.001

0.005

0.01

0.05

0.1
Charge = ± 1

                 Activity coefficient (γ\gamma  )
H+H^{+}

900

 0.967 0.933 0.914 0.86

0.83
(C6H5)2CHCO2−,(C3H7)4N+(C_{6}H_{5})_{2}CHCO_{2}^{-} , (C_{3}H_{7})_{4}N^{+}

800

 0.966 0.931 0.912 0.85

0.82
(O2N)3C6H2O−,(C3H7)3NH+,CH3OC6H4CO2−,(O_{2}N)_{3}C_{6}H_{2}O^{-} , (C_{3}H_{7})_{3}NH^{+} , CH_{3}OC_{6}H_{4}CO_{2}^{-} ,

700

 0.965 0.930 0.909 0.845

0.81
Li+,C6H5CO2−,HOC6H4CO2−,ClC6H4CO2−,C6H5CH2CO2−, CH2==CHCH2CO2−,(CH3)2CHCH2CO2−,(CH3CH2)4N+,(C3H7)2NH2+Li^{+} , C_{6}H_{5}CO_{2}^{-} , HOC_{6}H_{4}CO_{2}^{-} , ClC_{6}H_{4}CO_{2}^{-} , C_{6}H_{5}CH_{2}CO_{2}^{-} ,  CH_{2}==CHCH_{2}CO_{2}^{-} , (CH_{3})_{2}CHCH_{2}CO_{2}^{-} , (CH_{3}CH_{2})_{4}N^{+} , (C_{3}H_{7})_{2}NH_{2}^{+}

600

 0.965 0.929 0.907 0.835

0.80
Cl2CHCO2−,Cl3CCO2−,(CH3CH2)3NH+,(C3H7)2NH3+Cl_{2}CHCO_{2}^{-} , Cl_{3}CCO_{2}^{-} , (CH_{3}CH_{2})_{3}NH^{+} , (C_{3}H_{7})_{2}NH_{3}^{+}

500

 0.964 0.928 0.904 0.83

0.79
Na+,CdCl+,ClO2−,IO3−,HCO3−,H2PO4−,HSO3−,H2AsO4−,CO(NH3)4(NO2)2+,CH3CO2−,ClCH2CO2−,(CH3)4N+,(CH3CH2)2NH2+,H2NCH2CO2−Na^{+} , CdCl^{+} , ClO_{2}^{-} , IO_{3}^{-} , HCO_{3}^{-} , H_{2}PO_{4}^{-} , HSO_{3}^{-} , H_{2}AsO_{4}^{-} , CO(NH_{3})_{4}(NO_{2})_{2}^{+} , CH_{3}CO_{2}^{−} , ClCH_{2}CO_{2}^{−} , (CH_{3})_{4}N^{+} , (CH_{3}CH_{2})_{2}NH_{2}^{+} , H_{2}NCH_{2}CO_{2}^{-}

450

 0.964 0.928 0.902 0.82

0.775
+H3NCH2CO2H,(CH3)3NH+,CH3CH2NH3+^{+}H_{3}NCH_{2}CO_{2}H, (CH_{3})_{3}NH^{+} , CH_{3}CH_{2}NH_{3}^{+}

400

 0.964 0.927 0.901 0.815

0.77
OH−,F−,SCN−,OCN−,HS−,ClO3−,ClO4−,BrO3−,IO4−,MnO4−,HCO2−,H2citrate−,CH3NH3+,(CH3)2NH2+OH^{-} , F^{−} , SCN^{−} , OCN^{−} , HS^{−} , ClO_{3}^{−} , ClO_{4}^{−} , BrO_{3}^{−} , IO_{4}^{−} , MnO_{4}^{−} , HCO_{2}^{−} , H_{2citrate}^{−} , CH_{3}NH_{3}^{+} , (CH_{3})_{2}NH_{2}^{+}

350

 0.964 0.926 0.900 0.81

0.76
K+,Cl−,Br−,I−,CN−,NO2−,NO3−K^{+} , Cl^{−} , Br^{−} , I^{−} , CN^{−} , NO_{2}^{−} , NO_{3}^{−}

300

 0.964 0.925 0.899 0.805

0.755
Rb+,Cs+,NH4+,Tl+,Ag+Rb^{+} , Cs^{+} , NH_{4}^{+} , Tl^{+} , Ag^{+}

250

 0.964 0.924 0.898 0.80

0.75
Charge = ±2

     Activity coefficient ( γ\gamma )
Mg2+,Be2+Mg^{2+} , Be^{2+}

800

 0.872 0.755 0.69 0.52

0.45
CH2(CH2CH2CO2−)2,(CH2CH2CH2CO2−)2CH_{2}(CH_{2}CH_{2}CO_{2}^{−})_{2} , (CH_{2}CH_{2}CH_{2}CO_{2}^{−})_{2}

700

 0.872 0.755 0.685 0.50

0.425
Ca2+,Cu2+,Zn2+,Sn2+,Mn2+,Fe2+,Ni2+,Co2+,C6H4(CO2−)2,H2C(CH2CO2−)2,(CH2CH2CO2−)2Ca^{2+} , Cu^{2+} , Zn^{2+} , Sn^{2+} , Mn^{2+} , Fe^{2+} , Ni^{2+} , Co^{2+} , C_{6}H_{4}(CO_{2}^{−})_{2} , H_{2}C(CH_{2}CO_{2}^{−})_{2} , (CH_{2}CH_{2}CO_{2}^{−})_{2}

600

 0.870 0.749 0.675 0.485

0.405
Sr2+,Ba2+,Cd2+,Hg2+,S2−,S2O42−,WO42−,H2C(CO2−)2,(CH2CO2−)2,(CHOHCO2−)2Sr^{2+} , Ba^{2+} , Cd^{2+} , Hg^{2+} , S^{2−} , S_{2}O_{4}^{2−} , WO_{4}^{2−} , H_{2}C(CO_{2}^{−})_{2} , (CH_{2}CO_{2}^{−})_{2} , (CHOHCO_{2}^{−})_{2}

500

 0.868 0.744 0.67 0.465

0.38
Pb2+,CO32−,SO32−,MoO42−,Co(NH3)5Cl2+,Fe(CN)5NO2−,C2O42−,Hcitrate2−Pb^{2+} , CO_{3}^{2−} , SO_{3}^{2−} , MoO_{4}^{2−} , Co(NH_{3})_{5}Cl^{2+} , Fe(CN)_{5}NO^{2−} , C_{2}O_{4}^{2−} , Hcitrate^{2−}

450

 0.867 0.742 0.665 0.455

0.37
Hg22+,SO42−,S2O32−,S2O62−,S2O82−,SeO42−,CrO42−,HPO42−Hg_{2}^{2+} , SO_{4}^{2−} , S_{2}O_{3}^{2−} , S_{2}O_{6}^{2−} , S_{2}O_{8}^{2−} , SeO_{4}^{2−} , CrO_{4}^{2−} , HPO_{4}^{2−}

400

 0.867 0.740 0.660 0.445

0.355
Charge = ±3

    Activity coefficient (γ\gamma  )
Al3+,Fe3+,Cr3+,Sc3+,Y3+,In3+,lanthanidesaAl^{3+} , Fe^{3+} , Cr^{3+} , Sc^{3+} , Y^{3+} , In^{3+} , lanthanides^{a}

900

 0.738 0.54 0.445 0.245

0.18
citrate3−citrate^{3−}

500

 0.728 0.51 0.405 0.18

0.115
PO43−,Fe(CN)63−,Cr(NH3)63+,Co(NH3)5H2O3+PO_{4}^{3−} , Fe(CN)_{6}^{3−} , Cr(NH_{3})_{6}^{3+} , Co(NH_{3})_{5}H_{2}O^{3+}

400

 0.725 0.505 0.395 0.16

0.095
Charge = ±4

     Activity coefficient ( γ\gamma )
Th4+,Zr4+,Ce4+,Sn4+Th^{4+} , Zr^{4+} , Ce^{4+} , Sn^{4+}

1 100

 0.588 0.35 0.255 0.10

0.065
Fe(CN)64−Fe(CN)_{6}^{4−}

500

 0.57 0.31 0.20 0.048

0.021

a. Lanthanides are elements 57–71 in the periodic table.

AH+=Kw[OH−]γOH−=1.0×10−14(1.0×10−2)(0.900)\mathcal{A}_{H^{+}} = \frac{K_{w}}{[OH^{−}]\gamma _{OH^{−}}} = \frac{1.0 × 10^{−14}}{(1.0 × 10^{−2})(0.900)}
=1.11 × 10−1210^{−12} ⇒ pH = −log⁡AH+−\log \mathcal{A}_{H^{+}} = 11.95

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