Chapter 6. (a) The equilibrium constant epression is [NH4 [NH 3[H3O To find the equilibrium constant s value, we note that the overall reaction is the sum of two reactions, each with a standard equilibrium constant; thus NH3( aq) HO () l? OH ( aq) NH4 ( aq) HO 3 ( aq) OH ( aq)? HO () l ( ) w b,nh3 # w # a,nh w 9 0 75. # 0 570. # 0 (b) The equilibrium constant epression is [I [S 4 a,nh4 To find the equilibrium constant s value, we note that the overall reaction is the sum of two reactions, each with a standard equilibrium constant; thus PbI() s? Pb ( aq) I ( aq) Pb ( aq) S ( aq)? PbS() s sp,pbi # ( sp,pbs) 9 (. 79# 0 ) # 8 3# 0 3 # 0 (c) The equilibrium constant epression is [Cd(CN) [Y 4 4 4 [CdY [CN To find the equilibrium constant s value, we note that the overall reaction is the sum of two reactions, each with a standard equilibrium constant; thus 4 CdY ( aq)? Cd ( aq) Y ( aq) 9 Chapter 6 Equilibrium Chemistry By standard equilibrium constant, we mean one of the following: an acid dissociation constant, a base dissociation constant, a solubility product, a stepwise or an overall formation constant, or a solvent dissociation constant. 5 Cd ( aq) 4CN ( aq)? Cd(CN) 4 ( aq) ( f,cdy ) # b4,cd(cn) 4 7 6 #(. 83# 0 ) 89. 88. # 0 where b 4 is equal to 3 4. From Appendi, we have log 6.0, log 5., log 3 4.53, and log 4.7. Adding together these four values gives logb 4 as 7.9 and b 4 as 8.3 0 7.
5 Solutions Manual for Analytical Chemistry. (d) The equilibrium constant epression is [Ag(NH 3)[Cl [NH 3 To find the equilibrium constant s value, we note that the overall reaction is the sum of two reactions, each with a standard equilibrium constant; thus AgCl() s? Ag ( aq) Cl ( aq) From Appendi, we have log 3.3 and log 3.9. Adding together these four values gives logb as 7. and b as.66 0 7. Ag ( aq) NH3( aq)? Ag(NH 3) ( aq) # b sp,agcl, Ag(NH 3) (. 8# 0 )#( 66. # 0 ) 3. 0# 0 0 7 3 (e) The equilibrium constant epression is [Ba [HCO 3 [H3O To find the equilibrium constant s value, we note that the overall reaction is the sum of five reactions, each with a standard equilibrium constant; thus BaCO3() s? Ba ( aq) CO3 ( aq) CO 3 ( aq) HO () l? OH ( aq) HCO 3 ( aq) HCO 3 ( aq) HO() l? OH ( aq) HCO3( aq) HO 3 ( aq) OH ( aq)? HO () l PO 4 3 p a.35 HPO 4 p a 7.0 H PO 4 p a.5 H 3 PO 4 ph F p a 3.7 HF Figure SM6. Ladder diagram showing the areas of predominance for H 3 PO 4 on the left and the areas of predominance for HF on the right. HO 3 ( aq) OH ( aq)? HO () l _ sp,baco3 # b,co3 # b,hco3 # ( w) w w sp,baco3 # # # a,hco3 a,hco3 ( w) 9 (. 50# 0 ) # # 7 4. # 0 469. # 0 445. # 0. Figure SM6. shows the ladder diagram for H 3 PO 4 and for HF. From the ladder diagram, we predict that a reaction between H 3 PO 4 and F is favorable because their respective areas of predominance do not overlap. On the other hand, a reaction between HPO 4 and F, which must take place if the final product is to include HPO 4, is unfavorable because the areas of predominance for HPO 4 and F, overlap. To find the equilibrium constant for the first reaction, we note that it is the sum of three reactions, each with a standard equilibrium constant; thus HPO 3 4( aq) HO() l? HPO4( aq) HO 3 ( aq) 8
Chapter 6 Equilibrium Chemistry 53 F ( aq) H O() l? OH ( aq) HF( aq) HO 3 ( aq) OH ( aq)? HO () l ( ) w a,h3po4# b,f # w a,h3po4# # a,hf 3 7. # 0 4 5 68. # 0 Because is greater than, we know that the reaction is favorable. To find the equilibrium constant for the second reaction, we note that it is the sum of si reactions, each with a standard equilibrium constant; thus HPO 3 4( aq) HO() l? HPO4( aq) HO 3 ( aq) HPO 4 HO? H3O ( aq) HPO4 ( aq) F ( aq) H O() l? OH ( aq) HF( aq) F ( aq) H O() l? OH ( aq) HF( aq) HO 3 ( aq) OH ( aq)? HO () l w HO 3 ( aq) OH ( aq)? HO () l a,h3po4# a,hpo4 #( ) ( ) b,f # w w a,h3po4# a,hpo4 # a k # a a,hf k w 3 8 (. 7# 0 )#( 6. 3 # 0 )# a 4 k 97. # 0 68. # 0 Because is less than, we know that the reaction is unfavorable. 3. To calculate the potential we use the Nernst equation; thus 4 o o E ( E E ) 0596 [Sn [Fe 3 Fe /Fe 4 Sn /Sn log 3 [Sn [Fe (.. ). (. )(. ) 0 77 0 54 0 0596 0 00 0 030 log (. 0 05)( 050) 66 V 4. We can balance these reactions in a variety of ways; here we will identify the balanced halfreactions from Appendi 3 and add them together after adjusting the stoichiometric coefficients so that all electrons released in the oidation reaction are consumed in the reduction reaction. (a) The two halfreactions are MnO 4 ( aq) 8H ( aq) 5e? Mn ( aq) 4HO() l HSO 3( aq) HO() l? SO 4 ( aq) 4H ( aq) e 4 Within the contet of this problem, we do not need to balance the reactions; instead, we simply need to identify the two halfreactions and subtract their standard state reduction potentials to arrive at the reaction s standard state potential. Nevertheless, it is useful to be able to write the balanced overall reaction from the halfreactions as this information is needed if, as in Problem 3, we seek the reaction s potential under nonstandard state conditions. which combine to give an overall reaction of
54 Solutions Manual for Analytical Chemistry. MnO 4 ( aq) 5HSO3( aq)? Mn ( aq) 5SO 4 ( aq) 4H ( aq) 3HO() l Using the Nernst equation, the standard state potential is o E ( E o o MnO 4 /Mn E ) 5. 7 SO 4 /HSO3. 338 V. 34. V and an equilibrium constant of o ne / 0596 ( )(. )/ 0596 0 0 47. # 0 (b) The two halfreactions are 0 338 6 IO3 ( aq) 6H ( aq) 5e? I() s 3HO() l I ( aq)? I() s e which combine to give an overall reaction of IO3 ( aq) 5I ( aq) 6H ( aq)? 3I() s 3HO() l Using the Nernst equation, the standard state potential is o E ( E o E o IO 3/I I /I ). 95 5355 6595 V. 660 V and an equilibrium constant of o /. ()(. )/. 0 0 548. # 0 (c) The two halfreactions are ne 0 0596 5 06595 0 0596 55 ClO ( aq) H O() l e? Cl ( aq) OH ( aq) I ( aq) 6OH ( aq)? IO3 ( aq) 3HO() l 6e which combine to give an overall reaction of 3ClO ( aq) I ( aq)? 3Cl ( aq) IO 3 ( aq) Using the Nernst equation, the standard state potential is o E ( E o ClO /Cl E o IO 3 /I ) 890 57 633 V and an equilibrium constant of 0 ne o /. ()(. )/. 0 0596 0 06 0 0596 5. 8# 0 6 33 64 5. (a) Because SO 4 is a weak base, decreasing the solution s ph, which makes the solution more acidic, converts some of the SO 4 to HSO 4. Decreasing the concentration of SO 4 shifts the solubility reaction to the right, increasing the solubility of BaSO 4. (b) Adding BaCl, which is a soluble salt, increases the concentration of Ba in solution, pushing the solubility reaction to the left and decreasing the solubility of BaSO 4.
Chapter 6 Equilibrium Chemistry 55 (c) Increasing the solution s volume by adding water decreases the concentration of both Ba and of SO 4, which, in turn, pushes the solubility reaction to the right, increasing the solubility of BaSO 4. 6. (a) A solution of NaCl contains the following species: Na, Cl, H 3 O, and OH. The charge balance equation is [H3 O [Na [Cl [OH and the mass balance equations are 0 M [Na 0 M [Cl (b) A solution of HCl contains the following species: Cl, H 3 O, and OH. The charge balance equation is and the mass balance equation is [H3 O [Cl [ OH 0 M [Cl (c) A solution of HF contains the following species: HF, F, H 3 O, and OH. The charge balance equation is and the mass balance equation is [H3 O [F [OH 0 M [HF [F (d) A solution of NaH PO 4 contains the following species: Na, H 3 PO 4, HPO 3 4, HPO 4, PO 4, H 3 O, and OH. The charge balance equation is [Na [H3O [OH [HPO 4 #[HPO 4 3 #[PO 3 4 and the mass balance equations are 0 M [Na 3 0 M [HPO 3 4 [HPO 4 [HPO 4 [PO 4 (e) A saturated solution of MgCO 3 contains the following species: Mg, CO 3, HCO 3, H CO 3, H 3 O, and OH. The charge balance equation is #[Mg [H3O [OH [HCO 3 #[CO 3 and the mass balance equation is [Mg [HCO 3 [HCO 3 [CO 3 (f) A solution of Ag(CN) prepared using AgNO 3 and CN contains the following ions: Ag, NO 3,, CN, Ag(CN), HCN, H 3 O, and OH. The charge balance equation is A solution of HCl will contain some undissociated HCl(aq); however, because HCl is a strong acid, the concentration of HCl(aq) is so small that we can safely ignore it when writing the mass balance equation for chlorine. For a saturated solution of MgCO 3, we know that the concentration of Mg must equal the combined concentration of carbonate in all three of its forms.
56 Solutions Manual for Analytical Chemistry. [Ag [ [HO 3 [OH [NO 3 [CN [Ag(CN) and the mass balance equations are [NO 3 [Ag [Ag(CN) [ [CN [HCN # [Ag(CN) (g) A solution of HCl and NaNO contains the following ions: H 3 O, OH, Cl, Na, NO, and HNO. The charge balance equation is [Na [H3O [OH [NO [Cl and the mass balance equations are 0 M [Cl 050 M [Na 050 M [NO [HNO 7. (a) Perchloric acid, HClO 4, is a strong acid, a solution of which contains the following species: H 3 O, OH, and ClO 4. The composition of the solution is defined by a charge balance equation and a mass balance equation for ClO 4 [H3O [OH [ClO 4 [ClO 4 and by the w epression for water. 050 M [ HO[ OH 3 w Because HClO 4 is a strong acid and its concentration of 050 M is relatively large, we can assume that and that [OH << [ClO 4 [H3O [ClO 4 050 M The ph, therefore is,.3 To check our assumption, we note that a ph of.30 corresponds to a poh of.70 and to a [OH of.0 0 3 M. As this is less than 5% of 050 M, our assumption that is reasonable. [OH << [ClO 4 (b) Hydrochloric acid, HCl, is a strong acid, a solution of which contains the following species: H 3 O, OH, and Cl. The composition of the solution is defined by a charge balance equation and a mass balance equation for Cl
Chapter 6 Equilibrium Chemistry 57 [H3 O [OH [Cl [Cl.00 # 0 7 M and by the w epression for water. [ HO 3 [ OH w 00. # 0 4 Although HCl is a strong acid, its concentration of.00 0 7 M is relatively small such that we likely cannot assume that [OH << [Cl To find the ph, therefore, we substitute the mass balance equation for Cl into the charge balance equation and rearrange to solve for the concentration of OH [ OH [ HO 3 00. # 0 7 and then substitute this into the w epression for the dissociation of water [ HO 3 "[ HO 3 00. # 0,. 00 # 0 7 4 [ HO (. 00 0 7 )[ HO. 00 0 4 3 # 3 # 0 Solving the quadratic equation gives [H 3 O as.6 0 7 and the ph as 6.79. (c) Hypochlorous acid, HOCl, is a weak acid, a solution of which contains the following species: H 3 O, OH, HOCl, and ClO. The composition of the solution is defined by a charge balance equation and a mass balance equation for HOCl [H3 O [OH [OCl Here, and elsewhere in this tetbook, we assume that you have access to a calculator or other tool that can solve the quadratic equation. Be sure that you eamine both roots to the equation and that you choose the root that makes chemical sense. [HOCl [ClO 05 M and by the a and w epressions for HOCl and water, respectively [H3O[OCl 8 a 3.0 # 0 [HOCl [ HO 3 [ OH w 00. # 0 4 Because the solution is acidic, let s assume that [OH << [H3O which reduces the charge balance equation to [H3 O [ClO Net, we substitute this equation for [ClO into the mass balance equation and solve for [HOCl [HOCl 05
58 Solutions Manual for Analytical Chemistry. Having defined the concentrations of all three species in terms of a single variable, we substitute them back into the a epression for HOCl [H3O[OCl 8 a 05 30. # 0 [HOCl which we can solve using the quadratic equation. Alternatively, we can simplify further by recognizing that because HOCl is a weak acid, likely is significantly smaller than 05 and 05 05 8 05 30. # 0 which gives as.74 0 5 and the ph as 4.56. Checking our assumptions, we note that both are reasonable:.74 0 5 is less than 5% of 05 and [OH, which is 3.6 0 0 is less than 5% of [H 3 O, which is.74 0 5. (d) Formic acid, HCOOH, is a weak acid, a solution of which contains the following species: H 3 O, OH, HCOOH, and HCOO. The composition of the solution is defined by a charge balance equation and a mass balance equation for HCOOH [H O 3 [OH [ HCOO [HCOOH [HCOO 00 M and the a and w epressions for HCOOH and water, respectively [H3O[HCOO [HCOOH 80. 0 4 a # [ HO 3 [ OH w 00. # 0 4 Because the solution is acidic, let s assume that [OH << [H3O which reduces the charge balance equation to [H3 O [HCOO Net, we substitute this equation for [HCOO into the mass balance equation and solve for [HCOOH [ HCOOH 00 Having defined the concentrations of all three species in terms of a single variable, we substitute them back into the a epression for HCOOH [H3O[HCOO 4 a [HCOOH 00 80. # 0 and solve for. In Problem 8b we simplified this equation further by assuming that is significantly smaller than the initial concentration
Chapter 6 Equilibrium Chemistry 59 of the weak acid; this likely is not the case here because HCOOH is a stronger weak acid than HOCl and, therefore, more likely to dissociate. Solving for using the quadratic equation gives its value as.5 0 3 and the ph as.9 Checking our one assumption, we note that it is reasonable: the [OH, which is 8.00 0, is less than 5% of [H 3 O, which is.5 0 3. (e) Barium hydroide, Ba(OH), is a strong base, a solution of which contains the following species: H 3 O, OH, and Ba. The composition of the solution is defined by a charge balance equation and a mass balance equation for Ba [Ba # [HO 3 [OH nowing when an approimation likely is reasonable is a skill you learn with practice. There is no harm in making an assumption that fails, as long as you are careful to check the assumption after solving for. There is no harm, as well, in not making an assumption and solving the equation directly. [Ba 050 M and by the w epression for water. [ HO 3 [ OH w 00. # 0 4 Because Ba(OH) is a strong base and its concentration of 050 M is relatively large, we can assume that and that [H3O<< [ OH [OH #[ Ba # ( 050 M) 0 M The poh, therefore is,.00 and the ph is 3.0 To check our assumption, we note that a ph of 3.00 corresponds to a [H 3 O of.0 0 3 M. As this is less than 5% of 0 M, our assumption that is reasonable. [H3O<< [ OH (f) Pyridine, C 5 H 5 N, is a weak base, a solution of which contains the following species: H 3 O, OH, C 5 H 5 N, and C 5 H 5 NH. The composition of the solution is defined by a charge balance equation and a mass balance equation for C 5 H 5 N [H3O [C5HNH 5 [OH [ CHN 5 5 [C5HNH 5 00 M and the b and w epressions for C 5 H 5 N and water, respectively [OH [C5HNH 5 [C HN 69. 0 9 b # 5 5 [ HO 3 [ OH w 00. # 0 4 Because the solution is basic, let s assume that
60 Solutions Manual for Analytical Chemistry. [H3O<< [OH which reduces the charge balance equation to [ CHN 5 5 [OH Net, we substitute this equation for [C 5 H 5 NH into the mass balance equation and solve for [C 5 H 5 N [ CHN 5 5 00 ph A HA H A p a 6.33 p a.90 Figure SM6. Ladder diagram for maleic acid showing the ph values for which H A, HA, and A are the predominate species. ph A HA H A p a 5.696 p a.847 Figure SM6.3 Ladder diagram for malonic acid showing the ph values for which H A, HA, and A are the predominate species. Having defined the concentrations of all three species in terms of a single variable, we substitute them back into the b epression for C 5 H 5 NH [OH [C5HNH 5 9 b [C HN 00. 0 69. # 0 5 5 which we can solve using the quadratic equation. Alternatively, we can simplify further by recognizing that because C 5 H 5 N is a weak base, likely is significantly smaller than 00 and 00 00 9 00 69. # 0 which gives as 4. 0 6, the poh as 5.39, and the ph as 8.6. Checking our assumptions, we note that both are reasonable: 4. 0 6 is less than 5% of 00 and [H 3 O, which is.43 0 9 is less than 5% of [OH, which is 4. 0 6. 8. (a) Figure SM6. shows a ladder diagram for maleic acid. A solution of 0 M H A will contain more H A than HA, and have a ph of less than.9 Maleic acid is a relatively strong weak acid ( a is 03) and is likely to dissociate to an appreciable etent; thus, a reasonable estimate is that the solution s ph falls in the acidic portion of maleic acid s buffer region, perhaps between.4 and.6. A solution of 0 M NaHA will contain more HA than H A or A, and have a ph between.90 and 6.33. A reasonable estimate is that the ph is near the middle of the predominance region for HL, or approimately 4.. A solution of 0 M Na A will contain more A than H A or HA, and, because A is a weak base, will have a ph greater than 7. Although more difficult to estimate, a ph between 9 and 0 is a reasonable guess. (b) Figure SM6.3 shows a ladder diagram for malonic acid. A solution of 0 M H A will contain more H A than HA, and have a ph of less than.847. Malonic acid is a relatively strong weak acid ( a is.4 0 3 ) and is likely to dissociate to an appreciable etent, but less than for maleic acid; thus, a reasonable estimate is that the solution s
Chapter 6 Equilibrium Chemistry 6 ph falls close to the bottom of the acidic portion of malonic acid s buffer region, perhaps between.8 and. A solution of 0 M NaHA will contain more HA than H A or A, and have a ph between.847 and 5.696. A reasonable estimate is that the ph is near the middle of the predominance region for HA, or approimately 4.3. A solution of 0 M Na A will contain more A than H A or HA, and, because A is a weak base, will have a ph greater than 7. Although more difficult to estimate, a ph between 9 and 0 is a reasonable guess. (c) Figure SM6.4 shows a ladder diagram for succinic acid. A solution of 0 M H A will contain more H A than HA, and have a ph of less than 4.07. Maleic acid is not a relatively strong weak acid ( a is 6. 0 5 ); thus, a reasonable estimate is that the solution s ph falls below maleic acid s buffer region, perhaps between.5 and 3. A solution of 0 M NaHA will contain more HA than H A or A, and have a ph between 4.07 and 5.636. A reasonable estimate is that the ph is near the middle of the predominance region for HA, or approimately 4.9. A solution of 0 M Na A will contain more A than H A or HA, and, because A is a weak base, will have a ph greater than 7. Although more difficult to estimate, a ph between 9 and 0 is a reasonable guess. 9. (a) Malonic acid, H A, is a diprotic weak acid, a solution of which contains the following species: H 3 O, OH, H A, HA, and A. From its ladder diagram (see Figure SM6.3), we assume that [ A << [H A ph A HA H A p a 5.636 p a 4.07 Figure SM6.4 Ladder diagram for succinic acid showing the ph values for which H A, HA, and A are the predominate species. which means we can treat a solution of H L as if it is a monoprotic weak acid. Assuming that [ OH << [ HO 3 then we know that [H3O[H A a [H A 00. 4. # 0 which we solve using the quadratic equation, finding that is 0 and that the ph is.95, which is within our estimated range of.8.0 from Problem 8. Checking our assumptions, we note that the concentration of OH, which is 8.93 0 3, is less than 5% of [H 3 O ; thus, this assumption is reasonable. To evaluate the assumption that we can ignore A, we use a to determine its concentration [H3O[ A (0)[ A 6 a [H A [ A 0. # 0 (0) 3 To review how we arrived at this equation, see Section 6G.4 of the tet or the solution to Problem 7d.
6 Solutions Manual for Analytical Chemistry. finding that it is less than 5% of [HA ; thus, this assumption is reasonable as well. (b) A solution of monohydrogen malonate, NaHA, contains the following species: H 3 O, OH, H A, and HA, and A. From its ladder diagram (see Figure SM6.3), we assume that [H A << [HA and [A << [HA To review the derivation of this equation, see Section 6G.5. Under these conditions, the [H 3 O is given by the equation [ HO C C A a a a NaH a w NaHA 3 Substituting in values for a, a, w, and C NaHA gives [H 3 O as 5.30 0 5, or a ph of 4.8, which is very close to our estimate of 4.3 from Problem 8. To evaluate the assumption that we can ignore H A, we use a to calculate its concentration 5 [ HO 3 [ HA (. 530# 0 )( 0) 3 [H A 3 373. # 0 a 4. # 0 finding that it is less than 5% of [HA C NaHA 0 M. To evaluate the assumption that we can ignore A, we use a to determine its concentration 6 a[h A (. 0# 0 )( 0) 3 [ A 5 379. # 0 [ HO 3 530. # 0 finding that it, too, is less than 5% of [HA C NaHA 0 M. (c) Sodium malonate, Na A, is a diprotic weak base, a solution of which contains the following species: H 3 O, OH, Na, H A, HA, and A. From its ladder diagram (see Figure SM6.3), we assume that [H A<<[H A which means we can treat a solution of A as if it is a monoprotic weak base. Assuming that [ HO 3 << [ OH then we know that [OH [H A 9 b 00. 498. # 0 [ A which we solve using the quadratic equation, finding that is.3 0 5, that the poh is 4.65, and that the ph is 9.35, which is within our estimated range of 9 0 from Problem 8. Checking our assumptions, we note that the concentration of H 3 O, which is 4.48 0 0, is less than 5% of [OH ; thus, this assumption is reasonable. To evaluate the assumption that we can ignore H A, we use b to determine its concentration
Chapter 6 Equilibrium Chemistry 63 b[h A [H A [OH 5 (. 704# 0 )(. 3 # 0 ) 5 (. 3# 0 ) 704. # 0 finding that it is less than 5% of [HA ; thus, this assumption is reasonable as well. For a simple solubility reaction without any complications from acid base chemistry or from compleation chemistry, the composition of the system at equilibrium is determined by the solubility reaction Hg Br () s Hg? ( aq) Br ( aq) its corresponding solubility product sp [Hg [Br 56. # 0 3 and the stoichiometry of the solubility reaction. (a) For a simple saturated solution of Hg Br, the following table defines the equilibrium concentrations of Hg and Br in terms of their concentrations in the solution prior to adding Hg Br and the change in their concentrations due to the solubility reaction. HgBr() s? Hg ( aq) Br ( aq) intial 0 0 change equilibrium Substituting the equilibrium concentrations into the sp epression 3 3 sp [Hg [Br ()( ) 4 56. # 0 and solving gives as.4 0 8. The molar solubility of Hg Br is the concentration of Hg, which is or.4 0 8 M. (b) For a saturated solution of Hg Br in 05 M Hg (NO 3 ), the following table defines the equilibrium concentrations of Hg and Br in terms of their concentrations in the solution prior to adding Hg Br, and the change in their concentrations due to the solubility reaction. HgBr() s? Hg ( aq) Br ( aq) I C E 05 05 0 Substituting the equilibrium concentrations into the sp epression sp [Hg [Br (. 0 05 )( ) 5. 6# 0 3 Although perhaps not obvious, the approach we are taking here is equivalent to the systematic approach to solving equilibrium problems described in Section 6G.3 that combines a charge balance equation and/or a mass balance equation with equilibrium constant epressions. For part (a), a charge balance equation requires that # [Hg [Cl If we define the concentration of Hg as, then the concentration of Cl is, which is the stoichiometric relationship shown in the table and leads to the same equation sp 4 3 5.6 0 3 Note that we ignore the presence of H 3 O and OH when writing this charge balance equation because the solution has a neutral ph and the concentrations of H 3 O and OH are identical. The same argument holds true for parts (b) and (c), although you may need to do a little work to convince yourself of this. To save space, we use I to label the row of initial concentrations, C to label the row showing changes in concentration, and E to label the row of equilibrium concentrations. For obvious reasons, these tables sometimes are called ICE tables. leaves us with a cubic equation that is difficult to solve. We can simplify the problem if we assume that is small relative to 05 M, an
64 Solutions Manual for Analytical Chemistry. assumption that seems reasonable given that the molar solubility of Hg Br in water is just.4 0 8 M; thus sp (. 0 05 )( ). ( 05)( ) 5. 6# 0 3 Solving gives as.4 0, a result that clearly is significantly less than 05. The molar solubility of Hg Br is the concentration of Hg from the Hg Br, which is or.4 0 M. (c) For a saturated solution of Hg Br in 050 M NaBr, the following table defines the equilibrium concentrations of Hg and Br in terms of their concentrations in the solution prior to adding Hg Br and the change in their concentrations due to the solubility reaction. HgBr() s? Hg ( aq) Br ( aq) I 0 050 C E 050 Substituting the equilibrium concentrations into the sp epression 3 sp [Hg [Br ()(. 0 050 ) 5. 6# 0 leaves us with a cubic equation that is difficult to solve. We can simplify the problem if we assume that is small relative to 050 M, an assumption that seems reasonable given that the molar solubility of Hg Br in water is just.4 0 8 M; thus ()(. 0 050 ) ( )( 050) 56. 0 3 sp. # Solving gives as. 0 0, a result that clearly makes significantly less than 05 The molar solubility of Hg Br is the concentration of Hg, which is or. 0 0 M.. Because F is a weak base, the molar solubility of CaF depends on the solution s ph and whether fluorine is present as F or as HF. The ladder diagram for HF, which is included in Figure SM6., shows that F is the only significant form of fluorine at a ph of 7.00, which means the solubility of CaF is determined by the reaction CaF () s? Ca ( aq) F ( aq) for which the equilibrium constant epression is sp [Ca [F 39. # 0 The following table defines the equilibrium concentrations of Ca and F in terms of their concentrations in the solution prior to adding CaF, and the change in their concentrations due to the solubility reaction.
Chapter 6 Equilibrium Chemistry 65 I C E CaF () s? Ca ( aq) F ( aq) 0 0 Substituting the equilibrium concentrations into the sp epression 3 sp [ Ca [F ()( ) 4 39. # 0 and solving gives as. 0 4. The molar solubility of CaF at a ph of 7.00 is the concentration of Ca, which is or. 0 4 M. Our solution here assumes that we can ignore the presence of HF; as a check on this assumption, we note that 7 4 [ HO 3 [F (. 0# 0 )(. # 0 ) 8 [HF 4 3. # 0 a,hf 68. # 0 a concentration that is negligible when compared to the concentration of F. At a ph of.00, the only significant form of fluorine is HF, which means we must write the solubility reaction for CaF in terms of HF instead of F ; thus CaF() s HO 3 ( aq)? Ca ( aq) HF( aq) HO () l To determine the equilibrium constant for this reaction, we note that it is the sum of five reactions, each with a standard equilibrium constant; thus CaF () s? Ca ( aq) F ( aq) Note that the molar solubility of CaF is independent of ph for any ph level greater than approimately p a,hf 4.. This is because at these ph levels the solubility reaction does not include any acidbase chemistry. We also can write this reaction as CaF () s H O() l? Ca ( aq) F ( aq) OH ( aq) and then make appropriate changes to the equations that follow. The final answer is the same. F ( aq) H O() l? HF( aq) OH ( aq) F ( aq) H O() l? HF( aq) OH ( aq) HO 3 ( aq) OH ( aq)? HO () l HO 3 ( aq) OH ( aq)? HO () l sp,caf #( b,f ) #( w) w sp,caf # a k # a k a,hf w sp,caf 39. # 0 5 4 84. # 0 ( a,hf) (. 68# 0 ) The following table defines the equilibrium concentrations of Ca, HF, and H 3 O in terms of their concentrations in the solution prior to adding CaF, and the change in their concentrations due to the solubility reaction; for H 3 O, note that its concentration does not change because the solution is buffered.
66 Solutions Manual for Analytical Chemistry. To display this table within the available space, we identify the physical state only those species that are not present as aqueous ions or molecules. I C E CaF () s HO Ca 3? HF HO() l 00 0 0 00 Substituting the equilibrium concentrations into the reaction s equilibrium constant epression 3 [Ca [HF ()( ) 4 5 84. # 0 [H3O (. 0 00) (. 0 00) and solving gives as.3 0 3. The molar solubility of CaF at a ph of.00 is the concentration of Ca, which is or.3 0 3 M. Our solution here assumes that we can ignore the presence of F ; as a check on this assumption, we note that 4 3 a,hf [HF (. 68# 0 )( 3. # 0 ) 5 [F [ 00. 88. # 0 HO 3 a concentration that is negligible when compared to the concentration of HF.. The solubility of Mg(OH) is determined by the following reaction Mg(OH) () s? Mg ( aq) OH ( aq) for which the equilibrium constant epression is sp [Mg [ OH 7. # 0 The following table defines the equilibrium concentrations of Mg and OH in terms of their concentrations in the solution prior to adding CaF and the change in their concentrations due to the solubility reaction; note that the concentration of OH is fied by the buffer. I C E Mg(OH) () s Mg ( aq) OH ( aq) 0 7 0. # 0 7 0. # 0? Substituting the equilibrium concentrations into the sp epression 7 sp [Mg [OH ()(. 0# 0 ) 7. # 0 and solving gives as 7 The molar solubility of Mg(OH) at a ph of 7.00 is the concentration of Mg, which is or 70 M; clearly, Mg(OH) is very soluble in a ph 7.00 buffer. If the solution is not buffered, then we have
Chapter 6 Equilibrium Chemistry 67 I C E Mg(OH) () s Mg ( aq) OH ( aq) 0 7 0. # 0 7 (. 0# 0 )? and substituting into the equilibrium constant epression sp [Mg [OH (){( 0. # 0 ) )} 7. # 0 7 leaves us with an equation that is not easy to solve eactly. To simplify the problem, lets assume that is sufficiently large that (. 0# 0 7 ) c Substituting back sp [Mg [OH () ( ) 4 7. # 0 3 and solving gives as. 0 4. Checking our one assumption, we note that it is reasonable:.0 0 7 is less than 5% of.the molar solubility of Mg(OH) is the concentration of Mg, which is or. 0 4 M; clearly, Mg(OH) is much less soluble in the unbuffered solution. 3 3. Because PO 4 is a weak base, the molar solubility of Ag 3 PO 4 depends on the solution s ph and the specific form of phosphate present. The ladder diagram for H 3 PO 4, which is included in Figure SM6., shows that HPO 4 is the only significant form of phosphate at a ph of 9.00, which means the solubility of Ag 3 PO 4 is determined by the reaction Ag3PO4() s HO 3 ( aq)? 3Ag ( aq) HPO 4 ( aq) HO() l To determine the equilibrium constant for this reaction, we note that it is the sum of three reactions, each with a standard equilibrium constant; thus 3 Ag3PO4() s? 3Ag ( aq) PO4 ( aq) PO 3 4 ( aq) HO () l? OH ( aq) HPO 4 ( aq) HO 3 ( aq) OH ( aq)? HO () l ( ) w 3 sp,ag3po4# b,po4 # w sp,ag3po4# # a,hpo4 8 sp,ag3po4 8. # 0 6 3 6. # 0 a,hpo4 45. # 0 The following table defines the equilibrium concentrations of Ag, HPO 4, and H 3 O in terms of their concentrations in the solution prior to adding Ag 3 PO 4, and the change in their concentrations due to the solubility reaction; for H 3 O, note that its concentration does not change because the solution is buffered. w
68 Solutions Manual for Analytical Chemistry. You may wonder why our approach to this problem does not involve setting up an ICE table. We can use an ICE table to organize our work if there is one and only one reaction that describes the equilibrium system. This is the case, for eample, in Problem 0 where the solubility reaction for Hg Br is the only reaction in solution, and the case in Problem because only one reaction contributes significantly to the equilibrium condition. For more complicated systems, such as Problem 4 and several that follow, we must work with multiple equations, often solving them simultaneously. I C E Ag3PO4() s HO 3? 3Ag HPO4 HO() l 9 0. # 0 0 0 9 0. # 0 3 3 Substituting the equilibrium concentrations into the reaction s equilibrium constant epression 3 [Ag [ HPO 3 ( 3) () 4 4 7 6 9 9 6. 0 [H3O # 0. # 0 0. # 0 and solving gives as. 0 4. The molar solubility of Ag 3 PO 4 at a ph of.00 is the concentration of HPO 4, which is or. 0 4 M. Our solution here assumes that we can ignore the presence of other phosphate species; as a check on this assumption, we note that [HPO [PO 3 a,hpo4 4 4 [H3O 3 4 (. 45# 0 )(. # 0 ) 8 9 54. # 0 0. # 0 and that [HPO [H PO 4 [HO 3 4 a,hpo4 4 9 (. # 0 )( 0. # 0 ) 8 9. # 0 63. # 0 Both concentrations are less than 5% of the concentration of HPO 4, so our assumptions are reasonable. Note that we do not need to check the assumption that the concentration of H 3 PO 4 is negligible as it must be smaller than the concentration of HPO 4. 4. The equilibrium constants for the three reactions are sp [Ag [Cl 8. # 0 0 [AgCl( aq) 50. # 0 [Ag [Cl [AgCl 83. [AgCl ( aq) [Cl The concentration of AgCl(aq) is easy to determine because the denominator of is simply sp ; thus 3 0 7 [AgCl ( aq) sp (. 50# 0 )(. 8# 0 ) 9. 0# 0 M To determine the concentration of the remaining species, we note that a charge balance equation requires that [Ag [Cl [AgCl 3 6
Chapter 6 Equilibrium Chemistry 69 Given that the concentration of AgCl(aq) is 9.0 0 7, it seems reasonable to assume that the concentration of AgCl is less than this and that we can simplify the charge balance equation to [Ag [Cl As we must form AgCl(aq) before we can form AgCl, the concentration of AgCl will be less than [AgCl(aq) unless there is a large ecess of Cl. Substituting into the sp equation sp [Ag [Cl ()() 8. # 0 0 and solving gives as.3 0 5 ; thus, both the [Ag and the [Cl are.3 0 5 M. To check our assumption, we note that [AgCl [AgCl ( aq) [Cl ( 83. )( 9. 0# 0 )(. 3# 0 ) 9. 7# 0 7 5 0 the concentration of AgCl is 9.7 0 0 M; thus, our assumption that we can ignore the concentration of AgCl is reasonable. 5. (a) A solution of 050 M NaCl is 050 M in Na and 050 M in Cl ; thus n "(. 0 050)( ) ( 050)( ), 050 M (b) A solution of 05 M CuCl is 05 M in Cu and 050 M in Cl ; thus n "(. 005)( ) ( 050)( ), 075M (c) A solution of 0 M Na SO 4 is 0 M in Na and 0 M in SO 4 ; thus n "(. 0 0)( ) ( 0)( ), 30M Note that we did not include H 3 O and OH when calculating the ionic strength of these solutions because their concentrations are sufficiently small that they will not affect the ionic strength within the limit of our significant figures. The same reasoning eplains why we did not consider the acidbase chemistry of SO 4 in part (c) as the concentrations of H 3 O, OH, and HSO 4 are sufficiently small that we can safely ignore them. 6. (a) From Problem 0a, we know that the molar solubility of Hg Br is sufficiently small that the solution s ionic strength is not altered significantly by the limited number of Hg and Br ions in solution. The molar solubility remains.4 0 8 M. (b) From Problem 0b we know that the molar solubility of Hg Br is sufficiently small that the solution s ionic strength is not altered significantly by the limited number of Br ions or Hg ions arising from the solubility reaction; however, we cannot ignore the contribution of Hg (NO 3 ) to the solution s ionic strength, which is n "(. 0 05)( ) ( 050)( ), 075 M Given the ionic strength, we net find the activity coefficients for Hg and for Br ; thus (. 05)( ) 075 log c Hg (.)( 3 3 40) 075 c Hg 389 40
70 Solutions Manual for Analytical Chemistry. (. 05)( ) 0075. log c Br 0 (.)( 3 3 30) 075 c Br 776 where values of alpha are from Table 6.. The ionic strengthadjusted sp for the solubility of Hg Br is sp [Hg [Br chg cbr 56. # 0 3 From here, we proceed as in Problem 0b; thus sp (. 0 05 )( )(. 0 389)(. 0 776) 56. # 0. (. 0 05)( )(. 0 389)(. 0 776) 56. # 0 3 3 finding that is 4.9 0 and that the molar solubility of Hg Br of 4.9 0 M is greater than the value of.4 0 M that we calculated when we ignored the affect on solubility of ionic strength. (c) From Problem 0c we know that the molar solubility of Hg Br is sufficiently small that the solution s ionic strength is not altered significantly by the limited number of Br ions or Hg ions arising from the solubility reaction; however, we cannot ignore the contribution of NaBr to the solution s ionic strength, which is n "(. 0 050)( ) ( 050)( ), 050 M Given the ionic strength, we net find the activity coefficients for Hg and for Br ; thus (. 05)( ) 050 log c Hg (.)( 3 3 40) 050 c Hg 444 (. 05)( ) 00. 50 log c Br (.)( 3 3 30) 00. 50 c Br 807 35 0934 where values of alpha are from Table 6.. The ionic strengthadjusted sp for the solubility of Hg Br is sp [Hg [Br chg cbr 56. # 0 3 From here, we proceed as in Problem 0c; thus sp ()(. 0 050 )(. 0 444)(. 0 807) 56. # 0 sp. ()(. 0 050)(. 0 444)(. 0 807) 56. # 0 3 3 finding that is 7.7 0 0 and that the molar solubility of Hg Br of 7.7 0 0 M is greater than the value of. 0 0 M that we calculated when we ignored the affect on solubility of ionic strength.
Chapter 6 Equilibrium Chemistry 7 7. Because phosphate is a weak base, the solubility of Ca 3 (PO 4 ) will increase at lower ph levels as the predominate phosphate species 3 transitions from PO 4 to HPO 4 to HPO 4 to H 3 PO 4. A ladder diagram for phosphate is included in Figure SM6. and shows that 3 PO 4 is the predominate species for ph levels greater than.35; thus, to minimize the solubility of Ca 3 (PO 4 ) we need to maintain the ph above.35. 8. (a) Figure SM6. shows a ladder diagram for HF and H 3 PO 4. Based on this ladder diagram, we epect that the weak acid HF will react 3 with the weak bases PO 4 and HPO 4 as their areas of predominance do not overlap with HF; thus HF(aq) PO 3 (aq) HPO 4? 4 (aq) F (aq) HF(aq) PO 3 4 (aq)? HPO4 (aq) F (aq) HF(aq) HPO 4 (aq)? HPO4 (aq) F (aq) We also epect that the weak base F will react with the weak acid H 3 PO 4 ; thus F ( aq) H3PO4( aq)? HPO 4 ( aq) HF( aq) (b) Figure SM6.5 shows a ladder diagram for the cyano complees of Ag, Ni, and Fe. Based on this ladder diagram, we epect that Ag will displace Ni from the Ni(CN) 4 comple, that Ag will displace Fe 4 from the Fe(CN) 6 comple, and that Ni will displace Fe 4 from the Fe(CN) 6 ; thus Ag (aq) Ni(CN) 4 (aq)? Ag(CN) (aq) Ni (aq) 3Ag (aq) Fe(CN) 4 6 (aq)? 3Ag(CN) (aq) Fe (aq) 3Ni (aq) Fe(CN) 4 6 (aq)? 3Ni(CN) 4 (aq) Fe (aq) 3 (c) Figure SM6.6 shows a ladder diagram for the CrO 7 /Cr and the Fe 3 /Fe redo halfreactions. Based on this ladder diagram, we epect that Fe will reduce Cr O 7 to Cr 3 ; thus Cr O ( aq) 6Fe 7 ( aq) 4H ( aq)? 3 3 Cr ( aq) 6Fe ( aq) 7HO() l 9. The ph of a buffer that contains a weak acid, HA, and its conjugate weak base, A, is given by equation 6.60 ph p log C C A a HA which holds if the concentrations of OH and of H 3 O are significantly smaller than the concentrations of HA, C HA, and of A,C A. (a) The ph of the buffer is ph 3.745 log 05 05 35. 4 logb 4 7.56 E o.36 V Ni Ni(CN) 4 Cr O 7 Cr 3 pcn Ag Ag(CN) Fe Fe(CN) 6 4 E o logb 4 6 logb 6 5.90 Figure SM6.5 Ladder diagram for Problem 8b showing the areas of predominance for Ag, Ni, and Fe 3 and their cyano complees. Fe 3 Fe E o 77 V Figure SM6.6 Ladder diagram for Problem 3 8c showing the CrO 7 /Cr and the Fe 3 /Fe redo halfreactions.
7 Solutions Manual for Analytical Chemistry. With a ph of 3.5, the [H 3 O is 3.0 0 4 and the [OH is 3.3 0 ); thus, the assumptions inherent in equation 6.60 hold. (b) A miture consisting of an ecess of a weak base, NH 3, and a limiting amount of a strong acid, HCl, will react to convert some of the NH 3 to its conjugate weak acid form, NH 4 ; thus The moles of NH 4 NH3( aq) HCl( aq) $ NH4 ( aq) Cl ( aq) formed are mol NH 4 MHClVHCl (.0 M)(00350 L) 3.50 # 0 3 which leaves the moles of NH 3 as molnh3 MNH3VNH3MHClVHCl (. 0 M)( 05000 L) (. 0M)( 00350 L) 3 50. # 0 The total volume is 53.50 ml, which gives the concentrations of NH 4 and of NH 3 as 3 [NH 3.50 0 mol 4 # 0535 L 0654 M 3 [NH.50 0 mol 3 # 0535 L 0467 M and a ph of ph 9.44 log 0467 9.0 0654 With a ph of 9.0, the [H 3 O is 8.0 0 0 and the [OH is.3 0 5 ); thus, the assumptions inherent in equation 6.60 hold. (c) To calculate the ph we first determine the concentration of the weak acid, HCO 3, and the weak base, CO 3 mol HCO3 5.00 gnahco3 # 84.007 gnahco 3 [ HCO3 00 L 595 M mol CO3 5.00 gnaco 3 # 05. 99 gnaco 3 [ CO3 00 L 47 M and then the ph ph 39 log 47 595 3 With a ph of 3, the [H 3 O is 5.9 0 and the [OH is.7 0 4 ); thus, the assumptions inherent in equation 6.60 hold. Adding 5.0 0 4 mol of HCl converts 5.0 0 4 mol of the buffer s conjugate weak base, A, to its conjugate weak acid, HA. To simplify the calculations, we note that we can replace the concentrations of HA and of A in equation 6.60 with their respective moles as both
Chapter 6 Equilibrium Chemistry 73 HA and A are in the same solution and, therefore, share the same volume. (a) The ph after adding 5.0 0 4 mol of HCl is 4 (. 0 05)( 00) 500. # 0 ph 3. 745 log 4 37. (. 0 05)( 00) 500. # 0 With a ph of 3.7, the [H 3 O is 5.4 0 4 and the [OH is.9 0 ); thus, the assumptions inherent in equation 6.60 hold. (b) The ph after adding 5.0 0 4 mol of HCl is 4 (. 0 0467)( 0535) 500. # 0 ph 9. 44 log 4 894. (. 0 0654)( 0535) 500. # 0 With a ph of 8.94, the [H 3 O is. 0 9 and the [OH is 9. 0 6 ); thus, the assumptions inherent in equation 6.60 hold. (c) The ph after adding 5.0 0 4 mol of HCl is 4 (. 0 47)( 00) 500. # 0 ph 39 log 4 (. 0 595)( 00) 500. # 0 With a ph of, the [H 3 O is 6.0 0 and the [OH is.7 0 4 ); thus, the assumptions inherent in equation 6.60 hold.. Adding 5.0 0 4 mol of NaOH converts 5.0 0 4 mol of the buffer s conjugate weak base, HA, to its conjugate weak acid, A. To simplify the calculations, we note that we can replace the concentrations of HA and of A in equation 6.60 with their respective moles as both HA and A are in the same solution and, therefore, share the same volume. (a) The ph after adding 5.0 0 4 mol of NaOH is 4 (. 0 05)( 00) 500. # 0 ph 3. 745 log 4 374. (. 0 05)( 00) 500. # 0 With a ph of 3.74, the [H 3 O is.8 0 4 and the [OH is 5.5 0 ); thus, the assumptions inherent in equation 6.60 hold. (b) The ph after adding 5.0 0 4 mol of NaOH is 4 (. 0 0467)( 0535) 500. # 0 ph 9. 44 log 4 94. (. 0 0654)( 0535) 500. # 0 With a ph of 9.4, the [H 3 O is 5.8 0 0 and the [OH is.7 0 5 ); thus, the assumptions inherent in equation 6.60 hold. (c) The ph after adding 5.0 0 4 mol of NaOH is 4 (. 0 47)( 00) 500. # 0 ph 39 log 4 4 (. 0 595)( 00) 500. # 0 With a ph of 4, the [H 3 O is 5.8 0 and the [OH is.7 0 4 ); thus, the assumptions inherent in equation 6.60 hold.. (a)the equilibrium constant for the reaction is [ML [M[L As epected, adding HCl makes the solution more acidic, with the ph decreasing from 3.5 to 3.7. As epected, adding HCl makes the solution more acidic, with the ph decreasing from 9.0 to 8.94. As epected, adding HCl makes the solution more acidic, with the ph decreasing from 3 to ; the change in ph is smaller here because the concentration of the buffering agents is larger. As epected, adding NaOH makes the solution more basic, with the ph increasing from 3.5 to 3.74. As epected, adding NaOH makes the solution more basic, with the ph increasing from 9.0 to 9.4. As epected, adding NaOH makes the solution more basic, with the ph increasing from 3 to ; the change in ph is smaller here because the concentration of the buffering agents is larger.
74 Solutions Manual for Analytical Chemistry. Taking the log of both sides of this equation gives [ML [ML log log [L log[m log [L pm which, upon rearranging, gives the desired equation [ML pm log log [L For the case where is.5 0 8 we have 8 [ML pm log(. 5# 0 ) log [L 8. 8 [ML log [L (b) Because the reaction between M and L is very favorable, we epect that all of M, which is the limiting reagent, is converted to ML, consuming an equivalent amount of L. Once equilibrium is reached, 00 mol of L remain and 00 mol of ML are formed, which gives pm 8.8 log 00 00 88. (c) Adding an additional 00 mol M converts an additional 00 mol of L to ML; thus, we now have 0 mol ML and 008 mol L, and pm is pm 8.8 log 0 008 800. 3. The potential of a redo buffer is given by the Nernst equation o [Fe E E 3 Fe /Fe 0596log [Fe 3 Because Fe and Fe 3 are in the same solution, we can replace their concentrations in the Nernst equation with moles; thus o E E 0596log molfe 3 Fe /Fe molfe 3 77 0596log 05 00 76 V After converting 00 mol Fe to Fe 3, the solution contains 03 mol Fe and 0 mol Fe 3 ; the potential, therefore, is E 77 0596log 03 00. 769 V 4. A general approach to each problem is provided here, but more specific details of setting up an Ecel spreadsheet or writing a function in R are left to you; see Section 6J for more details. (a) To find the solubility of CaF we first write down all relevant equilibrium reactions; these are CaF () s? Ca ( aq) F ( aq) HF( aq) HO () l? HO 3 ( aq) F ( aq) HO() l? H3O ( aq) OH ( aq)
Chapter 6 Equilibrium Chemistry 75 There are five species whose concentrations define this system (Ca, F, HF, H 3 O, and OH ), which means we need five equations that relate the concentrations of these species to each other; these are the three equilibrium constant epressions a charge balance equation sp [Ca [F 39. # 0 [H3O[F 4 a [HF 68. # 0 w [H3 O[OH 00. # 0 4 [Ca [HO 3 [OH [F and a mass balance equation # [Ca [HF [F Be sure you understand why the concentration of Ca is multiplied by. To solve this system of five equations, we make a guess for [Ca, and then use sp to calculate [F, the mass balance equation to calculate [HF, a to calculate [H 3 O, and w to calculate [OH. We evaluate each guess by rewriting the charge balance equation as an error function error # [Ca [H3O[OH [F searching for a [Ca that gives an error sufficiently close to zero. Successive iterations over a narrower range of concentrations for Ca will lead you to a equilibrium molar solubility of. 0 4 M. (b) To find the solubility of AgCl we first write down all relevant equilibrium reactions; these are AgCl() s? Ag ( aq) Cl ( aq) Ag ( aq) Cl ( aq)? AgCl( aq) AgCl( aq) Cl ( aq)? AgCl ( aq) AgCl ( aq) Cl ( aq)? AgCl 3( aq) AgCl 3( aq) Cl ( aq)? AgCl 4( aq) There are si species whose concentrations define this system (Ag, Cl, AgCl(aq), AgCl, AgCl 3, and AgCl 3 4 ), which means we need si equations that relate the concentrations of these species to each other; these are the five equilibrium constant epressions sp [Ag [Cl 8. # 0 0 ( aq) [AgCl 50. 0 [Ag [Cl 3 #
76 Solutions Manual for Analytical Chemistry. [AgCl 83. [AgCl ( aq) [Cl [AgCl 3 3 603. [AgCl [Cl [AgCl 3 4 4 50 [AgCl 3 [Cl You can substitute a mass balance equation for the charge balance equation, but the latter is easier to write in this case. and a charge balance equation [Ag [Cl [AgCl #[AgCl 3 3 #[AgCl 3 4 To solve this system of five equations, we make a guess for [Ag, and then use sp to calculate [Cl, to calculate [AgCl(aq), to calculate [AgCl 3, 3 to calculate [AgCl 3, and 4 to calculate [AgCl 4. We evaluate each guess by rewriting the charge balance equation as an error function error [Ag [Cl [AgCl #[AgCl 3 3 #[AgCl 3 4 searching for a [Ag that gives an error sufficiently close to zero. Successive iterations over a narrower range of concentrations for Ag will lead you to a equilibrium molar solubility of.3 0 5 M. (c) To find the ph of 0 M fumaric acid we first write down all relevant equilibrium reactions; letting H A represent fumaric acid, these are HA ( aq) HO () l? HO 3 ( aq) HA ( aq) HA ( aq) HO() l HO( aq) A? 3 ( aq) HO() l? H3O ( aq) OH ( aq) There are five species whose concentrations define this system (H A, HA, A, H 3 O, and OH ), which means we need five equations that relate the concentrations of these species to each other; these are the three equilibrium constant epressions [H3O[HA 4 a [H A 885. # 0 a charge balance equation [H3O[A a [HA 3. # 0 w [ HO 3 [ OH 00. # 0 5 4 [H O 3 [OH [HA # [ A and a mass balance equation 0 M [H A [H A [ A
Chapter 6 Equilibrium Chemistry 77 To solve this system of five equations, we make a guess for the ph, calculate [H 3 O and use w to calculate [OH. Because each of the remaining equations include at least two of the remaining species, we must combine one or more of these equations to isolate a single species. There are several ways to accomplish this, one of which is to use a to epress [HA in terms of [H A, and to use a and a to epress [A in terms of [H A a[ha [HA [ HO 3 a[ha aa[ha [A [ HO 3 [ HO 3 and then substitute both into the mass balance equation a[ha aa[ha 0 M [HA [ HO 3 [ HO 3 [H A a aa # & 0 [ HO 3 [ HO 3 which we use to calculate [H A. Finally, we calculate [HA using a and [A using a. We evaluate each guess by rewriting the charge balance equation as an error function error [HO 3 [OH [HA # [A searching for a ph that gives an error sufficiently close to zero. Successive iterations over a narrower range of ph values will lead you to a equilibrium ph of.05. 5. The four equations that describe the composition of an equilibrium solution of HF are the a and w equilibrium constant epressions [H3O[F a [HF a charge balance equation and a mass balance equation w [ HO 3 [ OH [H3 O [OH [F C HF [HF [F To combine the equations, we first use the mass balance equation to epress [HF in terms of C HF and [ F [HF CHF [F and then substitute this into the a epression [ HO[F 3 a C HF [F
78 Solutions Manual for Analytical Chemistry. which we then solve for [F C a HF [F a [ HO[F 3 C [ HO[F [F a HF 3 a [F "[ HO, C 3 a a HF C a HFa [F [ HO 3 a Net, we solve w for [OH [ w OH [ HO 3 and then substitute this and the equation for [F into the charge balance equation w C a HF [H3O [ HO 3 [ HO 3 a Rearranging this equation w C a HFa [H3O 0 [ HO 3 [ HO 3 a multiplying through by [H 3 O C a HF[ HO 3 [H3O w 0 [ HO 3 a multiplying through by [H 3 O a 3 [H3O a[h3o w[ HO 3 a w C a HF[ HO 3 0 and gathering terms leaves us with the final equation 3 [H3O a[h3o ( achf w)[ HO 3 a w 0