The determination of equilibrium
constants
• Self-test 7.11 Calculate the solubility constant (the equilibrium
constant for reaction Hg2Cl2(s) ↔ Hg22+(aq) + 2Cl-(aq)) and the
solubility of mercury(I) chloride at 298.15K. The mercury(I) ion is the
diatomic species Hg22+.
• Answer: This chemical process does not involve electron transfer,
i.e. is not a redox reaction.
Choosing cathode reaction as: Hg2Cl2(s) + 2e → 2Hg(l) + 2Cl-(aq)
from reference table 7.2, Eθ = 0.27 V
the anode reaction can be obtained through R – Cell
Hg22+(aq) + 2e → 2Hg(l)
from reference table 7.2, Eθ = 0.79 V
Therefore the standard cell potential = 0.27 – 0.79 = -0.52 V
lnK = Eө*v/25.7mV,
here v = 2
lnK = - 40.467
K = 2.62 x 10-18
• K = (aHg(I) * a2cl-)/aHg2cl2
• Using equation
= b*(2b)2/1 = 4*b3 = 2.62 x 10-18
therefore b = 8.68 x 10-7 mol/kg
Species-selective electrodes:
Measuring pH
• The half reaction at a hydrogen electrode:
H+(aq) + e- → ½ H2(g)
 1/ 2
(
f
(
H
)
/
P
)
2
Here v= 1,
Q=
aH 
The potential
E(H+/H
E(H+/H
2)
2)
=
=
Eө
-
RT ( f ( H 2 ) / P )1 / 2
ln(
)
vF
aH 
 1/ 2
(
f
(
H
)
/
P
)
RT
2
ln(
)
vF
aH 
Assuming that the pressure of H2 gas equals 1 bar
E(H+/H
2)
=
RT
1
- ln( )
vF
aH 
E(H+/H2) =
RT
ln( a H  ) = RT ln(10 ) log a 
=
H
F
F
- RT ln(10) pH
F
• Two electrodes are required to build up an electrochemical cell. This
is why we need a reference electrode when measuring the pH of a
solution.
• A regular reference electrode is calomel (Hg2Cl2(s)).
• The hydrogen electrode is used as the right hand electrode, i.e.
cathode.
• E(cell) = E(H+/H2) - E(ref.)
• E(cell) + E(ref) = -
RT
ln(10 ) pH
F
• Why do we need to calibrate the pH electrode before its usage?
Thermodynamic function
• Nernst equation is a bridge connecting the thermodynamic quantify,
Gibbs energy, and the electromotive force.
• Consider: Pt(s)|H2(g)|H+(aq)|Ag+(aq)|Ag(s)
the ΔfGө(Ag+(aq)).
Eө = 0.80V. Calculate
Solution: First, write down the two reduction half reactions and then
do a simple subtraction (R-L) to get the cell reaction.
Cell :
R:
Ag+(aq) + eL:
H+(aq) + eAg+(aq) + 1/2 H2(g)
→ Ag(s)
→ 1/2 H2(g)
→ Ag(s) + H+(aq)
Continued
ΔrGө = ΔfGө(Ag(s)) + ΔfGө (H+) - ΔfGө (Ag+) - (1/2)ΔfGө(H2(g))
Δ rGө = 0
+
0 - ΔfGө (Ag+) - 0
Since
ΔrGө = -νFEө
ΔfGө(Ag+) = vFEө = 9.648 x 104 C mol-1 * 0.80V
= 7.715 x 104 CV mol-1
= 7.715 x 104 J mol-1 ( 1CV = 1J)
Temperature dependence of emf
•
ΔrGө = -νFEө
take the derivate of temperature for both sides
d r G   vFdE

dT
dT
•
dE
dT
d r G 
dE

dT
 vF dT
is called temperature coefficient of standard cell emf.
• Because
d r G 
  r S
dT



S
dE
one gets
 r
dT
vF
therefore, one can use electrochemical method to obtain reaction
entropy and relate them to entropies of ions in solution.
• noncalorimetric method of measuring ΔrHө
ΔrHө = ΔrGө + TΔrSө = -vFEө + T(vF
dE
dT
) =
-vF(Eө
-T
dE
dT
)
• Example: The standard electromotive force of the cell
Pt(s)|H2(g)|H+(aq)||Ag+(aq)|Ag(s)
was measured over a broad range of temperatures, and the data were
fitted to the following polynomial:
Eө/V = 0.07 – 4.11x10-4(T/K – 298) – 3.2x10-6(T/K -298)2
Evaluate the standard reaction Gibbs energy, enthalpy, and entropy at 298K.
Solution: The standard reaction Gibbs energy can be calculated
once we know the standard emf of the above cell:
At 298K,
Eө/V = 0.07- 4.11x10-4(298K/K – 298) – 3.2x10-6(298K/K 298)2
Eө/V = 0.07
Eө = 0.07 V
• to identify the value of v, we need to write down the cell reaction
Ag+ + 1/2H2(g) → H+(aq) + Ag(s)
ΔrGө = -vFEө = - (1) x 9.6485 x104 C mol-1 x (0.07V)
= - 6.754 x 103 CVmol-1
= - 6.754 x 103 J mol-1
Calculate the temperature coefficient of the cell electromotive
force
dE
dT
= – 4.11x10-4 K-1V – 3.2x10-6x2x(T/K -298) K-1V
= - 4.11x10-4 K-1V (at 298K)
then
ΔrSө = vF
dE
dT
= 1 x 9.6485 x 104 C mol-1 x (- 4.11x10-4 K-1V )
= - 39.66 CV K-1 mol-1
= - 39.66 J K-1 mol-1
to calculate the standard reaction enthalpy:
ΔrHө = ΔrGө + TΔrSө = -6.754kJ mol-1 + 298K (- 39.66 J K-1
mol-1)
= -18.572 kJ mol-1
Example: The standard cell potential of
Pt(s)|H2(g)|HCl(aq)| Hg2Cl2(s)|Hg(l)|Ag(s)
was found to be +0.269 V at 293 K and + 0.266 V at 303K. Evaluate the
standard reaction Gibbs function, enthalpy, and entropy at 298K.
Solution: Write the cell reaction
Hg2Cl2(s) + H2(g) → 2Hg(l) + 2HCl(aq)
So v = 2,
To find the ΔrGө at 298 K, one needs to know the standard emf at 298K,
which can be obtained by linear interpolation between the two temperatures.
Eө = 0.2675 V
ΔrGө = -2F Eө = -51.8 kJ mol-1
The standard reaction entropy can then be calculated from
dE
= (0.266V- 0.269V)/10K = -3.0x10-4 VK-1
dT

ΔrSө = 2F dE = - 58 JK-1 mol-1
dT
then
ΔrHө = ΔrGө + TΔrSө = -69 kJ mol-1
What is the quotient, Q, of the above cell reaction?
Evaluate the reaction potential from two
others
• Example 1. Calculate the standard potential of the Fe3+/Fe from the
values for the Fe3+/Fe2+ (+0.77V) and Fe2+/Fe( -0.44V).
• Solution: : first write down the half reactions for these three couples:
1)
Fe3+ + e- → Fe2+
2)
Fe2+ + 2e- → Fe
3)
Fe3+ + 3e- → Fe
Reaction 3 is the sum of 1 and 2, yet one cannot use E3 = E1+ E2
ΔrGө(1) = - 1x F x 0.77V
ΔrGө(2) = - 2x F x (-0.44)V
ΔrGө(3) = ΔrGө(1) + ΔrGө(2) = 0.11F V
ΔrGө(3) = - 3*F*E3 = 0.11 F V
E3 = -0.033V
A good practice of calculating the potential of a redox couple from other
redox coupled is going through the reaction Gibbs energy!
Example 2: Given that the standard potentials of the Cu2+/Cu and
Cu+/Cu couples are +0.340V and + 0.522V, respectively, evaluate
Eө(Cu2+, Cu+).
Solution: Again, we should go through the standard Gibbs energy to
calculate it.
First write the half-reactions:
(1) Cu2+(aq) + 2e(2) Cu+(aq) +
e(3)
Cu2+(aq) +
e-
→
→
Cu(s)
Cu(s)
→
Cu+
it can be identified easily that reaction (3) equals (1) - (2)
thus: ΔrGө(3) = ΔrGө(1) - ΔrGө(2) = (-2*F*0.340V) – (-1*F*0.522V)
ΔrGө(3) = -0.158F V
-1*F*Eө(Cu2+/Cu+) = -0.158F V
Eө(Cu2+/Cu+) = 0.158 V
Summary of chapter 7
•
•
•
•
•
•
•
•
Equilibrium constant,
Gibbs energy vs the extent of reaction
Reaction Gibbs energy
Standard reaction Gibbs energy and
thermodynamic equilibrium constant.
Le Chaterlier principle & van’t Hoff relationship.
Reactions in charged environment: Nerst
equation.
Cell potential and standard cell potential.
Connection between thermal and
electrochemical quantities.