The half-cell, Tl | Tl+(1 mol L-1), is connected to a Pt | H+ | H2(1 atm) half-cell in which the concentration of H+ is unknown. The measured cell voltage is 0.0841 V and the Pt | H2 | H+ half-cell is the cathode. Standard reduction potentials are given on the next page. (a) Draw a diagram of the electrochemical cell, labeling all electrodes and solutions. (b) What is the pH in the Pt | H+ | H2 half-cell? (c) What is the value of the equilibrium constant for the reaction occurring in this cell?
(a) In the electrochemical cell, we have two half-cells: Tl | Tl+(1 mol L-1) and Pt | H+ | H2(1 atm). The Tl | Tl+(1 mol L-1) half-cell is connected to the Pt | H+ | H2(1 atm) half-cell. The Pt electrode is common to both half-cells. The Tl electrode is in contact with a Tl+(1 mol L-1) solution, and the Pt electrode is in contact with an H+ solution and a H2 gas (1 atm) phase.
(b) To determine the pH in the Pt | H+ | H2 half-cell, we can use the Nernst equation. The Nernst equation relates the cell voltage to the concentrations of the reactants and products involved in the redox reaction.
The Nernst equation is given by:
Ecell = E°cell - (0.0592/n)log(Q),
where Ecell is the measured cell voltage, E°cell is the standard cell voltage at standard conditions, n is the number of electrons transferred in the balanced redox reaction, and Q is the reaction quotient.
In this case, the reaction occurring in the Pt | H+ | H2 half-cell is the reduction of H+ ions to H2 gas. The balanced equation for this reaction is:
2H+ + 2e- -> H2.
The standard reduction potential for this reaction is 0 V.
Plugging in the given value for Ecell (0.0841 V) and the standard reduction potential (E°cell = 0 V) into the Nernst equation, we can solve for the unknown, which is the concentration of H+ ions in the solution.
0.0841 V = 0 V - (0.0592/2)log(Q).
Simplifying the equation:
0.0841 V = -(0.0296)log(Q).
Taking the antilogarithm of both sides:
log(Q) = -0.0841 V / (-0.0296) = 2.8426.
Q = 10^2.8426 = 706.7363.
The reaction quotient (Q) for this reaction is 706.7363.
Since the reaction quotient is equal to the ratio of products over reactants, and we have a 1:2 ratio for H2 and H+, we can determine the concentration of H+ ions.
Let X be the concentration of H+ ions.
(1/2X)^2 / X^2 = Q.
1/4X^2 = 706.7363.
X^2 = 4/(706.7363).
X ≈ 0.0293 M.
Therefore, the concentration of H+ ions in the Pt | H+ | H2 half-cell is approximately 0.0293 M.
(c) The equilibrium constant (K) for the reaction occurring in this cell can be calculated using the Nernst equation and the standard reduction potentials (E°cell) of the half-reactions involved.
The balanced redox reaction for the cell is:
Tl+ + 1e- -> Tl.
The standard reduction potential for this reaction is given on the next page. Let's denote it as E°Tl.
The Nernst equation is given by:
Ecell = E°cell - (0.0592/n)log(Q),
where Ecell is the measured cell voltage, E°cell is the standard cell voltage at standard conditions, n is the number of electrons transferred in the balanced redox reaction, and Q is the reaction quotient.
In this case, Ecell is given as 0.0841 V, and E°cell can be calculated using the standard reduction potentials of the half-reactions:
E°cell = E°cathode - E°anode.
Since the Pt | H+ | H2 half-cell is the cathode and the Tl | Tl+(1 mol L-1) half-cell is the anode, we can use the known standard reduction potentials to calculate E°cell.
E°cathode = 0 V (from the reduction of H+ to H2).
E°anode = E°Tl.
Plugging these values into the equation:
E°cell = 0 V - E°Tl.
0.0841 V = -(0.0592/1)log(Q).
Simplifying the equation:
0.0841 V = -0.0592log(Q).
Taking the antilogarithm of both sides:
log(Q) = -0.0841 V / -0.0592 = 1.4216.
Q = 10^1.4216 = 25.133.
Since the reaction quotient (Q) is equal to the ratio of products over reactants, we can write the equilibrium expression for the reaction as:
K = ([Tl+]/[H+])^1.
Substituting the given concentration of Tl+ as 1 mol L-1 and the calculated concentration of H+ ions as 0.0293 M:
K = (1/0.0293)^1 = 34.14.
Therefore, the value of the equilibrium constant (K) for the reaction occurring in this cell is approximately 34.14.