To derive a single equation that describes how the measured cell potential varies with the concentration of ions in the cathode and anode half-cells, you can use the Nernst equation. The Nernst equation relates the cell potential to the concentration of ions involved in the electrochemical reaction.
The Nernst equation is given by:
E = E° - (0.0592/n) * log(Q)
Where:
E is the measured cell potential
E° is the standard cell potential
n is the number of electrons transferred in the balanced equation for the reaction
Q is the reaction quotient, which can be calculated using the concentrations of the species involved in the electrochemical reaction.
For the Ag vs. Cu concentration relationship, the cathode half-cell contains Ag ions, and the anode half-cell contains Cu ions. Therefore, the reaction for this cell can be written as:
Cu(s) + 2Ag+(aq) -> Cu2+(aq) + 2Ag(s)
Since 2 electrons are transferred in this reaction, n = 2.
Similarly, for the Zn vs. Cu concentration relationship, the cathode half-cell contains Zn ions, and the anode half-cell contains Cu ions. Therefore, the reaction for this cell can be written as:
Cu(s) + Zn2+(aq) -> Cu2+(aq) + Zn(s)
Since 2 electrons are transferred in this reaction, n = 2.
Now that you have the reactions and the values of n, you can rewrite the Nernst equation using the derived equations for each cell potential:
For the Ag vs. Cu concentration equation:
E1 = E1° - (0.0592/2) * log(Q1)
For the Zn vs. Cu concentration equation:
E2 = E2° - (0.0592/2) * log(Q2)
To derive a single equation, you can subtract the two equations:
(E1 - E2) = (E1° - E2°) - (0.0592/2) * (log(Q1) - log(Q2))
Now, you need to express the concentrations in terms of Q values. From the equations you provided, you can rearrange them to express x (concentration) in terms of y (cell potential):
For Ag vs. Cu concentration: x = (y - 0.5515) / -0.0452
For Zn vs. Cu concentration: x = (y - 0.7855) / 0.0372
Substituting these expressions for x in terms of y into the Q values, we get:
Q1 = [Ag+]^2 / [Cu+]^1
= [(y - 0.5515) / -0.0452]^2 / [Cu+]^1
Q2 = [Zn2+]^1 / [Cu+]^1
= [Zn+]^1 / [(y - 0.7855) / 0.0372]^1
Now, substitute Q1 and Q2 into the equation (E1 - E2) = (E1° - E2°) - (0.0592/2) * (log(Q1) - log(Q2)) and simplify to get the single equation that describes the variation of cell potential with ion concentration.
Note: It is essential to use the correct values for E1° and E2°, which correspond to the standard cell potentials for the respective reactions.