For what concentration of Ag+(aq.), will the emf of given cell be zero at \ 25 oC , if the concentration of Cu(s) | Cu2+(0.1M) || Ag+(aq.) | Ag(s)? Given , E0 Ag+/Ag=0.80V; E0 Cu2+/Cu=0.34V.

To determine the concentration of Ag⁺(aq) that will make the electromotive force (emf) of the given electrochemical cell zero at 25 °C, we can use the Nernst equation. The cell you described consists of a copper half-cell and a silver half-cell, and we need to find the conditions under which the overall cell potential becomes zero.

Understanding the Cell Reaction

The cell can be represented as:

Cu(s) | Cu²⁺(0.1 M) || Ag⁺(aq) | Ag(s)

In this setup, copper is oxidized, and silver is reduced. The half-reactions can be written as:

• Oxidation: Cu(s) → Cu²⁺(aq) + 2e^-
• Reduction: Ag⁺(aq) + e^- → Ag(s)

Standard Cell Potential Calculation

The standard cell potential (E⁰) can be calculated using the standard reduction potentials:

E⁰ = E⁰(Ag⁺/Ag) - E⁰(Cu²⁺/Cu)

Substituting the given values:

E⁰ = 0.80 V - 0.34 V = 0.46 V

Applying the Nernst Equation

The Nernst equation relates the cell potential to the concentrations of the reactants and products:

E = E⁰ - (RT/nF) ln(Q)

Where:

• R = universal gas constant (8.314 J/(mol·K))
• T = temperature in Kelvin (25 °C = 298 K)
• n = number of moles of electrons transferred (2 for this reaction)
• F = Faraday's constant (96485 C/mol)
• Q = reaction quotient

Calculating the Reaction Quotient (Q)

For our cell, the reaction quotient Q is given by:

Q = [Cu²⁺]/[Ag⁺]

Substituting the known concentration of Cu²⁺:

Q = 0.1/[Ag⁺]

Setting the Cell Potential to Zero

To find the concentration of Ag⁺ that makes the emf zero, we set E = 0:

0 = E⁰ - (RT/nF) ln(Q)

Rearranging gives:

ln(Q) = (nF/RT) E⁰

Substituting the values:

• n = 2
• F = 96485 C/mol
• R = 8.314 J/(mol·K)
• T = 298 K
• E⁰ = 0.46 V

Calculating the right side:

ln(Q) = (2 × 96485 / (8.314 × 298)) × 0.46

ln(Q) ≈ 0.59

Finding the Concentration of Ag⁺

Now we can solve for [Ag⁺]:

Q = e^0.59 ≈ 1.80

Substituting back into the expression for Q:

1.80 = 0.1/[Ag⁺]

Thus, [Ag⁺] = 0.1/1.80 ≈ 0.0556 M

Final Result

Therefore, the concentration of Ag⁺(aq) that will make the emf of the cell zero at 25 °C is approximately 0.0556 M. This means that at this concentration, the driving forces for the oxidation of copper and the reduction of silver are balanced, resulting in no net flow of electrons and thus an emf of zero.