Calculate ΔG at 298 K for the electrochemical cell consisting of Au metal in a 0.00500 M Au(NO3)3 solution and Cu metal in a 0.950 M Cu(NO3)2 solution.
Evaluate E for each half cell and calculate Ecell. Then DG = -nFEcell.
To calculate ΔG (change in Gibbs Free Energy) at 298 K for the given electrochemical cell, we need to use the Nernst equation:
ΔG = -nFE
Where:
ΔG = Change in Gibbs Free Energy
n = Number of moles of electrons transferred (in the balanced equation)
F = Faraday's constant (96485 C/mol)
E = Cell potential
To find the cell potential (E), we can use the Standard Reduction Potentials (E°) for the half-reactions involving Au and Cu.
The reduction half-reaction for Au is:
Au3+ + 3e- → Au E° = 1.498 V
The reduction half-reaction for Cu is:
Cu2+ + 2e- → Cu E° = 0.339 V
Since we are calculating ΔG at 298 K, we need to consider the effect of temperature on cell potential by using the Nernst equation:
E = E° - (RT/nF) ln(Q)
Where:
R = Universal Gas Constant (8.314 J/mol·K)
T = Temperature in Kelvin (298 K)
n = Number of moles of electrons transferred (in the balanced equation)
F = Faraday's constant (96485 C/mol)
Q = Reaction Quotient (concentrations of the products raised to the power of their stoichiometric coefficients divided by the concentrations of the reactants raised to the power of their stoichiometric coefficients)
Now, let's calculate Q for the given cell:
Q = [Au3+]/[Cu2+]
= (0.00500)/(0.950)
= 0.00526
Now we can calculate E for the given cell using the Nernst equation:
For Au:
E(Au) = 1.498 V - (8.314 J/mol·K * 298 K / (3 * 96485 C/mol)) * ln(0.00526)
For Cu:
E(Cu) = 0.339 V - (8.314 J/mol·K * 298 K / (2 * 96485 C/mol)) * ln(0.00526)
Finally, we can calculate ΔG using the formula mentioned earlier:
ΔG = -nFE
= -(3)(96485 C/mol)(E(Au)) - (2)(96485 C/mol)(E(Cu))
Simply substitute the values of E(Au) and E(Cu) into the equation to calculate ΔG.
To calculate the standard Gibbs free energy change (ΔG°) for the electrochemical cell, you need to use the Nernst equation:
ΔG° = -nFE°
Where:
- ΔG° is the standard Gibbs free energy change
- n is the number of moles of electrons transferred in the balanced redox equation
- F is the Faraday constant (96,485 C/mol)
- E° is the standard cell potential
To find the standard cell potential, you need to look up the standard reduction potentials for the half-cells involved in the electrochemical cell. You can find these values in a reference table or online. The half-reactions involved in this cell are:
Au3+ + 3e- → Au(s) (Reduction)
Cu2+ + 2e- → Cu(s) (Oxidation)
The standard reduction potentials for these half-reactions are:
E°(Au3+/Au) = +1.50 V
E°(Cu2+/Cu) = +0.34 V
To calculate the overall cell potential, you need to subtract the standard reduction potential of the oxidation half-reaction from the standard reduction potential of the reduction half-reaction:
E°cell = E°(Reduction) - E°(Oxidation)
E°cell = +1.50 V - (+0.34 V)
E°cell = +1.16 V
Now that you have the standard cell potential (E°cell), you can calculate the standard Gibbs free energy change (ΔG°) using the Nernst equation. The Nernst equation relates the standard Gibbs free energy change to the standard cell potential:
ΔG° = -nFE°
In this case, the number of moles of electrons transferred (n) is determined by the balanced equation of the cell. From the half-reactions provided, 3 moles of electrons are transferred for Au and 2 moles of electrons for Cu:
ΔG° = -nFE°
ΔG° = -(3 mol e- * 96,485 C/mol * 1.16 V + 2 mol e- * 96,485 C/mol * 1.16 V)
Finally, you can substitute the appropriate values into the equation and solve for ΔG° to get the answer.