E degrees (V)
A galvanic cell based on the following half reaction s Au3+ 3e- ----> Au 1.50
Mg2+ 2e- ------> Mg -2.37
The cell is set up at 25 C with [Mg2+] = 1.00 x 10^-5 M
The cell potential is observed to be 4.01 V. Calculate the [Au3+] that must be present.
Au3+ = ____________M
To calculate the concentration of Au3+ ([Au3+]), we can use the Nernst equation, which relates the cell potential (Ecell) to the standard cell potential (E°cell) and the concentration of the species involved in the redox reaction.
The Nernst equation is given as:
Ecell = E°cell - (RT / nF) * ln(Q)
Where:
- Ecell is the cell potential
- E°cell is the standard cell potential
- R is the gas constant (8.314 J/(mol·K))
- T is the temperature in Kelvin (25°C + 273 = 298 K)
- n is the number of electrons transferred in the balanced equation (3 in this case, based on the Au3+ half-reaction)
- F is the Faraday constant (96,485 C/mol)
- Q is the reaction quotient, which is equal to the ratio of the concentrations of the products to the reactants raised to their respective stoichiometric coefficients.
First, let's calculate the Q value based on the concentrations given:
Q = ([Au3+] / 1)^(3) / ([Mg2+] / 1)^(2)
Since [Au3+] is the only unknown, we can rewrite this equation as:
Q = ([Au3+]^3) / (1.00 x 10^-5)^(2)
Next, we can rearrange the Nernst equation to solve for [Au3+]:
Ecell = E°cell - (RT / nF) * ln(Q)
(Ecell - E°cell) / -0.0592 = ln(Q)
Substituting the known values:
(4.01 - 1.50) / -0.0592 = ln(([Au3+]^3) / (1.00 x 10^-5)^(2))
Now, solve for [Au3+]:
ln(([Au3+]^3) / (1.00 x 10^-5)^(2)) = -54.23
Next, take the exponential of both sides:
(([Au3+]^3) / (1.00 x 10^-5)^(2)) = e^(-54.23)
Finally, solve for [Au3+]:
[Au3+]^3 = (1.00 x 10^-5)^(2) * e^(-54.23)
[Au3+] = (1.00 x 10^-5)^(2/3) * e^(-54.23/3)
[Au3+] ≈ 7.04 x 10^-13 M
Therefore, the concentration of Au3+ ([Au3+]) that must be present in the cell is approximately 7.04 x 10^-13 M.