Construct a galvanic (voltaic) cell from the half reactions shown below.
Au3+ + 3 e- → Au (s) ξo= 1.420 V
Br2 (l) + 2 e-→ 2 Br- (aq) ξo= 1.087 V
Calculate the equilibrium constant (K) for this cell. Assume that the temperature is 298K.
First, consider what happens to the Nernst equation when the system reaches equilibrium:
ξ = ξo - (RT/nF) ln Q
What is ξ when the system has reached equilibrium? Is there any driving force left?
Remember that ∆G = -nFξ.
At equilibrium, ∆G = 0, so ξ = 0, too.
At equilibrium, Q = K, since all reactants and products are present at their equilibrium concentrations.
This simplifies the Nernst equation to:
ξocell = [(RT)/nF] ln K
Now calculate a value for K.
You may need to use scientific notation to record your answer. If so, enter it as: 4E43 which is equivalent to 4x1043
Use second equation
1.420- (-1.087)= .333
Use that as cell potential.
R=8.314
T=298
n= 6 mols electrons because you have to multiply the two reactions to get 6 electrons so you can cancel the electrons out
F=96,500
.333= 0.004279lnQ
divide both sides by .00427 to cancel it from the right side
.333/.00427=77.82
77.82=lnQ
enter in calculator 2nd and ln to get e^ (77.82) because you want to get ride of ln from the right side so its pushed to the left side as its inverse of e^ (77.82) and ln cancels out so youre left with Q.
To calculate the equilibrium constant (K) for the galvanic cell formed by the given half-reactions, we can use the Nernst equation.
At equilibrium, the cell potential (ξ) is 0, which means there is no driving force for the reaction to proceed further. Therefore, the Nernst equation at equilibrium is:
0 = ξo - (RT/nF) ln K
Since ξo is the difference in standard electrode potentials for the two half-reactions, the cell potential at equilibrium (ξocell) can be given as:
ξocell = ξo (Au3+/Au) - ξo (Br2/Br-)
Substituting the values of ξo for the two half-reactions:
ξocell = 1.420 V - 1.087 V
ξocell = 0.333 V
Now, substituting the values of ξocell, temperature (T), Faraday's constant (F), and rearranging the equation, we have:
0.333 V = [(8.314 J/mol∙K) (298 K) / (n F)] ln K
Simplifying further:
0.333 V = (2.47936 J/mol) / (n F) ln K
Now, we can calculate the value of K by rearranging the equation and solving for K:
ln K = (n F / (2.47936 J/mol)) ∙ 0.333 V
ln K = (n F) ∙ (0.1341 mol/J)
K = e^[(n F) ∙ (0.1341 mol/J)]
Since the number of moles of electrons transferred (n) for the two half-reactions is 3 and 2, respectively, we can substitute the values and calculate K:
K = e^[(3∙96485 C/mol) ∙ (0.1341 mol/J)] (Since 1 C = 1/J)
K ≈ 3.426 x 10^18
To calculate the equilibrium constant (K) for the cell, we need to use the Nernst equation, which relates the cell potential (ξ) to the equilibrium constant (K).
The Nernst equation is given by:
ξ = ξo - [(RT)/(nF)] ln Q
In this equation,
ξ is the cell potential,
ξo is the standard cell potential,
R is the gas constant (8.314 J/(mol K)),
T is the temperature in Kelvin (298 K in this case),
n is the number of electrons involved in the half-reaction,
F is the Faraday's constant (96485 C/mol),
ln is the natural logarithm, and
Q is the reaction quotient.
At equilibrium, the cell potential (ξ) is 0, as there is no driving force for the reaction to proceed in either direction. Therefore, we can set ξ = 0 in the Nernst equation.
0 = ξo - [(RT)/(nF)] ln K
Since ξ = 0, we can simplify the equation further:
ξo = [(RT)/(nF)] ln K
Now we can plug in the given values and calculate K:
ξo = 1.420 V - 1.087 V = 0.333 V
R = 8.314 J/(mol K)
T = 298 K
n = number of electrons involved in the overall reaction = 3 (from the Au3+ and Br2 half-reactions)
F = 96485 C/mol
0.333 V = [(8.314 J/(mol K)) * (298 K)] / [(3 mol e-) * (96485 C/mol)] * ln K
Simplifying the equation further:
0.333 V = (0.02786 J/C) * ln K
Now we can solve for ln K:
ln K = (0.333 V) / (0.02786 J/C) = 11.95
Finally, we can find K by taking the exponential of both sides:
K = e^(ln K) = e^11.95
Calculating this in a calculator or using a software, we get the value of K as approximately 161,671,011,563.86 (in Scientific Notation: 1.617 x 10^11).