How can i derive this Gibbs energy equation?

[lioric]

Homework Statement

For an electrochemical cell Gibbs free energy is is given by G=-nFE
Gibbs free energy for a reaction at any moment in time and standard state free energy is given by G=Go + RT lnQ

Derive an expression relating standard state cell potential and equilibrium constant for a reaction

Homework Equations

ΔG=-nFE and ΔG=ΔG^θ + RT lnQ

The Attempt at a Solution

I actually don t have an idea
But i m guessing that i have to mix both equations up since ΔG is common and then I'm totally clueless

 

[Sunil Simha]

What will be the value of ΔG and Q at equilibrium? And if E^θ were the standard cell potential, how would it be related to ΔG^θ?

 

[lioric]

Q would be K G would be-nFE

 

[Sunil Simha]

Q would be K G would be-nFE

Q would be K. You are right there. However, ΔG^θ=nFE^θ (you have already written so). Now to the part which is key to solving this,

When you have the reactants and products at equilibrium, what do you think the net free energy change for the whole process ( forward +backward reactions) is?

Say A $\Leftrightarrow$ B. For the forward reaction, let ΔG be the free energy change. What will be the free energy change for the reverse reaction? So what will be net free energy change (free energy change for forward reaction + free energy change for the reverse reaction)?

If you figure this out, the answer to the original question just pops out.

 

[DeeNos]

To derive the Gibbs energy equation for an electrochemical cell, we can start with the definition of Gibbs free energy:

ΔG = ΔH - TΔS

Where ΔH is the enthalpy and ΔS is the entropy of the system. For an electrochemical cell, the enthalpy change is related to the electrical work done by the cell, which is given by:

ΔH = -nFE

Where n is the number of moles of electrons transferred and F is the Faraday constant (96,485 C/mol). Substituting this into the Gibbs free energy equation, we get:

ΔG = -nFE - TΔS

Next, we can use the relation between entropy change and standard state free energy:

ΔS = ΔSθ + R lnQ

Where ΔSθ is the standard state entropy and Q is the reaction quotient. Substituting this into the Gibbs free energy equation, we get:

ΔG = -nFE - T(ΔSθ + R lnQ)

Simplifying, we get:

ΔG = -nFE - RT lnQ + RTΔSθ

Finally, we can use the relation between standard state free energy and standard state cell potential:

ΔGθ = -nFEθ

Where Eθ is the standard state cell potential. Substituting this into the equation, we get:

ΔG = ΔGθ + RT lnQ + RTΔSθ

Rearranging, we get the desired expression:

ΔGθ = -RT lnK

Where K is the equilibrium constant for the reaction. This shows that the standard state cell potential is related to the equilibrium constant through the Gibbs free energy equation.