Which of the following relationships are not correct?

Options

A. pH of solutio in hydrogen electrode`=("Electrode potential")/(0.0591)` at 298K
B. `E_(cell)=(0.0591)/(n)logK_(c)`
C. Cell cnstant=Conductivity/Conductance
D. `DeltaG^(@)=nFE_(cell)^(@)`.

Explanation

To determine which of the given relationships are not correct, we need to analyze each statement based on electrochemical principles. Here’s a step-by-step breakdown of the analysis:

### Step 1: Analyze the pH of the hydrogen electrode
1. **Standard Hydrogen Electrode (SHE)**: The standard electrode potential (E°) for SHE is defined as 0 V.
2. **Electrode Reaction**: The reaction at the anode involves the oxidation of hydrogen gas (H₂) to protons (H⁺) and electrons (e⁻):
\[ H_2 \rightarrow 2H^+ + 2e^- \]
3. **Nernst Equation**: The Nernst equation for the oxidation potential (E°_OP) is given by:
\[ E_{OP} = E°_{OP} - \frac{0.0591}{n} \log \frac{[H^+]^2}{1} \]
Since E°_OP = 0, it simplifies to:
\[ E_{OP} = -\frac{0.0591}{2} \log [H^+]^2 \]
4. **Relating pH and [H⁺]**: The pH is defined as:
\[ \text{pH} = -\log[H^+] \]
Therefore, the relationship between pH and E can be derived.

### Step 2: Evaluate the relationship for cathode
1. **Using SHE as Cathode**: If SHE acts as a cathode, the potential is considered as:
\[ \text{pH} = -\frac{E_{electrode}}{0.0591} \]
This indicates that the relationship changes depending on whether SHE is used as an anode or cathode.

### Step 3: Analyze the E_cell relationship
1. **E_cell Equation**: The equation for cell potential is:
\[ E_{cell} = E°_{cell} - \frac{0.0591}{n} \log K_C \]
At equilibrium, E_cell = 0, leading to:
\[ E°_{cell} = \frac{0.0591}{n} \log K_C \]
This indicates that the original statement about E_cell being equal to E°_cell is incorrect at equilibrium.

### Step 4: Evaluate the cell constant relationship
1. **Cell Constant Definition**: The cell constant (k) is defined as:
\[ k = \frac{L}{A} \]
where L is the length of the cell and A is the area. The statement that cell constant equals conductivity times conductance is incorrect.

### Step 5: Analyze the ΔG° relationship
1. **Gibbs Free Energy**: The relationship is given by:
\[ \Delta G° = -nFE°_{cell} \]
The statement regarding ΔG° does not represent a negative value for E°; hence, it is incorrect.

### Conclusion
Based on the analysis, all the relationships presented in the question are incorrect.

### Summary of Incorrect Relationships:
1. The relationship regarding pH and electrode potential for SHE as an anode or cathode.
2. The relationship for E_cell at equilibrium.
3. The relationship for cell constant.
4. The relationship for ΔG°.