`Zn | Zn^(2+)(C_(1) || Zn^(2+)(C_(2)| Zn`. For this cell `DeltaG` is negative if:

Options

A. `C_(1) = C_(2)`
B. `C_(1) gt C_(2)`
C. `C_(2) gt C_(1)`
D. `C_(1) gt 2C_(2)`

Answer

To determine the conditions under which ΔG is negative for the given electrochemical cell, we can follow these steps:

### Step 1: Understand the Cell Configuration
The cell is represented as:
\[ \text{Zn} | \text{Zn}^{2+}(C_1) || \text{Zn}^{2+}(C_2) | \text{Zn} \]
This indicates that zinc is oxidized at concentration \( C_1 \) and reduced at concentration \( C_2 \).

### Step 2: Relate ΔG to EMF
The Gibbs free energy change (ΔG) for an electrochemical reaction is related to the cell potential (E) by the equation:
\[ \Delta G = -nFE \]
Where:
- \( n \) = number of moles of electrons exchanged
- \( F \) = Faraday's constant (approximately 96485 C/mol)
- \( E \) = cell potential (EMF)

For ΔG to be negative, E must be positive.

### Step 3: Use the Nernst Equation
The Nernst equation for this cell can be expressed as:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.059}{n} \log \left( \frac{C_1}{C_2} \right) \]
Where:
- \( E^\circ_{\text{cell}} \) is the standard cell potential (which is constant for a given reaction).
- \( C_1 \) is the concentration of the oxidized species (Zn at \( C_1 \)).
- \( C_2 \) is the concentration of the reduced species (Zn at \( C_2 \)).

### Step 4: Analyze Conditions for E to be Positive
For \( E_{\text{cell}} \) to be positive:
\[ E^\circ_{\text{cell}} - \frac{0.059}{n} \log \left( \frac{C_1}{C_2} \right) > 0 \]
Rearranging gives:
\[ E^\circ_{\text{cell}} > \frac{0.059}{n} \log \left( \frac{C_1}{C_2} \right) \]

### Step 5: Determine the Relationship Between C1 and C2
To ensure that the logarithmic term does not make the overall expression negative, we need:
\[ \frac{C_1}{C_2} < 1 \]
This implies:
\[ C_1 < C_2 \]

### Conclusion
Thus, for ΔG to be negative, the condition that must be satisfied is:
\[ C_2 > C_1 \]

### Final Answer
The correct condition for ΔG to be negative is when \( C_2 \) is greater than \( C_1 \).

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