`EMF` of the cell `Zn ZnSO_(4)(a =0.2)||ZnSO_(4)(a_(2))|Zn` is `-0.0088V` at `25^(@)C`. Calculate the value of `a_(2)`.

Explanation

To solve the problem of calculating the value of \( a_2 \) in the concentration cell described, we will use the Nernst equation. Here are the steps to find the solution:

### Step 1: Understand the Cell Reaction
In the given concentration cell, we have two zinc electrodes in different concentrations of zinc sulfate (\( ZnSO_4 \)). The reaction can be written as:
\[ Zn^{2+} (a_2) + 2e^- \leftrightarrow Zn (s) \]
\[ Zn (s) \leftrightarrow Zn^{2+} (0.2) + 2e^- \]

### Step 2: Identify the Nernst Equation
The Nernst equation is given by:
\[ E_{cell} = E^\circ_{cell} - \frac{0.0591}{n} \log \frac{[Zn^{2+}]_{cathode}}{[Zn^{2+}]_{anode}} \]

### Step 3: Determine the Standard Cell Potential
Since both electrodes are made of the same metal (zinc), the standard cell potential \( E^\circ_{cell} \) is zero:
\[ E^\circ_{cell} = 0 \]

### Step 4: Substitute Known Values into the Nernst Equation
We know:
- \( E_{cell} = -0.0088 \, V \)
- \( n = 2 \) (number of electrons transferred)
- Concentration at the anode (lower concentration) = \( 0.2 \)
- Concentration at the cathode (higher concentration) = \( a_2 \)

Substituting these values into the Nernst equation:
\[ -0.0088 = 0 - \frac{0.0591}{2} \log \frac{0.2}{a_2} \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ -0.0088 = -0.02955 \log \frac{0.2}{a_2} \]

### Step 6: Isolate the Logarithmic Term
Dividing both sides by \(-0.02955\):
\[ \frac{-0.0088}{-0.02955} = \log \frac{0.2}{a_2} \]
\[ 0.297 = \log \frac{0.2}{a_2} \]

### Step 7: Convert from Logarithmic to Exponential Form
Using the property of logarithms:
\[ \frac{0.2}{a_2} = 10^{0.297} \]

### Step 8: Calculate \( 10^{0.297} \)
Calculating \( 10^{0.297} \):
\[ 10^{0.297} \approx 1.989 \]

### Step 9: Solve for \( a_2 \)
Now substituting back:
\[ \frac{0.2}{a_2} = 1.989 \]
\[ a_2 = \frac{0.2}{1.989} \]
\[ a_2 \approx 0.1006 \]

### Final Answer
The value of \( a_2 \) is approximately \( 0.1006 \, M \).

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