The nature of curve of `E^(@)` cell against log `K_(C )` is:

Options

A. a straight line
B. parabola
C. a hyperbola
D. an elliptical curve

Answer

To determine the nature of the curve of \( E^\circ \) (standard cell potential) against \( \log K_c \) (equilibrium constant), we can follow these steps:

### Step 1: Understand the Nernst Equation
The Nernst equation relates the cell potential \( E \) to the concentrations of the reactants and products:
\[ E = E^\circ - \frac{2.303RT}{nF} \log \left( \frac{[\text{products}]}{[\text{reactants}]} \right) \]
Where:
- \( E \) is the cell potential
- \( E^\circ \) is the standard cell potential
- \( R \) is the universal gas constant (8.314 J/mol·K)
- \( T \) is the temperature in Kelvin
- \( n \) is the number of moles of electrons transferred
- \( F \) is Faraday's constant (96500 C/mol)

### Step 2: Analyze the Condition at Equilibrium
At equilibrium, the cell potential \( E \) becomes 0, and we can express this condition as:
\[ 0 = E^\circ - \frac{2.303RT}{nF} \log K_c \]
Rearranging this gives:
\[ E^\circ = \frac{2.303RT}{nF} \log K_c \]

### Step 3: Substitute Known Values
Substituting the values for \( R \), \( T \), and \( F \):
- \( R = 8.314 \, \text{J/mol·K} \)
- \( T = 298 \, \text{K} \) (25°C)
- \( F = 96500 \, \text{C/mol} \)

Calculating \( \frac{2.303RT}{nF} \):
\[ \frac{2.303 \times 8.314 \times 298}{n \times 96500} \]
This simplifies to:
\[ \frac{0.0591}{n} \]

### Step 4: Formulate the Equation of the Line
Now we can express \( E^\circ \) as:
\[ E^\circ = \frac{0.0591}{n} \log K_c \]
This equation is in the form of \( y = mx \), where:
- \( y = E^\circ \)
- \( m = \frac{0.0591}{n} \)
- \( x = \log K_c \)

### Step 5: Determine the Nature of the Curve
Since the relationship between \( E^\circ \) and \( \log K_c \) is linear, the nature of the curve is a straight line. The slope of the line is positive if \( n \) is positive, indicating that as \( K_c \) increases, \( E^\circ \) also increases.

### Final Answer
The nature of the curve of \( E^\circ \) cell against \( \log K_c \) is a **straight line**.

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