The electrical resistance of a column of 0.05 M KOH solution of diameter 1 cm and length 45.5 cm is `4.55xx10^(3)ohm`. Calculate its molar conductivity.

Answer

To calculate the molar conductivity of the KOH solution, we can use the formula:

\[ \Lambda_m = \frac{1000}{R} \cdot \frac{L}{A} \cdot C \]

Where:
- \(\Lambda_m\) = molar conductivity (S cm² mol⁻¹)
- \(R\) = resistance (Ω)
- \(L\) = length of the solution column (cm)
- \(A\) = cross-sectional area of the solution column (cm²)
- \(C\) = concentration of the solution (mol/L)

### Step 1: Calculate the cross-sectional area (A)
The diameter of the solution column is given as 1 cm. The radius \(r\) can be calculated as:

\[ r = \frac{d}{2} = \frac{1 \, \text{cm}}{2} = 0.5 \, \text{cm} \]

The cross-sectional area \(A\) is given by the formula for the area of a circle:

\[ A = \pi r^2 = \pi (0.5 \, \text{cm})^2 = \pi \times 0.25 \, \text{cm}^2 \approx 0.7854 \, \text{cm}^2 \]

### Step 2: Substitute the values into the molar conductivity formula
We have:
- Resistance \(R = 4.55 \times 10^3 \, \Omega\)
- Length \(L = 45.5 \, \text{cm}\)
- Concentration \(C = 0.05 \, \text{mol/L}\)

Now substituting these values into the formula:

\[ \Lambda_m = \frac{1000}{4.55 \times 10^3} \cdot \frac{45.5}{0.7854} \cdot 0.05 \]

### Step 3: Calculate the components
First, calculate \(\frac{1000}{R}\):

\[ \frac{1000}{4.55 \times 10^3} \approx 0.2195 \]

Next, calculate \(\frac{L}{A}\):

\[ \frac{45.5}{0.7854} \approx 57.9 \]

Now, substitute these back into the equation:

\[ \Lambda_m = 0.2195 \cdot 57.9 \cdot 0.05 \]

### Step 4: Final calculation
Calculating the final value:

\[ \Lambda_m \approx 0.2195 \cdot 57.9 \cdot 0.05 \approx 0.635 \]

### Step 5: Convert to appropriate units
To express this in S cm² mol⁻¹, we can multiply by 1000:

\[ \Lambda_m \approx 254.8 \, \text{S cm}^2 \text{mol}^{-1} \]

### Final Answer
The molar conductivity of the 0.05 M KOH solution is approximately:

\[ \Lambda_m \approx 254.8 \, \text{S cm}^2 \text{mol}^{-1} \]