Given the ionic conductance of `underset(underset(COO^(-))(|))(COO^(-)),K^(o+), ` and `Na^(o+)` are `74,50,` and `73cm^(2)ohm^(-1)eq^(-1)`, respectviley. The equivalent conductance at infinite dilution of the salt `underset(underset(COOK)(|))(COONa)` is

Options

A. `197 cm^(2)ohm^(-1)eq^(-1)`
B. `172cm^(2)ohm^(-1)eq^(-1)`
C. `135.5cm^(2)ohm^(-1)eq^(-1)`
D. `160.5cm^(2)ohm^(-1)eq^(-1)`

Explanation

To find the equivalent conductance at infinite dilution of the salt \( \text{COOK} \) and \( \text{COONa} \), we will use the given ionic conductances of the ions involved. The ionic conductance values provided are:

- \( \text{COO}^- \): \( 74 \, \text{cm}^2 \, \text{ohm}^{-1} \, \text{eq}^{-1} \)
- \( \text{K}^+ \): \( 50 \, \text{cm}^2 \, \text{ohm}^{-1} \, \text{eq}^{-1} \)
- \( \text{Na}^+ \): \( 73 \, \text{cm}^2 \, \text{ohm}^{-1} \, \text{eq}^{-1} \)

### Step 1: Determine the number of equivalents for each ion
The equivalent of an ion is determined by the formula:
\[ \text{Number of equivalents} = \frac{\text{Charge on ion}}{\text{Total charge}} \]

1. For \( \text{COO}^- \):
- Charge on ion = 1 (since \( \text{COO}^- \) has a charge of -1)
- Total charge = 1 (the total charge of the ion)
- Number of equivalents = \( \frac{1}{1} = 1 \)

2. For \( \text{K}^+ \):
- Charge on ion = 1
- Total charge = 1
- Number of equivalents = \( \frac{1}{1} = 1 \)

3. For \( \text{Na}^+ \):
- Charge on ion = 1
- Total charge = 1
- Number of equivalents = \( \frac{1}{1} = 1 \)

### Step 2: Calculate the equivalent conductance at infinite dilution
The formula for the equivalent conductance at infinite dilution \( \Lambda^0 \) of the salt \( \text{COOK} \) and \( \text{COONa} \) is given by:
\[ \Lambda^0 = \frac{1}{2} \Lambda^0_{K^+} + \frac{1}{2} \Lambda^0_{Na^+} + \Lambda^0_{COO^-} \]

Substituting the values:
\[ \Lambda^0 = \frac{1}{2} \times 50 + \frac{1}{2} \times 73 + 74 \]

### Step 3: Perform the calculations
1. Calculate \( \frac{1}{2} \times 50 = 25 \)
2. Calculate \( \frac{1}{2} \times 73 = 36.5 \)
3. Now sum these values:
\[ \Lambda^0 = 25 + 36.5 + 74 = 135.5 \, \text{cm}^2 \, \text{ohm}^{-1} \, \text{eq}^{-1} \]

### Final Answer
The equivalent conductance at infinite dilution of the salt \( \text{COOK} \) and \( \text{COONa} \) is:
\[ \Lambda^0 = 135.5 \, \text{cm}^2 \, \text{ohm}^{-1} \, \text{eq}^{-1} \]