Calculate the value of equivalent conductivity of `MgCl_(2)` at infinite dilution if `lambda^(oo)(Mg^(2+))= 106.12 "ohm"^(-1)cm^(-2)"mol"^(-1), lambda^(oo)(Cl^(-))=76.34"ohm"^(-1)cm^(2)"mol"^(-1)`.

Explanation

To calculate the equivalent conductivity of \( \text{MgCl}_2 \) at infinite dilution, we will follow these steps:

### Step 1: Understand the components of \( \text{MgCl}_2 \)
The dissociation of \( \text{MgCl}_2 \) in water can be represented as:
\[ \text{MgCl}_2 \rightarrow \text{Mg}^{2+} + 2 \text{Cl}^- \]
This means that for every mole of \( \text{MgCl}_2 \), we get 1 mole of \( \text{Mg}^{2+} \) ions and 2 moles of \( \text{Cl}^- \) ions.

### Step 2: Write the formula for equivalent conductivity
The equivalent conductivity \( \Lambda_{\text{eq}}^{\infty} \) at infinite dilution can be calculated using the formula:
\[ \Lambda_{\text{eq}}^{\infty} = \frac{\Lambda^{\infty}(\text{Mg}^{2+}) + 2 \Lambda^{\infty}(\text{Cl}^-)}{Z} \]
where \( Z \) is the total number of moles of ions produced from one mole of \( \text{MgCl}_2 \).

### Step 3: Identify the values given
From the problem, we have:
- \( \Lambda^{\infty}(\text{Mg}^{2+}) = 106.12 \, \text{ohm}^{-1} \text{cm}^2 \text{mol}^{-1} \)
- \( \Lambda^{\infty}(\text{Cl}^-) = 76.34 \, \text{ohm}^{-1} \text{cm}^2 \text{mol}^{-1} \)

### Step 4: Calculate the total conductivity
Substituting the values into the formula:
\[ \Lambda_{\text{eq}}^{\infty} = 106.12 + 2 \times 76.34 \]
Calculating \( 2 \times 76.34 \):
\[ 2 \times 76.34 = 152.68 \]
Now, add this to \( 106.12 \):
\[ \Lambda_{\text{eq}}^{\infty} = 106.12 + 152.68 = 258.80 \, \text{ohm}^{-1} \text{cm}^2 \text{mol}^{-1} \]

### Step 5: Determine the effective number of ions \( Z \)
For \( \text{MgCl}_2 \), the total number of ions produced is:
\[ Z = 1 + 2 = 3 \]

### Step 6: Calculate the equivalent conductivity
Now, we can calculate the equivalent conductivity:
\[ \Lambda_{\text{eq}}^{\infty} = \frac{258.80}{3} \]
Calculating this gives:
\[ \Lambda_{\text{eq}}^{\infty} = 86.27 \, \text{ohm}^{-1} \text{cm}^2 \text{mol}^{-1} \]

### Final Answer
The equivalent conductivity of \( \text{MgCl}_2 \) at infinite dilution is:
\[ \Lambda_{\text{eq}}^{\infty} = 86.27 \, \text{ohm}^{-1} \text{cm}^2 \text{mol}^{-1} \]