Which of the following correctly represent the equivalent conductivity `(^^_(N))` and molar conductivity `(^^_(M))` of aluminium sulphate where equivalent conductivities of `Al^(+3)` and `SO_(4)^(-2)` are `lamda_(1)` and `lamda_(2)` respectively?

Options

A. .^^_(N)=lamda_(1)+lamda_(2)`
B. .^^_(M)=3lamda_(1)+2lamda_(2)`
C. .^^_(N)=(lamda_(1))/(3)+(lamda_(2))/(2)`
D. .^^_(M)=6lamda_(1)+6lamda_(2)`

Answer

To solve the question regarding the equivalent conductivity (λₙ) and molar conductivity (λₘ) of aluminum sulfate (Al₂(SO₄)₃), we will follow these steps:

### Step 1: Understand the dissociation of aluminum sulfate
Aluminum sulfate dissociates in water to produce aluminum ions (Al³⁺) and sulfate ions (SO₄²⁻). The balanced equation for this dissociation is:
\[ \text{Al}_2(\text{SO}_4)_3 \rightarrow 2 \text{Al}^{3+} + 3 \text{SO}_4^{2-} \]

### Step 2: Define equivalent conductivity (λₙ)
Equivalent conductivity (λₙ) is defined as the conductivity of a solution containing one gram equivalent of the solute. For aluminum sulfate, it can be expressed as:
\[ \lambda_n = \lambda_1 + \lambda_2 \]
where:
- λ₁ = equivalent conductivity of Al³⁺
- λ₂ = equivalent conductivity of SO₄²⁻

### Step 3: Define molar conductivity (λₘ)
Molar conductivity (λₘ) is defined as the conductivity of a solution containing one mole of the solute. For aluminum sulfate, it can be expressed as:
\[ \lambda_m = 2\lambda_1 + 3\lambda_2 \]
Here, the coefficients (2 and 3) correspond to the number of moles of Al³⁺ and SO₄²⁻ produced from one mole of aluminum sulfate.

### Step 4: Write the final expressions
From the above definitions, we have:
- Equivalent conductivity:
\[ \lambda_n = \lambda_1 + \lambda_2 \]
- Molar conductivity:
\[ \lambda_m = 2\lambda_1 + 3\lambda_2 \]

### Step 5: Identify the correct option
Based on the derived expressions for equivalent and molar conductivities, we can check the options provided in the question to find the correct representation.

### Summary of the solution:
1. **Equivalent conductivity (λₙ)**: \( \lambda_n = \lambda_1 + \lambda_2 \)
2. **Molar conductivity (λₘ)**: \( \lambda_m = 2\lambda_1 + 3\lambda_2 \)