For the given electrolyte `A_(x)B_(y)`. The degree of dissociation `'alpha'` can be given as:

Options

A. (a) `alpha = (i-1)/((x+y-1)`
B. (b) `i = (1-alpha)+x alpha+ y alpha`
C. ( c) `alpha = (1-i)/((1-x-y))`
D. (d) Either of these

Explanation

To derive the expression for the degree of dissociation (α) of the electrolyte \( A_xB_y \), we can follow these steps:

### Step 1: Understand the Dissociation of the Electrolyte
The electrolyte \( A_xB_y \) dissociates into its ions when dissolved in a solvent. The dissociation can be represented as:
\[ A_xB_y \rightleftharpoons x A^+ + y B^- \]

### Step 2: Set Up Initial Conditions
Assume we start with 1 mole of the electrolyte \( A_xB_y \). Before dissociation, the initial amount of the electrolyte is:
- Moles of \( A_xB_y = 1 \)
- Moles of \( A^+ = 0 \)
- Moles of \( B^- = 0 \)

### Step 3: Define the Degree of Dissociation
Let \( \alpha \) be the degree of dissociation. After dissociation, the moles of the species can be expressed as:
- Moles of \( A_xB_y \) remaining = \( 1 - \alpha \)
- Moles of \( A^+ \) produced = \( x\alpha \)
- Moles of \( B^- \) produced = \( y\alpha \)

### Step 4: Calculate Total Moles at Equilibrium
The total moles at equilibrium can be calculated as:
\[ \text{Total moles} = (1 - \alpha) + (x\alpha) + (y\alpha) = 1 - \alpha + (x + y)\alpha \]
This simplifies to:
\[ \text{Total moles} = 1 + (x + y - 1)\alpha \]

### Step 5: Define the Van 't Hoff Factor (i)
The Van 't Hoff factor \( i \) is defined as the ratio of the total number of particles in solution to the number of formula units of solute initially dissolved. Thus:
\[ i = \frac{\text{Total moles at equilibrium}}{\text{Initial moles of solute}} = \frac{1 + (x + y - 1)\alpha}{1} \]
This simplifies to:
\[ i = 1 + (x + y - 1)\alpha \]

### Step 6: Rearranging to Find α
Rearranging the equation to isolate \( \alpha \):
\[ i - 1 = (x + y - 1)\alpha \]
Thus, we can express \( \alpha \) as:
\[ \alpha = \frac{i - 1}{x + y - 1} \]

### Final Expression
The degree of dissociation \( \alpha \) for the electrolyte \( A_xB_y \) is given by:
\[ \alpha = \frac{i - 1}{x + y - 1} \]