`AB_(3)(g)`is dissociates as `AB_(3)(g)hArrAB_(2)(g)+(1)/(2)B_(2)(g)`
When the initial pressure of `AB_(3)` is800 torr and the pressure developed at equilibrium is 900 torr , what fraction of `AB_(3)(g)` is dissociated?

Options

A. `10%`
B. `20%`
C. `25%`
D. `30%`

Answer

To solve the problem step by step, we will analyze the dissociation of the gas \( AB_3 \) and calculate the fraction that has dissociated.

### Step 1: Write the reaction and identify initial conditions
The dissociation of \( AB_3 \) can be represented as:
\[ AB_3(g) \rightleftharpoons AB_2(g) + \frac{1}{2} B_2(g) \]
- Initial pressure of \( AB_3 \) = 800 torr
- At equilibrium, total pressure = 900 torr

### Step 2: Set up the change in pressure
Let \( x \) be the pressure of \( AB_3 \) that dissociates. Then:
- Pressure of \( AB_3 \) at equilibrium = \( 800 - x \) torr
- Pressure of \( AB_2 \) formed = \( x \) torr
- Pressure of \( B_2 \) formed = \( \frac{x}{2} \) torr

### Step 3: Write the expression for total pressure at equilibrium
The total pressure at equilibrium can be expressed as:
\[ \text{Total Pressure} = \text{Pressure of } AB_3 + \text{Pressure of } AB_2 + \text{Pressure of } B_2 \]
Substituting the pressures:
\[ (800 - x) + x + \frac{x}{2} = 900 \]

### Step 4: Simplify the equation
Combine the terms:
\[ 800 - x + x + \frac{x}{2} = 900 \]
This simplifies to:
\[ 800 + \frac{x}{2} = 900 \]

### Step 5: Solve for \( x \)
Rearranging the equation gives:
\[ \frac{x}{2} = 900 - 800 \]
\[ \frac{x}{2} = 100 \]
Multiplying both sides by 2:
\[ x = 200 \text{ torr} \]

### Step 6: Calculate the fraction of \( AB_3 \) that is dissociated
The fraction of \( AB_3 \) that is dissociated is given by:
\[ \text{Fraction dissociated} = \frac{x}{\text{Initial pressure of } AB_3} = \frac{200}{800} \]
Calculating this gives:
\[ \text{Fraction dissociated} = \frac{1}{4} = 0.25 \]

### Step 7: Convert to percentage
To express this as a percentage:
\[ \text{Percentage dissociated} = 0.25 \times 100 = 25\% \]

### Final Answer
The fraction of \( AB_3(g) \) that is dissociated is **25%**.

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