Question

For the isothermal gas-phase reaction A $\to$ 3B, the fractional change in volume of the system between no conversion and complete conversion is ______

• A. 1
• B. 2
• C. 3
• D. 2/3

Hint:

For any gas-phase reaction of the form \( aA \to bB + cC \), the fractional volume change parameter is calculated as:
\[ \epsilon_A = y_{A0} \cdot \delta \quad \text{where} \quad \delta = \frac{b + c - a}{a} \]
Here, \( y_{A0} = 1 \) (pure A) and \( \delta = \frac{3 - 1}{1} = 2 \), giving \( \epsilon_A = 1 \cdot 2 = 2 \).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the fractional change in volume (\( \epsilon_A \)) for the isothermal, isobaric gas-phase reaction \( A \to 3B \), assuming the feed consists of pure reactant A.
This is a standard problem in chemical reaction engineering involving variable-volume gas systems.

Step 2: Key Formula or Approach:
The fractional change in volume (\( \epsilon_A \)) is defined as:
\[ \epsilon_A = \frac{V_{X_A=1} - V_{X_A=0}}{V_{X_A=0}} \]
For an ideal gas system at constant temperature and pressure, the volume is directly proportional to the total number of moles:
\[ \epsilon_A = \frac{N_{X_A=1} - N_{X_A=0}}{N_{X_A=0}} \]

Step 3: Detailed Explanation:

• Consider a starting basis of \( 1 \text{ mole} \) of pure A:
Initial moles of A, \( N_{A0} = 1 \)
Total initial moles, \( N_0 = N_{A0} = 1 \) (corresponding to \( X_A = 0 \))

• Write the stoichiometric table for the reaction \( A \to 3B \):
At complete conversion (\( X_A = 1 \)):
All \( 1 \text{ mole} \) of A is consumed to produce \( 3 \text{ moles} \) of B.
Total final moles, \( N_f = 3 \text{ moles} \) (corresponding to \( X_A = 1 \))

• Calculate the fractional change in volume (\( \epsilon_A \)):
\[ \epsilon_A = \frac{N_f - N_0}{N_0} = \frac{3 - 1}{1} = 2 \]

• This positive value (\( \epsilon_A = 2 \)) indicates that the system expands to three times its original volume at complete conversion.


Step 4: Final Answer:
The fractional change in volume of the system is 2.