For reaction, `2B+A rarr 2C`
Which of the following is correct ?

Options

A. Rate of disappearance of B is twice that of rate of disappearance of C
B. Rate of disappearance of A is twice that of rate of disappearance of C
C. Rate of appearance of C is twice that of rate of disappearance of A
D. All of these

Answer

To solve the problem regarding the reaction \( 2B + A \rightarrow 2C \), we need to analyze the rates of disappearance and appearance of the reactants and products involved in the reaction. Here's the step-by-step solution:

### Step 1: Write the Rate Expressions
For the reaction \( 2B + A \rightarrow 2C \), we can express the rates of disappearance of the reactants and the rates of appearance of the products as follows:

- The rate of disappearance of \( B \) is given by:
\[ -\frac{1}{2} \frac{d[B]}{dt} \]
(The coefficient of \( B \) is 2, hence we take \( \frac{1}{2} \) to balance the rate expression.)

- The rate of disappearance of \( A \) is given by:
\[ -\frac{d[A]}{dt} \]

- The rate of appearance of \( C \) is given by:
\[ \frac{1}{2} \frac{d[C]}{dt} \]
(Again, the coefficient of \( C \) is 2, hence we take \( \frac{1}{2} \) to balance the rate expression.)

### Step 2: Relate the Rates
From the stoichiometry of the reaction, we can relate these rates:
\[ -\frac{1}{2} \frac{d[B]}{dt} = -\frac{d[A]}{dt} = \frac{1}{2} \frac{d[C]}{dt} \]

### Step 3: Express Relationships
From the above relationships, we can derive:
1. The rate of disappearance of \( B \) is twice that of the rate of disappearance of \( A \):
\[ \frac{d[A]}{dt} = \frac{1}{2} \frac{d[B]}{dt} \]

2. The rate of appearance of \( C \) is equal to the rate of disappearance of \( A \):
\[ \frac{d[C]}{dt} = 2 \frac{d[A]}{dt} \]

### Step 4: Identify the Correct Statement
Now we can analyze the statements provided in the question. The correct relationships derived indicate that:
- The rate of appearance of \( C \) is twice the rate of disappearance of \( A \).

Thus, the correct statement is:
- The rate of appearance of \( C \) is twice that of the rate of disappearance of \( A \).

### Conclusion
The correct option is statement number 3: "The rate of appearance of \( C \) is twice that of the rate of disappearance of \( A \)."

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