For a chemical reaction, `m_1A+m_2B rarrn_1C+n_2D` The ratio of rate of disappearance of A to that of appearance of C is

Options

A. `m_1//m_2`
B. `m_2//m_1`
C. `n_1//m_1`
D. `m_1//n_1`

Answer

To solve the problem regarding the ratio of the rate of disappearance of A to that of the appearance of C in the reaction \( m_1A + m_2B \rightarrow n_1C + n_2D \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Reaction**:
The given reaction is \( m_1A + m_2B \rightarrow n_1C + n_2D \). Here, \( A \) and \( B \) are reactants while \( C \) and \( D \) are products. The coefficients \( m_1, m_2, n_1, \) and \( n_2 \) represent the stoichiometric coefficients of the respective substances.

2. **Define Rate of Reaction**:
The rate of a chemical reaction can be expressed in terms of the change in concentration of reactants and products over time. For reactant \( A \), the rate of disappearance can be expressed as:
\[ \text{Rate of disappearance of A} = -\frac{1}{m_1} \frac{d[A]}{dt} \]
where \([A]\) is the concentration of \( A \).

3. **Define Rate of Appearance of Product**:
For product \( C \), the rate of appearance can be expressed as:
\[ \text{Rate of appearance of C} = \frac{1}{n_1} \frac{d[C]}{dt} \]
where \([C]\) is the concentration of \( C \).

4. **Set Up the Ratio**:
We need to find the ratio of the rate of disappearance of \( A \) to the rate of appearance of \( C \):
\[ \text{Ratio} = \frac{\text{Rate of disappearance of A}}{\text{Rate of appearance of C}} = \frac{-\frac{1}{m_1} \frac{d[A]}{dt}}{\frac{1}{n_1} \frac{d[C]}{dt}} \]

5. **Simplify the Ratio**:
By simplifying the above expression, we get:
\[ \text{Ratio} = \frac{-\frac{d[A]}{dt}}{\frac{d[C]}{dt}} \cdot \frac{n_1}{m_1} \]
Since we are interested in the absolute values (disregarding the negative sign which indicates disappearance), we can write:
\[ \text{Ratio} = \frac{n_1}{m_1} \]

6. **Final Result**:
Therefore, the ratio of the rate of disappearance of \( A \) to that of the appearance of \( C \) is:
\[ \frac{m_1}{n_1} \]