For the reaction : `aA to bB`, it is given that
`log[(-dA)/(dt)] = log [(dB)/(dt)] + 0.6020`. What is `a : b` is ?

Options

A. 4
B. 2
C. 1.5
D. 0

Answer

To solve the problem, we need to analyze the given equation and derive the stoichiometric ratio \( a : b \) for the reaction \( aA \rightarrow bB \).

### Step-by-Step Solution:

1. **Understanding the Rate of Reaction**:
The rate of disappearance of A is given as:
\[ -\frac{d[A]}{dt} = k[A]^a \]
The rate of appearance of B is given as:
\[ \frac{d[B]}{dt} = k[B]^b \]

2. **Using the Given Equation**:
We are given:
\[ \log\left(-\frac{dA}{dt}\right) = \log\left(\frac{dB}{dt}\right) + 0.6020 \]
This can be rewritten using properties of logarithms:
\[ \log\left(-\frac{dA}{dt}\right) - \log\left(\frac{dB}{dt}\right) = 0.6020 \]
Which simplifies to:
\[ \log\left(\frac{-dA/dt}{dB/dt}\right) = 0.6020 \]

3. **Exponentiating Both Sides**:
By exponentiating both sides, we can eliminate the logarithm:
\[ \frac{-dA/dt}{dB/dt} = 10^{0.6020} \]
We know that \( 10^{0.6020} \) is approximately equal to 4 (since \( 10^{0.6020} \approx 4 \)).

4. **Setting Up the Ratio**:
Therefore, we have:
\[ \frac{-dA/dt}{dB/dt} = 4 \]
This implies:
\[ -dA = 4 dB \]
Hence, we can express this in terms of stoichiometric coefficients:
\[ \frac{dA}{dB} = \frac{1}{4} \]

5. **Finding the Stoichiometric Ratio**:
From the above relationship, we can conclude that:
\[ a : b = 4 : 1 \]

### Final Answer:
Thus, the ratio \( a : b \) is \( 4 : 1 \).