`aA+bBrarr"Product",dx//dt=k[A]^(a)[B]^(b)`. If conc. Of A is dounled, rate becomes four times. If conc. Of B is made four times, rate is doubled. What is the relation between rate of disappearance of A that B?

Options

A. `{-(d[A])/(dt)}={-(d[B])/(dt)}`
B. `{-(d[A])/(dt)}=4{-(d[B])/(dt)}`
C. `-4{(d[A])/(dt)}={-(d[B])/(dt)}`
D. None of the above

Answer

To solve the problem, we need to determine the relationship between the rate of disappearance of A and B based on the information given about the changes in concentration and their effects on the rate of reaction.

### Step-by-Step Solution:

1. **Understanding the Rate Law**:
The rate of the reaction is given by the equation:
\[ \frac{dx}{dt} = k[A]^a[B]^b \]
where \( k \) is the rate constant, \( [A] \) is the concentration of A, \( [B] \) is the concentration of B, and \( a \) and \( b \) are the orders of the reaction with respect to A and B, respectively.

2. **Effect of Doubling A**:
When the concentration of A is doubled (\([A] \to 2[A]\)), the rate becomes four times the original rate. Let’s denote the original rate as \( R_1 \):
\[ R_1 = k[A]^a[B]^b \]
The new rate \( R_2 \) when A is doubled is:
\[ R_2 = k(2[A])^a[B]^b = k \cdot 2^a [A]^a [B]^b \]
According to the problem, \( R_2 = 4R_1 \):
\[ k \cdot 2^a [A]^a [B]^b = 4(k[A]^a[B]^b) \]
Dividing both sides by \( k[A]^a[B]^b \):
\[ 2^a = 4 \]
This implies:
\[ a = 2 \]

3. **Effect of Quadrupling B**:
Now, when the concentration of B is quadrupled (\([B] \to 4[B]\)), the rate becomes double the original rate. The new rate \( R_3 \) is:
\[ R_3 = k[A]^a(4[B])^b = k[A]^a \cdot 4^b [B]^b \]
According to the problem, \( R_3 = 2R_1 \):
\[ k[A]^a \cdot 4^b [B]^b = 2(k[A]^a[B]^b) \]
Dividing both sides by \( k[A]^a[B]^b \):
\[ 4^b = 2 \]
This implies:
\[ b = \frac{1}{2} \]

4. **Final Reaction Equation**:
From the values of \( a \) and \( b \), we can write the balanced reaction:
\[ 2A + \frac{1}{2}B \rightarrow \text{Products} \]

5. **Rate of Disappearance Relation**:
The rate of disappearance of A and B can be expressed as:
\[ -\frac{d[A]}{dt} = \frac{1}{2} \cdot -\frac{d[B]}{dt} \]
Rearranging gives:
\[ -\frac{d[A]}{dt} = 4 \cdot -\frac{d[B]}{dt} \]

### Conclusion:
The relationship between the rate of disappearance of A and B is:
\[ -\frac{d[A]}{dt} = 4 \cdot -\frac{d[B]}{dt} \]