The rate expression for reaction `A(g) +B(g) rarr C(g)` is rate `=k[A]^(1//2)[B]^(2)`. What change in rate if initial concentration of `A` and B increases by factor 4 and 2 respectively ?

Options

A. 4
B. 6
C. 8
D. None of these

Explanation

To solve the problem, we need to analyze how the rate of the reaction changes when the concentrations of reactants A and B are altered. The rate expression given is:

\[ \text{Rate} = k[A]^{1/2}[B]^{2} \]

### Step 1: Identify the initial rate expression
The initial rate of the reaction can be expressed as:

\[ \text{Rate}_0 = k[A]^{1/2}[B]^{2} \]

### Step 2: Determine the new concentrations
The problem states that the concentration of A increases by a factor of 4, and the concentration of B increases by a factor of 2. Therefore, the new concentrations can be expressed as:

- New concentration of A: \( [A]_{new} = 4[A] \)
- New concentration of B: \( [B]_{new} = 2[B] \)

### Step 3: Substitute the new concentrations into the rate expression
Now, we substitute the new concentrations into the rate expression to find the new rate:

\[ \text{Rate}_1 = k[4A]^{1/2}[2B]^{2} \]

### Step 4: Simplify the new rate expression
Now, we simplify the expression:

1. Calculate \( [4A]^{1/2} \):
\[ [4A]^{1/2} = 4^{1/2}[A]^{1/2} = 2[A]^{1/2} \]

2. Calculate \( [2B]^{2} \):
\[ [2B]^{2} = 2^{2}[B]^{2} = 4[B]^{2} \]

Now, substituting these back into the rate expression:

\[ \text{Rate}_1 = k \cdot (2[A]^{1/2}) \cdot (4[B]^{2}) \]

### Step 5: Combine the factors
Now, we can combine the factors:

\[ \text{Rate}_1 = k \cdot 2 \cdot 4 \cdot [A]^{1/2} \cdot [B]^{2} = 8k[A]^{1/2}[B]^{2} \]

### Step 6: Relate the new rate to the initial rate
We can relate the new rate to the initial rate:

\[ \text{Rate}_1 = 8 \cdot \text{Rate}_0 \]

### Conclusion
Thus, the rate of the reaction increases by a factor of 8 when the concentration of A is increased by a factor of 4 and the concentration of B is increased by a factor of 2.

### Final Answer
The rate after the change is **8 times the initial rate**.

---