The rate expression for the reaction A(g) + B(g) `to` C(g) is rate = `KC_A^2 C_B^½`. What changes in the initial concentration of A and B will cause the rate of reaction increase by a factor of eight?

Options

A. `C_A times 2, C_B times 2`
B. `C_A times 2,C_B times 4`
C. `C_A times 1 ,C_B times 4`
D. `C_A times 4,C_B times 1`

Answer

To determine the changes in the initial concentrations of A and B that will cause the rate of the reaction \( A(g) + B(g) \to C(g) \) to increase by a factor of eight, we start with the given rate expression:

\[ \text{Rate} = k [C_A]^2 [C_B]^{1/2} \]

We want to find new concentrations \( [C_A'] \) and \( [C_B'] \) such that:

\[ \text{New Rate} = 8 \times \text{Rate} \]

This means:

\[ k [C_A']^2 [C_B']^{1/2} = 8 \times k [C_A]^2 [C_B]^{1/2} \]

We can simplify this by canceling \( k \) from both sides:

\[ [C_A']^2 [C_B']^{1/2} = 8 [C_A]^2 [C_B]^{1/2} \]

Next, we can express \( [C_A'] \) and \( [C_B'] \) in terms of the original concentrations \( [C_A] \) and \( [C_B] \):

Let:
\[ [C_A'] = x [C_A] \]
\[ [C_B'] = y [C_B] \]

Substituting these into the equation gives:

\[ (x [C_A])^2 (y [C_B])^{1/2} = 8 [C_A]^2 [C_B]^{1/2} \]

Expanding this, we get:

\[ x^2 [C_A]^2 y [C_B]^{1/2} = 8 [C_A]^2 [C_B]^{1/2} \]

Now, we can cancel \( [C_A]^2 \) and \( [C_B]^{1/2} \) from both sides (assuming they are not zero):

\[ x^2 y = 8 \]

Now, we need to find suitable values for \( x \) and \( y \) that satisfy this equation.

### Possible Combinations:

1. **Option 1**: Let \( x = 2 \) and \( y = 2 \):
\[ (2)^2 (2) = 4 \times 2 = 8 \quad \text{(Valid)} \]

2. **Option 2**: Let \( x = 2 \) and \( y = 4 \):
\[ (2)^2 (4) = 4 \times 4 = 16 \quad \text{(Not Valid)} \]

3. **Option 3**: Let \( x = 1 \) and \( y = 8 \):
\[ (1)^2 (8) = 1 \times 8 = 8 \quad \text{(Valid)} \]

4. **Option 4**: Let \( x = 4 \) and \( y = 1 \):
\[ (4)^2 (1) = 16 \times 1 = 16 \quad \text{(Not Valid)} \]

### Conclusion:

The valid combinations that will increase the rate by a factor of eight are:
- Doubling the concentration of A and doubling the concentration of B: \( [C_A'] = 2[C_A] \) and \( [C_B'] = 2[C_B] \).
- Keeping the concentration of A the same and increasing the concentration of B by a factor of 8: \( [C_A'] = [C_A] \) and \( [C_B'] = 8[C_B] \).