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 how the initial concentrations of A and B can be changed to increase the rate of the reaction by a factor of eight, we start with the given rate expression:

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

Where:
- \( [C_A] \) is the concentration of A,
- \( [C_B] \) is the concentration of B,
- \( k \) is the rate constant.

### Step 1: Understand the current rate expression
The rate of the reaction is dependent on the concentrations of A and B raised to their respective powers in the rate law.

### Step 2: Define the initial rate
Let the initial concentrations of A and B be \( [C_A] \) and \( [C_B] \). The initial rate \( R_0 \) can be expressed as:
\[ R_0 = k [C_A]^2 [C_B]^{1/2} \]

### Step 3: Determine the new rate for an increase by a factor of 8
To increase the rate by a factor of 8, we want:
\[ R = 8 R_0 \]

### Step 4: Set up the equation for the new concentrations
Let the new concentrations of A and B be \( [C_A'] \) and \( [C_B'] \). The new rate can be expressed as:
\[ R = k [C_A']^2 [C_B']^{1/2} \]

### Step 5: Relate the new rate to the initial rate
We want:
\[ k [C_A']^2 [C_B']^{1/2} = 8 k [C_A]^2 [C_B]^{1/2} \]

This simplifies to:
\[ [C_A']^2 [C_B']^{1/2} = 8 [C_A]^2 [C_B]^{1/2} \]

### Step 6: Consider possible changes in concentrations
Now we can analyze the changes in concentrations. We can express \( [C_A'] \) and \( [C_B'] \) in terms of their initial concentrations:
1. Let \( [C_A'] = x [C_A] \)
2. Let \( [C_B'] = y [C_B] \)

Substituting these into the equation gives:
\[ (x[C_A])^2 (y[C_B])^{1/2} = 8 [C_A]^2 [C_B]^{1/2} \]

This simplifies to:
\[ x^2 y^{1/2} [C_A]^2 [C_B]^{1/2} = 8 [C_A]^2 [C_B]^{1/2} \]

### Step 7: Cancel out the common terms
Since \( [C_A]^2 [C_B]^{1/2} \) is common on both sides, we can cancel it out:
\[ x^2 y^{1/2} = 8 \]

### Step 8: Solve for \( x \) and \( y \)
Now we need to find values of \( x \) and \( y \) that satisfy this equation.

1. If we double the concentration of A, \( x = 2 \):
\[ (2)^2 y^{1/2} = 8 \]
\[ 4y^{1/2} = 8 \]
\[ y^{1/2} = 2 \]
\[ y = 4 \]

Thus, if we double the concentration of A and quadruple the concentration of B, the rate will increase by a factor of 8.

### Conclusion
The concentrations of A and B should be changed as follows:
- **Double the concentration of A**: \( [C_A'] = 2[C_A] \)
- **Quadruple the concentration of B**: \( [C_B'] = 4[C_B] \)