Which of the following rate laws has an overall order of `0.5` for reaction involving substances `x, y` and `z` ?

Options

A. Rate=`k(c_x)(c_y)(c_z)`
B. Rate=`k(c_x)^0.5 (c_y)^0.5 (c_z)^0.5`
C. Rate=`k(c_x)^1.5 (c_y)^-1 (c_z)^0`
D. Rate=`k(c_x)(c_z)^-1 (c_y)^2`

Answer

To determine which rate law has an overall order of 0.5 for a reaction involving substances x, y, and z, we need to analyze the given options based on the definition of overall order in chemical kinetics.

### Step-by-Step Solution:

1. **Understanding Overall Order**:
The overall order of a reaction is defined as the sum of the powers of the concentration terms in the rate law expression. If the rate law is expressed as:
\[ \text{Rate} = k [A]^m [B]^n [C]^p \]
Then the overall order is given by:
\[ \text{Overall Order} = m + n + p \]

2. **Analyzing Each Option**:
We will evaluate each provided option to find the one that gives an overall order of 0.5.

- **Option 1**:
\[ \text{Rate} = k [C_x]^1 [C_y]^1 [C_z]^1 \]
Here, the overall order is:
\[ 1 + 1 + 1 = 3 \]
This is not correct.

- **Option 2**:
\[ \text{Rate} = k [C_x]^{0.5} [C_y]^{0.5} [C_z]^{0.5} \]
Here, the overall order is:
\[ 0.5 + 0.5 + 0.5 = 1.5 \]
This is not correct.

- **Option 3**:
\[ \text{Rate} = k [C_x]^{1.5} [C_y]^{-1} [C_z]^0 \]
Here, the overall order is:
\[ 1.5 - 1 + 0 = 0.5 \]
This is correct.

- **Option 4**:
\[ \text{Rate} = k [C_x]^1 [C_y]^2 [C_z]^{-1} \]
Here, the overall order is:
\[ 1 + 2 - 1 = 2 \]
This is not correct.

3. **Conclusion**:
The only option that results in an overall order of 0.5 is **Option 3**:
\[ \text{Rate} = k [C_x]^{1.5} [C_y]^{-1} [C_z]^0 \]

### Final Answer:
The rate law with an overall order of 0.5 is:
\[ \text{Rate} = k [C_x]^{1.5} [C_y]^{-1} [C_z]^0 \]