For the reaction: `X(g) rarr Y(g)+Z(g)`, the following data were obtained at `30^(@)C`:
`{:("Experiment",[X](mol L^(-1)),"Rate" (mol L^(-1) hr^(-1))),(I,0.17,0.05),(II,0.34,0.10),(III,0.68,0.20):}`
The rate constant of the above reaction is

Options

A. `0.588 hr^(-1)`
B. `0.294 hr^(-1)`
C. `0.123 hr^(-1)`
D. `0.210 hr^(-1)`

Explanation

To find the rate constant for the reaction \( X(g) \rightarrow Y(g) + Z(g) \), we will follow these steps:

### Step 1: Write the rate law expression
The rate law for the reaction can be expressed as:
\[ \text{Rate} = k [X]^a \]
where \( k \) is the rate constant, \( [X] \) is the concentration of reactant \( X \), and \( a \) is the order of the reaction with respect to \( X \).

### Step 2: Determine the order of the reaction (a)
We will use the initial rate method by comparing the rates and concentrations from the experiments provided.

From the data:
- Experiment I: \( [X] = 0.17 \, \text{mol L}^{-1}, \, \text{Rate} = 0.05 \, \text{mol L}^{-1} \text{hr}^{-1} \)
- Experiment II: \( [X] = 0.34 \, \text{mol L}^{-1}, \, \text{Rate} = 0.10 \, \text{mol L}^{-1} \text{hr}^{-1} \)

Using the rates from Experiment I and II:
\[ \frac{\text{Rate}_1}{\text{Rate}_2} = \frac{0.05}{0.10} = \frac{1}{2} \]
\[ \frac{[X]_1^a}{[X]_2^a} = \frac{(0.17)^a}{(0.34)^a} = \left(\frac{0.17}{0.34}\right)^a = \left(\frac{1}{2}\right)^a \]
Since both sides equal \( \frac{1}{2} \), we can equate:
\[ \frac{1}{2} = \left(\frac{1}{2}\right)^a \]
This implies \( a = 1 \).

### Step 3: Confirm the order using another pair of experiments
Now, let's confirm the order using Experiment II and III:
- Experiment II: \( [X] = 0.34 \, \text{mol L}^{-1}, \, \text{Rate} = 0.10 \, \text{mol L}^{-1} \text{hr}^{-1} \)
- Experiment III: \( [X] = 0.68 \, \text{mol L}^{-1}, \, \text{Rate} = 0.20 \, \text{mol L}^{-1} \text{hr}^{-1} \)

Using the rates from Experiment II and III:
\[ \frac{\text{Rate}_2}{\text{Rate}_3} = \frac{0.10}{0.20} = \frac{1}{2} \]
\[ \frac{[X]_2^a}{[X]_3^a} = \frac{(0.34)^a}{(0.68)^a} = \left(\frac{0.34}{0.68}\right)^a = \left(\frac{1}{2}\right)^a \]
Again, both sides equal \( \frac{1}{2} \), confirming \( a = 1 \).

### Step 4: Calculate the rate constant (k)
Now that we know the order of the reaction \( a = 1 \), we can use any of the experiments to find \( k \). Let's use Experiment I:
\[ \text{Rate} = k [X]^1 \]
Substituting the values from Experiment I:
\[ 0.05 = k \times 0.17 \]
Solving for \( k \):
\[ k = \frac{0.05}{0.17} \approx 0.294 \, \text{hr}^{-1} \]

### Final Answer
The rate constant \( k \) for the reaction is approximately \( 0.294 \, \text{hr}^{-1} \).

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