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):}`
In experiment `(I)`, what time will reactant `(X)` take to be reduced into `0.0425 M`?

Options

A. `2.4 hr`
B. `0.48 hr`
C. `3.1 hr`
D. `4.7 hr`

Explanation

To solve the problem step-by-step, we will follow the outlined process:

### Step 1: Determine the order of the reaction
We have the rate data from two experiments. We can use the rate law to find the order of the reaction.

Given:
- Experiment I: [X] = 0.17 M, Rate = 0.05 mol L^(-1) hr^(-1)
- Experiment II: [X] = 0.34 M, Rate = 0.10 mol L^(-1) hr^(-1)

Using the rate law:
\[ \text{Rate} = k [X]^a \]

Taking the ratio of the rates from experiments I and II:
\[ \frac{\text{Rate}_I}{\text{Rate}_{II}} = \frac{k [X]_I^a}{k [X]_{II}^a} \]
Substituting the values:
\[ \frac{0.05}{0.10} = \frac{[0.17]^a}{[0.34]^a} \]
This simplifies to:
\[ \frac{1}{2} = \left(\frac{0.17}{0.34}\right)^a = \left(\frac{1}{2}\right)^a \]
Thus, we find:
\[ \frac{1}{2} = \frac{1}{2}^a \implies a = 1 \]

### Step 2: Write the rate law
Since the order of the reaction \( a \) is 1, the rate law can be expressed as:
\[ \text{Rate} = k [X] \]

### Step 3: Calculate the rate constant \( k \)
Using the data from Experiment I:
\[ 0.05 = k \times 0.17 \]
Solving for \( k \):
\[ k = \frac{0.05}{0.17} \approx 0.294 \text{ hr}^{-1} \]

### Step 4: Use the first-order kinetics formula to find time
For a first-order reaction, the formula for time \( t \) is given by:
\[ t = \frac{2.303}{k} \log\left(\frac{[X]_0}{[X]}\right) \]
Where:
- \( [X]_0 \) is the initial concentration of \( X \) (0.17 M from Experiment I)
- \( [X] \) is the concentration of \( X \) at time \( t \) (0.0425 M)

Substituting the values:
\[ t = \frac{2.303}{0.294} \log\left(\frac{0.17}{0.0425}\right) \]

### Step 5: Calculate the logarithm
Calculating the ratio:
\[ \frac{0.17}{0.0425} \approx 4 \]
Thus:
\[ \log(4) \approx 0.602 \]

### Step 6: Calculate time \( t \)
Now substituting back into the time equation:
\[ t = \frac{2.303}{0.294} \times 0.602 \]
Calculating:
\[ t \approx \frac{2.303 \times 0.602}{0.294} \approx 4.7 \text{ hours} \]

### Final Answer
The time taken for the concentration of \( X \) to be reduced to 0.0425 M is approximately **4.7 hours**.

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