To solve the problem, we need to determine the initial rate of the reaction given the concentration data and the time required for the concentration of Z to increase by 0.005 mol/L. The reaction is:
\[ X + 2Y \rightarrow 3Z \]
### Step 1: Understanding the relationship between time and rate
The rate of a reaction is inversely proportional to the time taken for a certain change in concentration. Therefore, we can express the rate as:
\[ \text{Rate} = \frac{\Delta [Z]}{\Delta t} \]
Given that \(\Delta [Z] = 0.005 \, \text{mol/L}\), we can calculate the rate for each set of data.
### Step 2: Calculate the rate for each concentration set
1. For \([X_0] = 0.01 \, \text{mol/L}\) and \([Y_0] = 0.01 \, \text{mol/L}\):
- Time = 72 sec
- Rate = \(\frac{0.005}{72} = 6.94 \times 10^{-5} \, \text{mol/L/s}\)
2. For \([X_0] = 0.02 \, \text{mol/L}\) and \([Y_0] = 0.005 \, \text{mol/L}\):
- Time = 36 sec
- Rate = \(\frac{0.005}{36} = 1.39 \times 10^{-4} \, \text{mol/L/s}\)
3. For \([X_0] = 0.02 \, \text{mol/L}\) and \([Y_0] = 0.01 \, \text{mol/L}\):
- Time = 18 sec
- Rate = \(\frac{0.005}{18} = 2.78 \times 10^{-4} \, \text{mol/L/s}\)
### Step 3: Analyze the data to find the order of reaction with respect to X and Y
#### Finding the order with respect to X:
- Compare the first and third sets of data where \([Y]\) is constant (0.01 mol/L):
- When \([X]\) changes from 0.01 to 0.02 (doubles), the time decreases from 72 sec to 18 sec (4 times faster).
- Since time is inversely proportional to rate, the rate increases by a factor of 4.
- Therefore, if the concentration of X doubles and the rate increases by 4, the order with respect to X is 2.
#### Finding the order with respect to Y:
- Compare the second and third sets of data where \([X]\) is constant (0.02 mol/L):
- When \([Y]\) changes from 0.005 to 0.01 (doubles), the time decreases from 36 sec to 18 sec (2 times faster).
- Therefore, if the concentration of Y doubles and the rate increases by a factor of 2, the order with respect to Y is 1.
### Step 4: Write the rate law
From our findings, we can write the rate law as:
\[ \text{Rate} = k[X]^2[Y]^1 \]
### Step 5: Conclusion
The initial rate of the reaction can be calculated using the rate law once the rate constant \(k\) is known. However, since the question asks for the initial rate based on the data provided, we can summarize that the reaction is second order with respect to X and first order with respect to Y.
### Final Answer
The initial rate of the reaction is dependent on the concentrations of X and Y as described above.
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