To determine the order of the reaction and calculate the rate constant, we can follow these steps:
1. Given Data:
- Time (t) in minutes: [0, 5.0, 10.0, 15.0]
- Concentration of cyclopropane [tex]\([C_3H_6]\)[/tex] in mol/L: [0.098, 0.080, 0.066, 0.054]
2. Transformation and Linearization:
- To test for a first-order reaction, we will plot the natural logarithm (ln) of the concentration against time. If the reaction is first-order, this plot should yield a straight line.
3. Natural Logarithm of Concentrations:
- ln(0.098) ≈ -2.3228
- ln(0.080) ≈ -2.5257
- ln(0.066) ≈ -2.7181
- ln(0.054) ≈ -2.9188
Thus, the transformed data is:
- Time (t): [0, 5.0, 10.0, 15.0]
- ln([Cyclopropane]): [-2.3228, -2.5257, -2.7181, -2.9188]
4. Linear Regression:
- To determine the straight line that fits these points, we perform a linear regression on the ln(concentration) vs. time data. The slope of this line will help us determine the rate constant for the reaction, and the y-intercept can be used for further analysis if needed.
From the data:
- Slope (m) ≈ -0.0396
- Intercept (b) ≈ -2.3243
5. Determination of Reaction Order:
- Since the plot of ln([Cyclopropane]) vs. time is linear, and the slope is -0.0396, we can conclude that the reaction is first-order.
6. Calculation of the Rate Constant (k):
- For a first-order reaction, the relationship is given by:
[tex]\[ \ln([A]) = -kt + \ln([A]_0) \][/tex]
Here, the slope of the line [tex]\( m = -k \)[/tex].
- Therefore, the rate constant [tex]\( k \)[/tex] is:
[tex]\[ k = -(\text{slope}) = -(-0.0396) = 0.0396 \, \text{min}^{-1} \][/tex]
Therefore, the reaction is first-order, and the rate constant [tex]\( k \)[/tex] is [tex]\( 0.0396 \, \text{min}^{-1} \)[/tex].
