Determining Rate Laws and Rate Constants from Experimental Data

Chemical Kinetics

Determining Rate Laws from Experimental Data

Chemical kinetics involves studying the speed of chemical reactions and the factors that affect these rates. The rate law expresses the relationship between the rate of a reaction and the concentrations of reactants.

• Rate Law: An equation that relates the reaction rate to the concentrations of reactants, each raised to a power (the reaction order with respect to that reactant).

• General Form: For a reaction $aA + bB \rightarrow \text{products}$, the rate law is $\text{Rate} = k[A]^m[B]^n$ where $k$ is the rate constant, and $m$ and $n$ are the reaction orders with respect to $A$ and $B$.

Example: For the reaction $2X + Y \rightarrow Z$, the rate law can be written as:

$\text{Rate} = k[X]^m[Y]^n$

Using Experimental Data to Determine Reaction Orders

• Compare experiments where only one reactant concentration changes to determine the order with respect to that reactant.

• Use the ratio method: $\frac{\text{Rate}_2}{\text{Rate}_1} = \left(\frac{[X]_2}{[X]_1}\right)^m$ (if [Y] is constant).

Step-by-Step Solution:

1. Determine the order with respect to X: Compare experiments 1 and 2:

   • [X] doubles (0.20 to 0.40), [Y] doubles (0.10 to 0.20), Rate doubles (7.0×10-4 to 1.4×10-3).

   • Compare experiments 2 and 3: [X] constant, [Y] doubles (0.20 to 0.40), Rate doubles (1.4×10-3 to 2.8×10-3).

   • Compare experiments 3 and 4: [X] increases by 1.5× (0.40 to 0.60), [Y] increases by 1.5× (0.40 to 0.60), Rate increases by 1.5× (2.8×10-3 to 4.2×10-3).

   Conclusion: The reaction is first order in X and first order in Y.

2. Write the rate law: $\text{Rate} = k[X][Y]$

Calculating the Rate Constant and Its Units

• Substitute values from any experiment into the rate law to solve for $k$.

• Example (using experiment 1):

$7.0 \times 10^{-4} = k (0.20)(0.10)$ $k = \frac{7.0 \times 10^{-4}}{0.02} = 0.035\ \text{mol}^{-1}\text{L}\ \text{s}^{-1}$

• Units: For a second-order reaction (overall order = 2), the units of $k$ are $\text{mol}^{-1}\text{L}\ \text{s}^{-1}$.

Summary Table: Experimental Data

Initial Rate of Formation of Z (mol L-1 s-1)	Initial [X] (mol L-1)	Initial [Y] (mol L-1)
7.0×10-4	0.20	0.10
1.4×10-3	0.40	0.20
2.8×10-3	0.40	0.40
4.2×10-3	0.60	0.60

Key Points

• Rate law for the reaction: $\text{Rate} = k[X][Y]$

• Rate constant: $k = 0.035\ \text{mol}^{-1}\text{L}\ \text{s}^{-1}$

• Units of k: $\text{mol}^{-1}\text{L}\ \text{s}^{-1}$ (for a second-order reaction)

Additional info: The reaction order is determined experimentally and does not necessarily match the stoichiometric coefficients in the balanced equation.