Kinetics: The Change of Concentration with Time (Integrated Rate Laws)

4.4 The Change of Concentration with Time

Introduction to Rate Laws and Reaction Order

Chemical kinetics studies the rates of chemical reactions and how they change with varying conditions. Rate laws express the relationship between the concentration of reactants and the rate of reaction. These laws can be converted into equations that relate reactant concentrations to time, allowing us to predict how concentrations change as a reaction proceeds.

• Rate Law: An equation that links the rate of a reaction to the concentration(s) of reactant(s).

• Reaction Order: The power to which the concentration of a reactant is raised in the rate law.

First-Order Reactions

A first-order reaction is one whose rate depends on the concentration of a single reactant raised to the first power.

• Differential Rate Law: $\text{Rate} = -\frac{\Delta [A]}{\Delta t} = k[A]$

• Integrated Rate Law: $\ln[A]_t = -kt + \ln[A]_0$

• This form relates the initial concentration $[A]_0$ to its concentration at any time $[A]_t$.

• Allows determination of:

  • Concentration of reactant remaining at any time

  • Time required for a given fraction of a sample to react

  • Time required for a reactant concentration to fall to a certain level

Example: If you plot $\ln[A]_t$ vs. time and obtain a straight line, the reaction is first order. The slope of the line equals $-k$.

Second-Order Reactions

A second-order reaction is one whose rate depends either on the reactant concentration raised to the second power or on the concentrations of two reactants each raised to the first power.

• Differential Rate Law: $\text{Rate} = -\frac{\Delta [A]}{\Delta t} = k[A]^2$

• Integrated Rate Law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

• This form relates the initial concentration $[A]_0$ to its concentration at any time $[A]_t$.

Example: If you plot $1/[A]_t$ vs. time and obtain a straight line, the reaction is second order. The slope of the line equals $k$.

Experimental Determination of Reaction Order

To determine whether a reaction is first or second order, plot both $\ln[A]_t$ vs. time and $1/[A]_t$ vs. time using experimental data. The plot that yields a straight line indicates the reaction order.

• First Order: Straight line for $\ln[A]_t$ vs. time

• Second Order: Straight line for $1/[A]_t$ vs. time

Example Data Table

Time (s)	[NO2] (M)	ln[NO2]	1/[NO2]
0	0.01000	-4.605	100
50	0.00787	-4.845	127
100	0.00649	-5.037	154
200	0.00481	-5.337	208
300	0.00380	-5.573	263

Graphical Analysis

• Plot $\ln[NO_2]_t$ vs. time: If not a straight line, reaction is not first order.

• Plot $1/[NO_2]_t$ vs. time: If a straight line, reaction is second order.

Summary Table: Identifying Reaction Order by Graphing

Order	Graph	Linear Relationship	Equation
First	$\ln[A]_t$ vs. time	Straight line	$\ln[A]_t = -kt + \ln[A]_0$
Second	$1/[A]_t$ vs. time	Straight line	$\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

Key Points

• Rate constant (k): Can be determined from the slope of the appropriate linear plot.

• Integrated rate laws: Allow calculation of reactant concentration at any time.

• Experimental data: Used to distinguish between first- and second-order reactions by graphical analysis.

Additional info: These notes cover the foundational concepts of integrated rate laws, graphical determination of reaction order, and the use of experimental data in chemical kinetics. The equations provided are essential for solving kinetics problems in General Chemistry.