Change in Concentration with Time (Integrated Rate Laws)

First-Order Reactions

The integrated rate law for a first-order reaction allows calculation of the concentration of a reactant at any time, given the initial concentration and the rate constant. For a reaction A → B that is first order in [A], the rate law is:

• Differential Rate Law:

• Integrated Rate Law:

• Linear Form:

This equation shows that a plot of versus time yields a straight line with a slope of .

Using Partial Pressure: For gas-phase reactions, partial pressure can be used in place of concentration because, at constant temperature, pressure is directly proportional to concentration ().

Second-Order Reactions

For a second-order reaction, the rate depends on the square of the concentration of one reactant. The integrated rate law is:

• Differential Rate Law:

• Integrated Rate Law:

• Linear Form:

A plot of versus time yields a straight line with a slope of .

Example: Determining Reaction Order

To determine if a reaction is first or second order, compare plots of vs. time and vs. time. The plot that is linear indicates the order.

Conclusion: Since vs. time is linear, the reaction is second order in NO2.

Zero-Order Reactions

For zero-order reactions, the rate is independent of the concentration of the reactant. The integrated rate law is:

• Differential Rate Law:

• Integrated Rate Law:

Zero-order reactions are often observed in catalytic processes where the catalyst surface is saturated.

Summary of Integrated Rate Laws

Half-Life of a Reaction

Definition and Equations

The half-life () is the time required for the concentration of a reactant to decrease to half its initial value. The half-life equations for different reaction orders are:

• Zero Order:

• First Order:

• Second Order:

Only first-order reactions have a half-life independent of the initial concentration.

Reaction Rate and Temperature

Collision Model

According to the collision model, molecules must collide with sufficient energy and proper orientation to react. The three main factors affecting reaction rate are:

• Frequency of collisions

• Orientation factor

• Activation energy ()

Energy Profile Diagram

The energy profile diagram shows the energy changes during a reaction. The activation energy () is the minimum energy required for a reaction to occur, while is the overall energy change.

Arrhenius Equation

The Arrhenius equation relates the rate constant to temperature and activation energy:

• Taking the natural logarithm:

Where is the frequency factor, is the gas constant (8.314 J/mol·K), and is the temperature in Kelvin.

Reaction Mechanisms

Elementary Reactions and Molecularity

An elementary reaction occurs in a single step and can be classified by the number of reactant molecules involved (molecularity):

Multistep Reactions and Rate-Determining Step

In multistep reactions, the overall rate is determined by the slowest step, known as the rate-determining step. Intermediates are species produced and consumed during the mechanism but do not appear in the overall equation.

Potential Energy Profiles and Mechanisms

Potential energy diagrams can be used to analyze the number of steps (elementary reactions), intermediates, and identify the rate-determining step in a reaction mechanism.

Additional info: The notes above include expanded academic context, definitions, and examples to ensure completeness and clarity for General Chemistry students.