Question

If the units for rate are \(\mathrm{M} \mathrm{s}^{-1}\), what are the units for the rate constant \((\mathrm{k})\), if the overall order of the reaction is three? a. \(\mathrm{M}^{-1} \mathrm{~s}^{-1}\) b. \(\mathrm{M}^{-2} \mathrm{~s}^{-1}\) c. \(\mathrm{s}^{-1}\) d. \(\mathrm{M}^{2} \mathrm{~s}^{-1}\)

Short answer

The units for the rate constant \( k \) are \( \mathrm{M}^{-2} \mathrm{s}^{-1} \). (Option b)

Step-by-step solution

Understand Reaction Rate Units

The rate of a reaction is given in units of \( \text{M} \cdot \text{s}^{-1} \). This means it is expressed as concentration per unit time.

Understand Rate Law Formula

For a reaction with a rate law given by \( \text{Rate} = k [A_1]^{m_1} [A_2]^{m_2} \cdots [A_n]^{m_n} \), the rate \( \text{Rate} \) has units of \( \text{M} \cdot \text{s}^{-1} \). Here, \( k \) is the rate constant and the sum of exponents \( m_1 + m_2 + \cdots + m_n \) is the overall order of the reaction.

Relate Reaction Order to Rate Constant Units

For an overall reaction order of three, the units of \( k \) need to cancel with the units contributed by the concentration terms to result in \( \text{M} \cdot \text{s}^{-1} \). Since \( [A_1], [A_2], \ldots \) have units of \( \text{M} \), we need to account for two more molecules in the reaction looking like \( \text{M}^{2} \) in the denominator of \( k \)’s units to cancel with the \( \text{M}^{3} \) from the concentrations.

Calculate Units for k

For a third-order reaction, using the expression \( k = \frac{\text{Rate}}{\text{M}^{3}} \), we write the units of \( k \) as \( \frac{\text{M} \cdot \text{s}^{-1}}{\text{M}^{3}} = \text{M}^{-2} \cdot \text{s}^{-1} \). Thus, the rate constant \( k \) for a third-order reaction has units of \( \text{M}^{-2} \cdot \text{s}^{-1} \).

Key concepts

Reaction Order

The reaction order is a fundamental concept in chemical kinetics. It indicates how the rate of a chemical reaction depends on the concentration of its reactants. Reaction orders are typically determined experimentally rather than being derived from the stoichiometric coefficients of the balanced chemical equation.

There are different types of reaction orders:

• **Zero-Order Reaction**: The rate is independent of the concentration of reactants. This means the reaction proceeds at a constant rate.
• **First-Order Reaction**: The rate is directly proportional to the concentration of one reactant. Doubling the concentration would double the reaction rate.
• **Second-Order Reaction**: The rate is proportional to the square of the concentration of one reactant, or to the product of the concentrations of two reactants.
• **Third-Order Reaction**: The rate depends on the concentration of one reactant cubed, or on combinations of concentrations that add up to three.

Understanding reaction orders is crucial as they help us comprehend how different factors affect the speed of chemical reactions.

Rate Constant Units

The rate constant, represented as 'k', is a proportionality factor in the rate law. Its units vary depending on the overall order of the reaction. This variability ensures the rate law has consistent units of concentration/time, typically expressed as molarity per second \(\mathrm{M}\cdot \mathrm{s}^{-1}\).

For example:

• In a **first-order reaction** (reaction order = 1), the units of \( k \) are \( \mathrm{s}^{-1} \).
• In a **second-order reaction** (reaction order = 2), the rate constant units are \( \mathrm{M}^{-1} \cdot \mathrm{s}^{-1} \).
• In a **third-order reaction** (reaction order = 3), like in the exercise, the units for \( k \) become \( \mathrm{M}^{-2} \cdot \mathrm{s}^{-1} \). This accounts for the concentration terms so that the final rate has units \( \mathrm{M}\cdot \mathrm{s}^{-1} \).

The rate constant's units are essential to comprehending how the reaction speed changes with concentration changes. They also provide insight into the interaction and complexity of the reaction's mechanism.

Rate Law

The rate law expresses the relation between the rate of a chemical reaction and the concentration of its reactants. It is an equation of the form:

• \( \text{Rate} = k [A_1]^{m_1} [A_2]^{m_2} \cdots [A_n]^{m_n} \)

In this expression:

• \( k \) is the rate constant.
• \([A_1], [A_2], \ldots [A_n]\) are the concentrations of the reactants.
• \(m_1, m_2, \ldots m_n\) are the reaction orders concerning each reactant, indicating how each reactant's concentration affects the rate.

The overall order of a reaction is the sum of these exponents: \(m_1 + m_2 + \ldots + m_n\). This defines how changes in concentrations will affect the reaction rate.

For example, in a third-order reaction discussed in the exercise, three moles of reactant concentrations will determine how the rate speeds up or slows down. Thus, the rate law is a critical aspect for scientists to predict how fast a reaction can proceed under various conditions.