Rate of reaction, collision theory and Boltzmann distribution

Introduction to Chemical Kinetics

Welcome to one of the most exciting and dynamic areas of chemistry: chemical kinetics! In chemical reactions, knowing what products can form is only half the story; we also need to know how fast the reaction happens and what factors control that speed.

Whether it is the rapid deployment of a car airbag in milliseconds or the slow rusting of an iron bridge over decades, the rate of reaction plays a crucial role in the world around us. In this chapter, we will explore what governs the speed of chemical reactions using two powerful models: collision theory and the Boltzmann distribution.

Note: Related topics such as catalysts, atmospheric radical kinetics, and rate equations with rate constants are covered in their dedicated companion chapters in this section.

1. What is the Rate of Reaction?

In simple terms, the rate of reaction is a measure of how quickly reactants are turned into products over time.

Mathematically, it is defined as the change in concentration of a reactant or a product per unit time:

\(\text{Rate of reaction} = \frac{\text{change in concentration}}{\text{time taken}}\)

The standard units for rate of reaction are typically moles per cubic decimetre per second: \(\text{mol dm}^{-3}\text{ s}^{-1}\).

Measuring Reaction Rates in the Laboratory

To measure how fast a reaction is proceeding, we monitor a physical or chemical property that changes as the reaction progresses. Common experimental techniques include:

• Monitoring gas volume: If a gas is produced (for example, hydrogen gas from a metal reacting with acid), we can collect the gas in a gas syringe or over water in an inverted measuring cylinder and record the volume at regular time intervals.
• Monitoring mass loss: When a heavy gas (such as carbon dioxide) escapes from an open flask on an electronic balance, the mass decreases over time.
• Colorimetry: If one of the reactants or products is coloured, a colorimeter measures the absorbance or transmission of light through the solution over time.
• Clock reactions: An initial rate method where we record the time taken, \(t\), for an observable change to appear (such as a colour change or a precipitate obscuring a cross). The initial rate is taken as proportional to \(\frac{1}{t}\).

Finding the Rate from Concentration–Time Graphs

When you plot concentration against time, the graph curves because the reaction starts fast, slows down as reactants are consumed, and eventually stops when a limiting reactant runs out.

• Initial rate: The rate right at the beginning of the reaction (\(t = 0\)). It is found by drawing a tangent to the curve at \(t = 0\) and calculating its gradient.
• Instantaneous rate: The rate at any specific time \(t\). It is found by drawing a tangent to the curve at that exact time and calculating its gradient:
\(\text{Gradient} = \frac{\Delta y}{\Delta x} = \frac{\Delta[\text{concentration}]}{\Delta t}\)

Key Takeaway: The rate of reaction is the change in concentration per unit time. The steeper the gradient of a concentration–time graph, the faster the reaction.

2. Collision Theory

Why do chemical reactions occur at different speeds? Collision theory provides the answer. It states that for a chemical reaction to occur, particles must collide with each other.

However, not every collision results in a chemical reaction! In fact, only a tiny fraction of collisions are successful. For a collision to be successful (effective), it must satisfy two essential conditions:

1. Sufficient Energy: The colliding particles must possess kinetic energy equal to or greater than the activation energy (\(E_{\text{a}}\)).
2. Correct Orientation: The reactive parts of the molecules must collide at the correct angle so that the necessary bonds can break and new bonds can form.

Activation Energy (\(E_{\text{a}}\))

Activation energy is defined as the minimum amount of energy required for a collision to be successful and start a reaction.

Think of activation energy as an energetic hurdle. If you roll a ball gently towards a steep hill, it simply rolls back down. Only a ball pushed with enough energy can clear the summit and roll down the other side.

On an enthalpy profile diagram:

• The activation energy \(E_{\text{a}}\) is always an arrow pointing upwards from the energy level of the reactants to the top of the energy barrier (the transition state).
• The overall enthalpy change, \(\Delta H\), is the difference between the energy level of the reactants and the energy level of the products.

Key Takeaway: A reaction happens only when particles collide with energy greater than or equal to \(E_{\text{a}}\) and with the correct molecular orientation.

3. Factors Affecting Rate: Concentration, Pressure, and Surface Area

We can alter the rate of a reaction by changing the frequency of collisions between particles.

Effect of Concentration (in solutions)

When you increase the concentration of a reactant in solution, there are more particles per unit volume. Because the particles are crowded closer together, they collide more frequently. This leads to a higher frequency of successful collisions, which increases the rate of reaction.

Effect of Pressure (in gaseous reactions)

Increasing the pressure of a gas mixture pushes the molecules into a smaller volume. Just like increasing concentration, this means there are more gas molecules per unit volume. As a result, the frequency of collisions increases, leading to an increased rate of reaction.

Effect of Surface Area (in heterogeneous solid reactions)

When a solid reactant is broken into smaller pieces or ground into a fine powder, its total surface area increases while its mass remains constant. More solid particles are exposed to the surrounding liquid or gas particles. Consequently, collisions occur more frequently at the exposed surface, increasing the frequency of successful collisions and thus increasing the rate of reaction.

Exam tip: Always write "increased frequency of collisions" (collisions per second) rather than just "more collisions". The time factor is essential!

Key Takeaway: Increasing concentration, pressure, or surface area increases the number of particles per unit volume (or exposed particles), resulting in a greater frequency of collisions and a faster rate.

4. The Boltzmann Distribution

In any gas or liquid sample at a given temperature, not all molecules move at the exact same speed. Some move very slowly, many travel at intermediate speeds, and a few move very fast. This spread of kinetic energies is described by the Maxwell–Boltzmann distribution.

Features of the Boltzmann Distribution Curve

When we plot the number of molecules (or fraction of molecules) on the y-axis against kinetic energy on the x-axis, the resulting curve has key characteristics that you must be able to describe and sketch:

• Starts at the origin \((0,0)\): No molecules have zero energy because all particles have some thermal motion.
• Peak of the curve: The highest point represents the most probable energy (\(E_{\text{mp}}\)) among the molecules. Note: this is not the average energy; the mean energy lies slightly to the right of the peak because the curve is asymmetrical.
• Asymptotic tail: The curve drops off towards high energies but never touches the x-axis. In theory, there is no upper limit to the energy a particle can possess.
• Total area under the curve: The area represents the total number of molecules in the sample. For a closed system, this total area remains constant.

Marking Activation Energy on the Distribution

We can mark the activation energy, \(E_{\text{a}}\), as a vertical line on the energy axis (usually towards the higher-energy tail on the right). The shaded area under the curve to the right of \(E_{\text{a}}\) represents the number of molecules with energy equal to or greater than the activation energy (\(E \ge E_{\text{a}}\)). Only collisions involving these energetic molecules have the potential to be successful!

Key Takeaway: The Boltzmann distribution shows the spread of molecular energies at a given temperature. Only the molecules in the shaded area to the right of \(E_{\text{a}}\) have enough energy to react upon collision.

5. Effect of Temperature on Reaction Rate and the Boltzmann Distribution

Why does even a small increase in temperature (say, by \(10\text{ K}\) or \(10\text{ }^\circ\text{C}\)) cause a dramatic increase in the rate of a chemical reaction, often doubling it?

Students often assume it is because particles move faster and collide more often. While it is true that particles move faster at higher temperatures, this increased collision frequency accounts for only a very small fraction of the rate increase!

The real secret lies in the Boltzmann distribution:

1. When temperature increases, the average kinetic energy of the particles increases.
2. The Boltzmann distribution curve changes shape:
  • The peak shifts to the right (to a higher energy value).
  • The peak becomes lower and broader.
  • The total area under the curve remains unchanged because the total number of particles has not changed.
  • The curve still starts at the origin and never touches the x-axis.
3. Because the curve flattens and shifts to the right, a significantly larger proportion of molecules now possess energy equal to or greater than the activation energy (\(E \ge E_{\text{a}}\)).
4. This causes a dramatic increase in the frequency of successful collisions, which produces a large increase in the reaction rate.

Did you know? The exponential relationship between temperature, activation energy, and the rate constant is described quantitatively by the Arrhenius equation, \(k = A e^{-E_{\text{a}}/RT}\), which appears on your Chemistry B Data Sheet and is studied in detail in later chapters!

Key Takeaway: Higher temperature flattens and shifts the Boltzmann curve to the right. This significantly increases the proportion of particles with \(E \ge E_{\text{a}}\), leading to a much greater frequency of successful collisions.

6. Summary of Common Exam Mistakes to Avoid

• Drawing Boltzmann curves incorrectly: Never start the curve above the origin, never let the high-energy tail touch or cross the x-axis, and ensure the higher-temperature curve has its peak to the right and lower than the original curve.
• Vague collision terminology: Do not write "more collisions occur". Always specify that there is an increased frequency of collisions (or more collisions per unit time) and, crucially, an increased frequency of successful collisions.
• Confusing \(E_{\text{a}}\) with temperature: Remember that changing the temperature does not change the activation energy \(E_{\text{a}}\)! The value of \(E_{\text{a}}\) is fixed for a given uncatalysed reaction. Temperature simply gives more particles enough energy to overcome that fixed barrier.