Table of Contents
Why Temperature Matters for Reaction Rates
Chemical reactions almost always speed up when the temperature is increased. In this chapter, we focus on how and how much reaction rates depend on temperature, and what this tells us about the underlying molecular processes.
You already know from chemical kinetics that a reaction rate describes how fast reactants are converted into products. Here we connect this speed to temperature quantitatively, mainly through the Arrhenius equation and the collision/transition-state picture.
Empirical Description: The Arrhenius Equation
For many reactions in a moderate temperature range, the dependence of the rate constant $k$ on temperature $T$ (in kelvin) is well described by the Arrhenius equation:
$$
k = A\, e^{-E_\mathrm{a}/(R T)}
$$
- $k$: rate constant (depends on the specific reaction and its order)
- $A$: pre-exponential factor (or frequency factor)
- $E_\mathrm{a}$: activation energy (J mol$^{-1}$)
- $R$: gas constant (8.314 J mol$^{-1}$ K$^{-1}$)
- $T$: absolute temperature (K)
Key qualitative consequences:
- Higher $T$ $\Rightarrow$ $E_\mathrm{a} / (R T)$ becomes smaller $\Rightarrow$ the exponential factor becomes less negative $\Rightarrow$ $k$ increases.
- Larger $E_\mathrm{a}$ $\Rightarrow$ stronger dependence of $k$ on $T$.
For comparison, a reaction with $E_\mathrm{a} = 10 \,\text{kJ mol}^{-1}$ is much less temperature-sensitive than one with $E_\mathrm{a} = 100 \,\text{kJ mol}^{-1}$.
The Pre-exponential Factor $A$
The factor $A$ summarizes contributions that are not exponentially sensitive to temperature:
- Collision frequency of reacting particles
- Orientation factor: fraction of collisions with a suitable geometry to react
- In more advanced treatments, contributions from molecular vibrations and entropy changes
$A$ often varies only mildly with temperature over a limited range and is sometimes treated as approximately constant.
Activation Energy and the Energy Barrier Concept
The activation energy $E_\mathrm{a}$ represents the energy barrier that must be overcome for reactants to be transformed into products.
- Reactant molecules typically must reach a high-energy, short-lived configuration called the transition state or activated complex.
- $E_\mathrm{a}$ is the difference in energy between reactants and the transition state (as described in the energy profile of a reaction).
Higher $E_\mathrm{a}$ means:
- Fewer molecules (at a given temperature) have enough energy to react on collision.
- The reaction is more strongly accelerated by increasing temperature.
Note: $E_\mathrm{a}$ is not the same as the enthalpy change $\Delta H$ of the reaction; it refers only to the height of the barrier, not the overall energy difference between reactants and products.
Linear Arrhenius Plots
The Arrhenius equation can be made linear by taking the natural logarithm of both sides:
$$
\ln k = \ln A - \frac{E_\mathrm{a}}{R} \cdot \frac{1}{T}
$$
This has the form of a straight line:
- $y = \ln k$
- $x = 1/T$
- Slope $m = -E_\mathrm{a}/R$
- Intercept $b = \ln A$
So, a plot of $\ln k$ vs. $1/T$ (an Arrhenius plot) should be approximately a straight line if the reaction follows Arrhenius behavior in that temperature range.
Determining Activation Energy Experimentally
- Measure the rate constant $k$ at several temperatures $T$.
- Calculate $\ln k$ and $1/T$ for each temperature.
- Plot $\ln k$ (vertical axis) vs. $1/T$ (horizontal axis).
- Fit a straight line; its slope $m$ is:
$$
m = -\frac{E_\mathrm{a}}{R}
$$
- Thus,
$$
E_\mathrm{a} = -m R
$$
- The intercept gives $\ln A$, so
$$
A = e^{\text{intercept}}
$$
This method is widely used to extract $E_\mathrm{a}$ from kinetic data.
Practical Rules of Thumb (and Their Limitations)
Many practical fields (e.g. food chemistry, pharmacy) use approximate rules to estimate how much a reaction speeds up with temperature. A common rule:
- “$Q_{10}$ rule”: For many processes, the rate roughly doubles when the temperature is increased by $10^\circ\mathrm{C}$ near room temperature.
Mathematically, this can be expressed as a temperature coefficient $Q_{10}$:
$$
Q_{10} = \frac{k(T + 10\,\mathrm{K})}{k(T)}
$$
Typical values:
- $Q_{10} \approx 2$: rate doubles per 10 K
- In reality, $Q_{10}$ depends on the reaction and the temperature range.
These rules are only approximations and should not replace a proper Arrhenius analysis when precise information is required.
Molecular Interpretation: Collision and Transition-State Ideas
The Arrhenius form can be rationalized using simple molecular models (without going deeply into the full theories from more advanced physical chemistry):
Collision Perspective
- Reactant molecules are in constant motion; they collide with each other.
- Only collisions in which the relative kinetic energy exceeds a threshold related to $E_\mathrm{a}$ can lead to reaction.
- The fraction of molecules with energy $\ge E_\mathrm{a}$ at temperature $T$ follows a Boltzmann-type factor $\propto e^{-E_\mathrm{a}/(R T)}$.
- This directly leads to an exponential dependence of $k$ on $1/T$.
Transition-State Perspective (Qualitative)
- During a reaction, molecules pass through a highly unstable transition state.
- The rate depends on:
- How often reactant molecules reach this configuration (frequency factor $A$).
- How high the free energy barrier to the transition state is (related to $E_\mathrm{a}$ under certain conditions).
- Raising $T$ increases the population of molecules able to cross this barrier.
These pictures give a physical meaning to $E_\mathrm{a}$ and $A$ beyond viewing them as mere empirical fitting parameters.
Deviations from Simple Arrhenius Behavior
Not all reactions follow the simple Arrhenius law perfectly over wide temperature ranges. Some observed deviations:
- Curved Arrhenius plots (non-linear $\ln k$ vs $1/T$):
- May indicate a change in mechanism at different temperatures.
- May reflect changes in phase (e.g. melting, boiling) or conformational changes in macromolecules (e.g. enzymes).
- Can arise from complex multi-step reactions where the rate-determining step changes with $T$.
In such cases, $E_\mathrm{a}$ may not be constant; instead, an apparent activation energy can be defined in a limited temperature region.
Temperature Limits and Practical Considerations
Raising temperature increases reaction rates but is not always beneficial or unlimited:
- At very high temperatures:
- Unwanted side reactions may become significant.
- Decomposition of reactants, products, or catalysts can occur.
- Materials of the reaction vessel or equipment may not withstand the conditions.
- In biological systems:
- Enzymes and other biomolecules can denature at elevated temperatures, drastically changing the kinetics.
- There is usually an optimal temperature range for enzyme-catalyzed reactions.
Thus, choosing the reaction temperature is usually a compromise between sufficient speed and stability/selectivity.
Summary of Key Points
- Reaction rates depend strongly on temperature; this is commonly expressed via the Arrhenius equation:
$$
k = A\, e^{-E_\mathrm{a}/(R T)}.
$$ - The activation energy $E_\mathrm{a}$ reflects the height of the energy barrier; higher $E_\mathrm{a}$ means greater temperature sensitivity.
- A plot of $\ln k$ against $1/T$ (Arrhenius plot) is approximately linear; from its slope, $E_\mathrm{a}$ can be determined.
- Practical rules (like “rate doubles per 10 K”) provide rough estimates but rest on the same basic exponential dependence.
- Molecular pictures (collisions, transition states) explain why only a fraction of molecules have enough energy to react, and why this fraction increases with temperature.
- Deviations from simple Arrhenius behavior may signal more complex mechanisms or changes in the reaction system.