2.8.7 · D5Chemical Kinetics
Question bank — Temperature dependence — Arrhenius equation k = A·e^(−Ea - RT)
Before we start, one reminder of the symbols so nothing here is used blind:
- = rate constant (how fast the reaction runs — see Rate Laws and Rate Constants).
- = pre-exponential factor (collision frequency × orientation).
- = activation energy (the barrier height — see Activation Energy and Transition State Theory).
- = gas constant ; = absolute temperature in Kelvin.
- The piece is the Boltzmann factor, the fraction of collisions energetic enough to react (from the Maxwell-Boltzmann Distribution).
True or false — justify
Doubling the temperature always doubles the rate constant.
False. depends on exponentially through , not linearly — a modest rise can multiply many-fold, and the factor is different for every reaction and every starting .
A reaction with a larger is more sensitive to temperature changes.
True. The slope of vs is , so a bigger makes climb more steeply as rises — high-barrier reactions respond dramatically to heating.
As , the rate constant approaches .
True. When is enormous, so , leaving — the maximum rate if energy were never the limiting factor, only collision frequency and orientation.
The Arrhenius equation predicts at absolute zero.
True (as a limit). At , so — no molecule has enough energy to clear the barrier, so the reaction effectively stops.
A catalyst increases in the Arrhenius equation.
False (mostly). A catalyst chiefly lowers by offering a new pathway, which raises far more powerfully than a change in would.
is a straight-line function of .
False. is straight against , not against . The linear form is , so the horizontal axis must be .
The pre-exponential factor has the same units as .
True. Since is dimensionless, forces to carry the units of , which depend on reaction order.
For an exothermic reaction the activation energy is negative.
False. is the barrier from reactants to the transition state and is essentially always positive; the overall energy change is a separate quantity — see Thermodynamics vs Kinetics.
If two reactions have the same they will have the same rate constant.
False. They also depend on ; two reactions can share a barrier height yet differ in collision frequency and orientation, giving different .
The Arrhenius equation applies only to gas-phase reactions.
False. It is broadly empirical and describes solution and surface reactions too, wherever an activation barrier governs the rate.
Spot the error
"At 25 °C, ."
The temperature must be absolute: . Using Celsius (or worse, a temperature that could be negative) breaks the dimensionless energy ratio and gives nonsense.
"."
The signs are flipped. The correct two-point form is ; heating () must give , which only the correct order delivers.
"The slope of an Arrhenius plot equals ."
The slope is (negative). is positive, so the line falls as increases (i.e. as drops).
"Since kJ/mol and J/(mol·K), ."
Units clash. Convert to joules first: J/mol, so . Mixing kJ with J-based makes the answer 1000× too small.
"A larger intercept on an Arrhenius plot means a larger ."
The intercept is , which is about , not . The slope carries ; the intercept is unrelated to the barrier.
"Because the reaction is exothermic, it must be fast."
Thermodynamics () and kinetics () are independent. A very exothermic reaction can still be immeasurably slow if is large — feasibility is not speed.
Why questions
Why does the exponential, not the term, dominate temperature sensitivity?
varies only mildly with , while changes very fast because sits inside an exponential — a small drop in produces a large multiplicative jump in .
Why do we use absolute temperature in Kelvin rather than Celsius?
The Boltzmann factor comes from statistical mechanics where measures the actual thermal energy scale from true zero; only Kelvin makes a meaningful, always-positive ratio of energies.
Why does raising increase the reacting fraction so much?
Higher fattens the high-energy tail of the Maxwell-Boltzmann Distribution, so far more molecules exceed ; the extra population in the tail grows exponentially, not proportionally.
Why can we eliminate using measurements at two temperatures?
Subtracting the two equations cancels the common term, leaving an expression in alone — so we can find without ever knowing .
Why is the steric/orientation factor bundled inside rather than the exponential?
Orientation is a geometric requirement independent of energy, so it multiplies the collision frequency (both temperature-weak); the exponential is reserved for the energy criterion alone.
Why does a catalyst speed a reaction without being consumed?
It provides an alternative route with lower , raising the Boltzmann factor ; being regenerated each cycle, it emerges unchanged — see Catalysis.
Edge cases
What does the Arrhenius equation predict when ?
The Boltzmann factor becomes , so and the rate is temperature-independent — every collision is energetic enough, so only collision frequency matters.
What happens to if is extremely large (very high barrier)?
becomes minuscule, so is tiny and the reaction is effectively frozen at ordinary temperatures until is raised enough to populate the tail.
Can the two-point formula ever give ?
Only if decreases with increasing ; genuine elementary reactions give , so a negative result usually flags a composite mechanism or measurement error.
If a reaction rate barely changes with temperature, what does that imply about ?
It implies a very small (a low or nearly flat barrier); with little energy required, warming adds few new successful collisions, so stays roughly constant — related to a small Q10 Temperature Coefficient.
What would a perfectly horizontal Arrhenius plot ( vs ) mean?
Zero slope means , i.e. : the rate constant does not depend on temperature at all, an idealized barrier-free process.
Recall Quick self-check
The slope of vs is ::: (negative, since ). As , ::: , because the Boltzmann factor tends to 1. Temperature in the Arrhenius equation must be in ::: Kelvin (absolute). A catalyst mainly changes ::: (lowers it), not the fundamental thermodynamics.