Section 42–4 Chemical kinetics

Idealizations / Approximations / Limitations

 

This section is related to the Arrhenius equation k = Ae^−Ea/RT, which provides a fundamental link between the activation energy E_a and the rate constant k of a chemical reaction. The rate constant is expressed as the product of a pre-exponential factor (or prefactor A) and an exponential factor. The prefactor represents the frequency of molecular collisions, while the exponential factor gives the fraction of molecules whose kinetic energy equals or exceeds the activation energy. Interestingly, although this equation is his most famous contribution to chemical kinetics, Svante Arrhenius was awarded the Nobel prize for his “electrolytic theory of dissociation.” He was the first to propose that compounds like salt do not merely dissolve in water, but spontaneously split into ions—an idea so radical at the time that his doctoral thesis was nearly rejected.

 

1. Idealizations:

“The same situation that we have just called “ionization” is also found in a chemical reaction. For instance, if two objects A and B combine into a compound AB, then if we think about it for a while we see that AB is what we have called an atom, B is what we call an electron, and A is what we call an ion. With these substitutions the equations of equilibrium are exactly the same in form: n_An_B/n_AB = ce^−W/kT (42.9) (Feynman et al., 1963).”

 

In standard chemical kinetics, the Arrhenius equation relies on several idealizations that simplify the complexity of molecular collisions. First, it assumes that reactants are in thermal equilibrium with their surroundings, so their kinetic energies follow a Maxwell-Boltzmann distribution characterized by a single temperature T. Second, it treats the activation energy as a fixed, temperature-independent threshold, assuming that a reaction occurs only when molecules possess energy exceeding the energy barrier, while neglecting quantum effects such as tunneling. Third, the prefactor A is taken to be constant or only weakly temperature-dependent, even though in reality it can vary significantly with molecular orientations and collision dynamics. Despite these simplifications, the Arrhenius equation provides a remarkably accurate empirical framework for a wide variety of "well-behaved" systems.

 

“This energy is called the activation energy—the energy needed to ‘activate’ the reaction. Call A^* the activation energy, the excess energy needed in a collision in order that the reaction may really occur. Then the rate R_f at which A and B produce AB would involve the number of atoms of A times the number of atoms of B, times the rate at which a single atom would strike a certain cross section σ_AB, times a factor e^−A*/kT which is the probability that they have enough energy: R_f = n_An_Bvσ_ABe^−A*/kT (Feynman et al., 1963).”

 

In collision theory, Feynman’s formulation  refines the prefactor by making it explicitly dependent on the average molecular speed v, rather than treating it as a constant. From the kinetic theory of gases, v µ ÖT, so the collision frequency—and hence the prefactor—becomes temperature-dependent. The Eyring–Polanyi equation develops this refinement further by showing that the prefactor is linearly proportional to temperature. Using transition state theory, the prefactor is expressed as kkT/h, where k is the transmission coefficient, k is the Boltzmann constant, T is the temperature, and h is the Planck’s constant. This model is empirically more accurate because it accounts for internal degrees of freedom—such as molecular vibrations and rotations—revealing that as temperature rises, the collision frequency for reactions increase even more.

 

2. Approximations:

“The interesting thing is that the rate of the reaction also varies as e^−const/kT, although the constant is not the same as that which governs the concentrations; the activation energy A∗ is quite different from the energy W. W governs the proportions of A, B, and AB that we have in equilibrium, but if we want to know how fast A+B goes to AB, that is not a question of equilibrium, and here a different energy, the activation energy, governs the rate of reaction through an exponential factor (Feynman et al,, 1963).”

 

Feynman’s statement that “activation energy A* is quite different from the energy W” can be misleading if it is interpreted to mean that the two are unrelated. In fact, they are closely connected through the reaction pathway: the forward and reverse activation energies differ precisely by the energy change W. Specifically, the forward activation energy is A_f = A*, while the reverse activation energy is A_b = A* + W, so that A_b - A_f = W. A helpful analogy is to think of W as the height difference between the reactants (A + B) and the product (AB), but A* is the height of the fence you must climb to go from reactants to products. In this theory, W determines equilibrium proportions (how much A, B, and AB are present after a long time), whereas A* governs the reaction rate (how fast A and B convert into AB). The molecular energies follow the Maxwell-Boltzmann distribution—an approximation that breaks down under non-equilibrium conditions, which can be influenced by endothermic or exothermic energy release.

The Arrhenius equation is a central formula in chemical kinetics that expresses how reaction rates increase rapidly with rising temperature or lower activation energy. While earlier studies focused on reactant concentrations, Arrhenius (1889) showed that the effect of temperature could be captured in a remarkably simple equation. By analyzing eight sets of empirical data and equations, he expressed the rate constants at two temperatures as: k(T₁) = k(T₀)exp{C(T₁ - T₀)/T₁T₀}, where T₀ and T₁ are absolute temperatures and C is a constant characteristic of the reaction. This expression is mathematically equivalent to the modern form k = Aexp(-E_a/RT), in which the constant C is related to the activation energy. Arrhenius’s key insight was that vastly different chemical processes could all be modeled with a common exponential equation, even though each reaction has its own characteristic parameters that depend on its mechanism and temperature range over which it is studied (see below).

(Source: Logan, 1982)

 

3. Limitations

“Or if we put in a third kind of object it may change the rate very much; some things produce enormous changes in rate simply by changing the A* a little bit—they are called catalysts. A reaction might practically not occur at all because A* is too big at the given temperature, but when we put in this special stuff, the catalyst, then the reaction goes very fast indeed, because A* is reduced (Feynman et al., 1963).”

 

While the Arrhenius equation remains a useful framework for describing catalyzed reactions, its parameters require reinterpretation, and its apparent simplicity can be misleading. A catalyst does not simply lower an existing energy barrier; it introduces an alternative pathway with a reduced activation energy. Because E_a appears in the exponential term, even a modest decrease produces a large increase in the reaction rate—this is the primary effect of catalysis. However, catalyzed reactions are often multi-step mechanisms involving intermediate processes. Furthermore, the prefactor A is not truly constant: a catalyst can influence molecular orientation and adsorption probabilities, making A temperature-dependent. In essence, the Arrhenius equation still provides a convenient empirical description for catalyzed reactions, particularly over limited temperature ranges, but a deeper understanding requires the insights of Transition State Theory and, in some cases, quantum tunneling effect.

 

“The forward rate would involve the product n_An_Bn_C, and it might seem that our formula is going wrong, but no! …... The law of equilibrium, (42.9), which we first wrote down is absolutely guaranteed to be true, no matter what the mechanism of the reaction may be! (Feynman et al., 1963).”

 

It can be misleading when Feynman suggests that the reaction-rate equation is guaranteed to hold regardless of the reaction mechanism. The standard Arrhenius  equation is only an approximation that can be refined into the modified form k = ATⁿe^−Ea/RT to account for the temperature-dependence of the prefactor. The Tⁿ term—typically with -1 < n < 1—expresses how molecular speeds, collision frequencies, and orientation effects vary with thermal energy. In logarithmic form, ln k = ln A + nln T − E_a/RT, this expression shows that a standard Arrhenius plot (ln k vs. 1/T) is not strictly linear; the nln T term introduces a slight curvature, reflecting the dynamic nature of the prefactor. Over a limited temperature ranges, this deviation is often neglected, allowing researchers to extract an effective activation energy from the local slope. Thus, the modified formulation makes explicit what the simple Arrhenius equation conceals: the exponential term still represents the statistical probability of overcoming the energy barrier, while the prefactor includes the temperature-dependent collision frequency of the reaction attempts.

 

While the Arrhenius equation provides a useful first-order approximation for reaction rates, it begins to break down under certain conditions. A clear example is diffusion-limited reactions, where the activation energy is negligible (E_a » 0). In this limit, the reaction rate is governed by how quickly reactants can diffuse through the medium to encounter one another. The kinetics are therefore controlled by the physical properties of the solvent—especially the diffusion coefficient, which depends essentially on temperature and viscosity rather than the nature of chemical bond. Although the rate constant can still be written in an Arrhenius-like form, the exponential term is no longer dominant; instead, the temperature dependence is expressed in the prefactor. In short, diffusion-limited reactions represent the opposite extreme of the classical Arrhenius picture: rather than being barrier-limited, the kinetics are governed by transport, more appropriately described by the Stokes-Einstein relation (To be discussed in the next chapter).

 

Review Questions:

1. What are the key idealizations underlying the Arrhenius equation for reaction rates?

2. How would you interpret Feynman’s statement that “activation energy A* is quite different from the energy W”?

3. To what extent do you agree with Feynman’s claim that the reaction-rate equation is guaranteed to hold regardless of the reaction mechanism? How would you evaluate the physical limitations of this claim?”?

 

Key Takeaways:

1. Temperature is an Exponential Accelerator

The Arrhenius equation is more than a formula, it’s a statement about the power of the Boltzmann Distribution. Because temperature is in the denominator of a negative exponent, even a small increase in temperature leads to a large, non-linear rise in the number of molecules with enough energy to react. This explains why some food spoils wihin days at room temperature, yet can last for months in a freezer.

 

2. The "Constant" Prefactor is an approximation

Although Arrhenius originally treated the prefactor A as a constant, it is in fact a temperature-dependent variable. Even Feynman’s refinement only partially capture this dependence. Within the Transition State Theory, the prefactor accounts for the reaction mechanism: how fast molecules move, how often they collide, and molecular orientations. As temperature varies, these factors change, often producing a slight curvature in Arrhenius plots.

 

3. Catalysis: Work Smarter, Not Harder

A catalyst does not increase the energy of molecules; instead, it provides an alternative reaction pathway with a lower activation barrier. By effectively lowering the "fence," it allows the existing thermal energy distribution to drive reactions that would require higher temperatures. In this sense, catalysis reveals a limitation of the simple Arrhenius picture: reaction-rate rates are governed not only by energy barriers but also by the specific mechanisms through which those barriers are traversed.

 

The Moral of the Lesson: Vitamin C and Cancer

In the following letter, Linus Pauling advises Richard Feynman to immediately begin a high daily intake of vitamin C (20 g or more) for his abdominal malignancy, while avoiding chemotherapy and sugar, emphasizing that vitamin C works by boosting immune mechanisms and should not be stopped once started.

 

LINUS PAULING TO RICHARD P. FEYNMAN, JUNE 28, 1978 

Dear Dick: 

I have learned from Linda that you have had a malignant tumor removed. These abdominal malignancies are serious. The 5-year survival fraction is rather small. Chemotherapy has little value—in Britain it is rarely used for these cancers. 

I think that the best thing to do is to begin immediately a high intake of vitamin C—20 g. per day or more. I am corresponding with a man who had extensive abdominal cancer, and who took 60 g. per day for 3 months. He is now much better, and is down to 35 g. per day. 

Enclosed are a couple of papers, with references to more. Linda can tell you where to get pure ascorbic acid and sodium ascorbate and how to take it. 

Vitamin C works largely by potentiating the body’s immune mechanisms. The cytotoxic drugs destroy them, and probably decrease the effectiveness of the vitamin C. On the other hand, immune stimulants, such as BCG may well be compatible with vitamin C. 

It is very important not to stop the intake of vitamin C, once you have started. 

We have another paper in press in PNAS. Also, Morishita and Murata in Japan have got similar results. 

 

Best wishes, 

Linus 

P.S. Also no sugar, little meat, lots of fresh vegetables, vegetable juice & fruit juice. 

 

A Critical Post-Script:

While Pauling’s advocacy of vitamin C was influential, modern oncology and pharmacokinetics point to three critical caveats.

 

1. Vitamin C as a pro-oxidant:

Pauling explains that vitamin C works largely by potentiating the body’s immune mechanisms. In general, Vitamin C (ascorbic acid) acts primarily as a potent antioxidant, protecting cells from free radicals and boosting immune function. However, at sufficiently high doses (concentrations) and in the presence of catalysts or free metal ions (such as iron or copper), it can act as a pro-oxidant, generating reactive oxygen species. Specifically, through the Fenton reaction, it can produce hydrogen peroxide, which may exert cytotoxic effects on cancer cells under certain conditions.

 

2. Intravenous (IV) versus oral administration:

An oral intake of 20 grams of vitamin C per day or more may be unsafe due to gastrointestinal side effects. In contrast, many patients have received intravenous (IV) administration of vitamin C at an average dose of 0.5 g/kg without significant side effects. Clinical studies, such as those by Mikirova and colleagues (2007), have explored IV doses in the range of 5 to 25 grams, achieving pharmacological concentrations that may enable pro-oxidant activity.

 

3. The Sugar Paradox:

Pauling’s suggestion to consume large amounts of fruit juice introduces a potential complication. Cancer cells are “glucose-hungry”—a phenomenon often associated with the Warburg Effect. Importantly, fruit juice can increase blood glucose, which provides fuel for cancer cells. Thus, excessive sugar intake could counteract efforts to deliver vitamin C effectively for cancer treatment.

 

Summary

Pauling was correct about the potential of vitamin C to act as a cytotoxic agent. However, his proposed method—oral intake alongside fruit juices—may not achieve the specific concentrations and chemical environment required to trigger significant pro-oxidant activity within Feynman’s tumor.

 

p.s. Heat destroys vitamin C; cold preserves it. This is why prolonged cooking or high-heat processing significantly lowers the vitamin C content in foods.

 

Case Study: Manuka Honey and H. pylori

Manuka honey is more than a natural sweetener; it has been studied for its antibacterial activity, including against Helicobacter pylori, a microbe associated with stomach ulcers and a leading risk factor for gastric cancer. Its effectiveness, however, depends on how temperature influences its chemical components.

 

1. The Chemical "Ripening" (DHA to MGO)

Manuka honey’s antibacterial strength is linked to its Non-Peroxide Activity (NPA), primarily Methylglyoxal (MGO). Fresh nectar contains relatively lesser MGO; instead, it forms gradually from the precursor Dihydroxyacetone (DHA) during storage. This conversion is a temperature-dependent process: at low temperatures, the reaction proceeds very slowly, while excessive thermal energy can accelerate competing reactions that degrade MGO. Thus, an optimal temperature range is required to balance formation and stability.

 

2. The Thermal Breaking Point

Temperature acts as a limiting factor for Manuka honey’s medicinal properties. While MGO is relatively stable, other components—particularly enzymes such as Glucose oxidase—are heat-sensitive proteins. Elevated temperature can denature these enzymes, disrupting their structure and reducing their ability to generate additional antibacterial agents. In this sense, excessive thermal energy can diminish the medicinal properties of the honey.

 

3. Practical Application:

To preserve the medicinal properties of Manuka honey—whether for soothing the throat or supporting digestive health—it is advisable to avoid high temperatures. Adding honey to your tea or water at a “drinkable" temperature (roughly 40–50 °C) helps maintain MGO levels and protects heat-sensitive enzymes. In addition to its chemical activity, honey also exerts osmotic effects: it is known to draw water out of microbial cells, causing them to dehydrate and die. Together, these mechanisms show how both chemical reactions and temperature control determine whether Manuka honey functions as therapeutic agent or simply as a sweetener.

 

References:

Feynman, R. P. (2005). Perfectly reasonable deviations from the Beaten track: The letters of Richard P. Feynman (M. Feynman, ed.). New York: Basic Books.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Logan, S. R. (1982). The origin and status of the Arrhenius equation. Journal of Chemical Education, 59(4), 279-281.

Mikirova, N., Casciari, J., Riordan, N., & Hunninghake, R. (2013). Clinical experience with intravenous administration of ascorbic acid: achievable levels in blood for different states of inflammation and disease in cancer patients. Journal of translational medicine, 11(1), 191.