Turnover Number and Turnover Frequency

Introduction

Catalysts are valuable because one active site can transform many molecules. Turnover number (TON) counts accumulated product per site; turnover frequency (TOF) measures how quickly those turnovers occur. The arithmetic is simple, but the denominator is not. Total metal atoms, surface atoms, probe-accessible sites and true operating sites can be very different populations. A fair comparison states exactly which count was used.

Core explanation

For a one-product, one-site cycle, TON = moles of desired product formed divided by moles of counted active sites over the stated run. TOF = moles of desired product per unit time divided by moles of counted active sites, with units such as s⁻¹ or h⁻¹. If rate changes over time, a quoted TOF must specify whether it is initial, average over an interval or measured at a particular time. TON is cumulative and can continue rising even as instantaneous TOF declines due to deactivation.

In a homogeneous catalyst, the concentration of molecular metal is often known, but not every molecule may be active; precatalyst activation, resting reservoirs and decomposition matter. In a heterogeneous catalyst, many metal atoms are buried, and accessible surface atoms may not all be the relevant motif. Chemisorption, electrochemical titration, spectroscopy and microscopy can estimate different site populations. An ACS discussion of nanoparticle-geometry effects on TOF shows why dispersion and geometry introduce uncertainty. Report a metric as apparent TOF when the site count is a proxy.

Use complementary normalisations. Product per gram catalyst indicates mass productivity; product per geometric electrode area indicates device output; product per metal mass can illuminate precious-metal utilisation. These are useful even when true active-site counts are uncertain. They should not be relabelled as intrinsic TOF. A catalyst with exceptional per-site TOF but very few sites per reactor volume can have low total throughput. Conversely, high per-gram activity can result from high site density rather than faster sites.

TON claims require product verification and time boundaries. A large TON calculated from a tiny, uncertain catalyst loading may be inflated if reaction proceeds from a contaminant, support or sacrificial reagent. Blank tests and atom balances are essential. Regeneration complicates interpretation: report whether repeated cycles reuse the same sites or include replacement metal. For electrochemical studies, electron stoichiometry and Faradaic efficiency are needed to convert charge into product moles.

Step-by-step reasoning

1. Define the desired product and confirm its measured amount. 2. Choose and disclose a site-counting method with uncertainty. 3. Divide cumulative product by sites for TON and product rate by sites for TOF. 4. State time, operating conditions and whether rate is initial or average. 5. Report mass, area and stability metrics alongside site-normalised values.

Visual explanation

Draw a product-versus-time curve that starts steep and flattens as catalyst deactivates. The curve's height divided by site count is TON; the tangent slope divided by site count is instantaneous TOF. A separate bar indicates site density per gram. The diagram distinguishes cumulative output, current speed and amount of machinery.

Real-world analogy

An employee's completed tasks over a year resemble cumulative turnover number, while tasks per hour resemble turnover frequency. A team with few fast workers may complete less total work than a larger team of slower workers. The analogy breaks down if “employee count” includes people who cannot do the task; that is precisely the active-site counting problem.

Real-world example

Two catalysts produce the same amount of desired product in an hour. One uses a small amount of expensive metal with high apparent TOF; the other uses a larger amount of cheap metal. Process selection also considers site lifetime, catalyst separation and supply cost. A single TOF ranking cannot decide which is better for the plant.

Why?

Why can a catalyst have high TON but low final TOF? It may have operated rapidly for a long period and then deactivated. The cumulative product remains in the TON numerator, while the instantaneous rate at the end is small. Reporting both tells more about durability than either alone.

Common misconception

“TON and TOF are synonyms” confuses accumulated output with rate. “All metal atoms are active sites” is often false. “A large TOF guarantees high productivity” ignores site density. “A rate divided by total metal is always a true intrinsic TOF” is better described as apparent when the active fraction is unknown.

Worked example

A catalyst sample contains 2.0 µmol total metal, but a site probe counts 0.50 µmol accessible sites. During the first minute it forms 1.5 µmol desired product; over a 60-minute run it forms 30 µmol. Initial TOF per probe-counted site is 1.5/0.50 = 3.0 min⁻¹, while average TOF is 30/(0.50 × 60) = 1.0 min⁻¹. TON over the run is 30/0.50 = 60. If total metal is used instead, apparent initial TOF is 0.75 min⁻¹ and TON is 15. Both calculations are arithmetically correct but describe different denominators. The lower average TOF suggests the rate may have declined or started slowly; a time course resolves which.

Quick check

1. What is the unit difference between TON and TOF? Answer: TON is a dimensionless product-per-site count; TOF includes inverse time, such as min⁻¹.

Exam focus

Calculate TON and both initial and average TOF with units and explicit site denominator. Explain why site-counting uncertainty affects intrinsic claims. Connect TOF, site density and total reactor productivity without conflating them.

Advanced insight

For multi-atom active ensembles, dividing by individual metal atoms can misrepresent a cycle performed by a pair or interface. A valid site count should match the proposed mechanism. If that motif is not measurable, report robust process metrics and a range of apparent TOFs under plausible counting assumptions rather than false precision.

Summary

TON measures cumulative turnovers per counted site and TOF measures their rate. The site denominator, time window, product verification and catalyst stability determine whether the numbers support an intrinsic or only an apparent performance comparison.

Practice questions

1. A catalyst forms 100 mmol product using 2 mmol counted sites. What is TON? Answer: 100/2 = 50. 2. If those 100 mmol form uniformly over 10 hours, what is average TOF? Answer: 100/(2 × 10) = 5 h⁻¹. 3. Why might rate per gram rise while TOF per site falls? Answer: More sites per gram can outweigh slower turnover at each counted site. 4. What should be reported when the true active-site count is uncertain? Answer: The proxy and its uncertainty, an apparent TOF, and complementary mass or area productivity.