Ventus W636 Weather Station with outdoor sensor

If your weather station displays barometric pressure, temperature and relative humidity like the one pictured above, you can calculate the amount of water vapor in the air expressed either as grams of water vapor per kilogram of dry air (known as Mixing Ratio) or as grams of water vapor per kilogram of vapor-containing air (known as Specific Humidity). The two measures are very similar for cooler air; differences only become apparent for warmer air.

Formulas for calculating Mixing Ratio and Specific Humidity

In the formulas below, barometric pressure P is expressed in hectopascals (hPa), temperature T is expressed in degrees Celsius, relative humidity rh is expressed in %, and e is the Euler number 2.71828 [raised to the power of the contents of the square brackets]:

The decimal separator is shown as a full point (.) In developing these formulas, the following textbook was consulted: Atmospheric Thermodynamics by Grant W. Petty, Sundog Publishing, Madison Wisconsin. ISBN-10: 0-9729033-2-1

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P Mander, June 2020

Photo acknowledgement:

For citizens like me, old enough and perhaps wise enough to take a cautious view of mingling again with one’s fellow man after a period of lockdown in a time of pandemic, it’s natural to wonder about the risk of contact with infectious individuals as one re-establishes one’s daily routines. I thought this over and figured that if I knew the proportion of a population carrying infection, I could use probability theory to quantify the risk.

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How many random contacts are required to have a 50% chance of at least one contact with an infectious individual?

Knowing the answer to this question provides a way of relating a level of human interaction to a level of risk. And this little formula supplies the answer:

where n is the number of random contacts and q is calculated as follows:

N – is the number of people in the local population
a – is the number of people carrying infection

Example 1

You live in a city of 10 million people. Diagnostic testing projects that around 125,000 are carrying infection.

Assuming uniform distribution of infection in the population, probability theory indicates that 55 random contacts with people in your city are required to have a 50% chance of at least one contact with an infectious individual.

Example 2

You are a storeworker in a town where it is estimated that 0.25% of the population is carrying infection. Assuming the same percentage applies to customers visiting your store:

Probability theory indicates that 277 random contacts with people in your store are required to have a 50% chance of at least one contact with an infectious individual.

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Background to the formula

Calculating the probability of one of two possible outcomes in n independent trials is an application of probability theory known as a binomial experiment where p is the probability of one outcome (success) and q is the probability of the other (failure).

A simple formula exists for calculating the probability of at least one successful outcome in n trials

In the present context, n is the number of random contacts and q is the probability of an individual chosen at random not carrying infection. Setting P(X≥1) = ½ and solving for n

By the change of base rule logab = logxb/logxa

where the log can be expressed to any base.

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The value of testing

Diagnostic testing is an important tool for controlling the spread of infection as the graph below helps to illustrate. If the level of infection in the population rises above 1 percent, the number of random contacts necessary to promote spread is not large. But below 1 percent the picture quickly changes as n climbs, with the likelihood of transmission diminishing accordingly.

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P Mander, May 2020

Active cases are those who are currently infected with coronavirus. The number of active cases reflects the constantly changing size of the epidemic in your location, as new cases continuously add to the number while recoveries and deaths subtract from it.

The number of active cases is never static. The figure rises if the number of new cases is greater than the number of recoveries and deaths in a given period of time (N > R + D).

Conversely the figure falls if the number of new cases is less than the number of recoveries and deaths in a given period of time (N < R + D).

How to find the current number of Active Cases in your location

1. Go to The Johns Hopkins Coronavirus Resource Center (Give the page time to load)
2. Below the map, toggle Cumulative Confirmed Cases to the right to display Active Cases.
3. Click on the orange circle denoting your location on the world map.
4. A data box will appear showing the current number of active cases

Keeping a note of active cases will enable you to chart the growth/decline of the unfolding coronavirus epidemic in your location over the coming days and weeks.

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UPDATE 10 MAY 2020
I have charted active cases for several European countries over the last six weeks. Assuming the data from Johns Hopkins is trustworthy, it shows some remarkable differences and some striking similarities in the national dynamics of the coronavirus pandemic.

The above diagram is taken from a paper[ref] published in 1995 by the Hungarian biochemist Gaspar Banfalvi in which he introduced circular graphs to map energy changes in metabolic cycles – in this particular case showing the relationship between the Gibbs free energy (dotted line) and the average carbon oxidation state (solid line) of intermediates in the Krebs cycle.

This post adopts Banfalvi’s innovative approach in order to further explore the redox behavior of Krebs cycle intermediates. But before we start, a brief section on calculating average carbon oxidation states.

Calculating Average Carbon Oxidation States

The overall oxidation state of a molecule is its charge magnitude taking sign into account:
Charge magnitude = sum of carbon oxidation states + sum of oxidation states of other atoms

The sum of oxidation states of other atoms is computed by assigning a value to each of these atoms in the molecule as appropriate e.g. H = +1, O = -2, N = -3 in amino group [-NH2], P = +5 in phosphate group [-O-PO3]^2-, and summing them. Subtracting this sum from the charge magnitude gives the sum of carbon oxidation states. Dividing this by the number of carbon atoms in the molecule gives the average carbon oxidation state.

Example 1: Citrate

Charge magnitude = -3
Sum of oxidation states of other atoms Hx5, Ox7 = +5 -14 = -9
Sum of carbon oxidation states = -3 – (-9) = +6
Number of carbon atoms in molecule = 6
Average carbon oxidation state = +6/6 = +1

Example 2: Urea

Charge magnitude = 0
Sum of oxidation states of other atoms Hx4, Ox1 , Nx2 = +4 -2 -6 = -4
Sum of carbon oxidation states = 0 – (-4) = +4
Number of carbon atoms in molecule = 1
Carbon oxidation state = +4/1 = +4

Example 3: 1,3-diphosphoglycerate

Charge magnitude = -4
Sum of oxidation states of other atoms Hx4, Ox10, Px2 = +4 – 20 + 10 = -6
Sum of carbon oxidation states = -4 – (-6) = +2
Number of carbon atoms in molecule = 3
Average carbon oxidation state = +2/3

Useful links for determining oxidation states
– Sulfur and Phosphorus
– Nitrogen

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Carbon Oxidation States and Krebs Cycle

OK so here’s a different way of illustrating the Krebs Cycle. This diagram shows the cycle in terms of the number of carbon atoms in the input, output and cycle intermediate molecules, together with their average oxidation states. The numbers inscribed in blue show the changes in oxidation state of the intermediates taking place around the cycle.

The diagram reveals a feature that is not easily discernible in conventional depictions of Krebs Cycle – namely that the cycle intermediates undergo a progressive oxidation state reduction of -1 from oxaloacetate (C4,+1½) to succinyl-CoA (C4,+½), followed by a progressive oxidation state increase of +1 from succinyl-CoA back to oxaloacetate. Since a lower carbon oxidation state reflects more energy residing in chemical bonds, these changes indicate that cycle intermediates store energy in the first half of the cycle and liberate it in the second half.

Within this overall movement of energy lies a more detailed redox picture. Consider the steps in the first half where carbon dioxide is released. Decarboxylation of isocitrate per se (C6H5O7^3-) → [C5H5O5^3-] equates to a reduction of carbon oxidation state from +1 to +2/5 in the theoretical residue shown in [ ]. But isocitrate also undergoes oxidation, which we can notate as the theoretical residue transferring charge and a hydrogen atom to the NAD+ cofactor [C5H5O5^3-] → (C5H4O5^2-) resulting in a lesser overall reduction of carbon oxidation state from +1 to +4/5. Isocitrate thus combines both reductive and oxidative roles.

Similarly, decarboxylation of α-ketoglutarate per se (C5H4O5^2-) → [C4H4O3^2-] equates to a reduction of carbon oxidation state from +4/5 to 0 in the theoretical residue. But α-ketoglutarate also undergoes two oxidative processes, which we can notate as the theoretical residue adding an -SCoA group to the molecule and transferring charge to the NAD+* cofactor [C4H4O3SCoA^2-] → (C4H4O3S-CoA^1-) resulting in a lesser overall reduction of carbon oxidation state from +4/5 to +½. Thus α-ketoglutarate also combines both reductive and oxidative roles.

Two intermediates combine reductive and oxidative roles in the second half of the cycle. Succinyl-CoA undergoes two reductive processes: the deesterification of succinyl-CoA per se and receiving charge from a phosphate ion (C4H4O3S-CoA^1-) → [C4H4O3^2-] equates to a reduction of carbon oxidation state from +½ to +0 in the theoretical residue. Succinyl-CoA also undergoes oxidation, which we can notate as the theoretical residue receiving an oxygen atom from the same phosphate ion [C4H4O3^2-] → (C4H4O4^2-) resulting in an increase of carbon oxidation state from 0 to +½. Thus succinyl-CoA exhibits equal reductive and oxidative capacity resulting in no change in the average oxidation state of the next intermediate succinate.

Finally oxaloacetate undergoes two reductive processes: the addition of an acetyl group and receiving charge from the hydrogen of a water molecule (C4H2O5^2-) → [C6H5O6^3-] equates to a reduction of carbon oxidation state from +1½ to +2/3 in the theoretical residue. Oxaloacetate also undergoes oxidation, which we can notate as the theoretical residue receiving an oxygen atom from the same water molecule [C6H5O6^3-] → (C6H5O7^3-) resulting in a lesser overall reduction of carbon oxidation state from +1½ to +1.

*note that the hydrogen atom transferred to NADH comes from CoASH not α-ketoglutarate

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Krebs Cycle net equation restricted to cycle intermediates

There are plenty of sources out here on the web which will give you the Net Equation of Krebs Cycle showing every single substance involved, and there are lots of them. What I want to do here is write a simple net equation which shows only the inputs and outputs that are directly incorporated into or sourced from the structure of the cycle intermediate molecules themselves. The equation is a concise statement of what much of the foregoing has been trying to say – that the cycle intermediates have both oxidative and reductive roles:

Question. In overall terms two molecules of water enter the Krebs cycle, so why is there only one molecule of water shown on the input side? The answer is that while the water molecule which hydrates fumarate to malate is fully incorporated into the cycle intermediate structure, the water molecule involved in the conversion of oxaloacetate to citrate is not. Only the oxygen atom of this water molecule is incorporated into the citrate structure; one hydrogen atom is used to regenerate CoA-SH from acetyl-CoA while the other donates charge (C4^2- → C6^3-) and is oxidized to H+.

One of the two input oxygen atoms shown in the equation is now accounted for, but what about the other? The answer reveals a fascinating detail of the Krebs Cycle: it comes from the phosphate ion HPO4^2- involved in GTP/ATP formation associated with the conversion of succinyl-CoA to succinate. A PO3^1- moiety is used in the conversion of GDP to GTP while the hydrogen atom regenerates CoA-SH from succinyl-CoA, leaving a single oxygen which is incorporated into the succinate molecule together with a transfer of charge (C4^1- → C4^2-).

In summary we can say that at each turn of the Krebs Cycle, an acetyl group, a water molecule and two oxygen atoms – one from another water molecule and one from a phosphate ion – are incorporated into the structures of the cycle intermediates to facilitate the oxidative formation of carbon dioxide for release and hydrogen-reduced cofactors for onward transmission to the electron transport chain and ATP production.

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Krebs Cycle Thermodynamics

All cells that generate ATP by metabolizing acetyl-CoA to carbon dioxide make use of the Krebs cycle, which drives energy production in a way that shares characteristics with other more familiar engines operating in cycles. Like them, the Krebs cycle executes energy conversion and in so doing produces heat.

As with all energy conversion processes, metabolism is not particularly efficient and only around 40% of the available energy is converted to ATP; the rest is dissipated as heat. There are two aspects to this heat production that are worthy of note. The first is the effect that heat has on reaction kinetics; even a modest increase in temperature over that of the surroundings can speed up chemical processes significantly. Warmer organisms react faster, which is vital for survival.

The second aspect is the scale of heat release; the Gibbs free energy changes in the Krebs cycle reactions are very modest so that heat production and conduction of heat out of the cell can occur while maintaining thermal stability. Even in situations of high ATP demand, the mitochondrion won’t get fried.

Step Reaction ΔG’° kJ/mol Keq
1 Acetyl-CoA + Oxaloacetate → Citrate -32.2 2.65 x 10^5
2 Citrate → Isocitrate 13 6.47 x 10^-3
3 Isocitrate → α-ketoglutarate -8.4 2.60 x 10^1
4 α-ketoglutarate → Succinyl-CoA -33.5 4.38 x 10^5
5 Succinyl-CoA → Succinate -2.9 3.08
6 Succinate → Fumarate 0 1
7 Fumarate → Malate -3.8 4.36
8 Malate → Oxaloacetate 29.7 9.96 x 10^-6

The equilibrium constants of the Krebs cycle reactions exhibit a mixture of reactant-favored and product-favored processes. The second half of the cycle (Steps 5-8) is essentially reversible, with Steps 1 and 4 providing the product-favored momentum. It should be noted that Steps 3 and 4 involve the release of carbon dioxide which continuously diffuses out of the cell to neighborhoods of lower concentration, thereby lowering the reaction quotient Q and enhancing product favorability.

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Structure and Nomenclature of Krebs Cycle Intermediates

The sequence of intermediates in the Krebs cycle was worked out long ago, and part of that history is preserved in their esoteric names. For example succinic acid used to be obtained from amber by distillation, which is why it is named after the Latin word for amber – succinum. All very interesting, but it doesn’t help to visualize what the molecule looks like. A better way is to learn the carbon chain shapes of the cycle intermediates, and the scientific names that describe them. The advantage of doing this is that one reinforces the other; once you have memorized these scientific names you can draw molecules directly from them.

Try this out for yourself. You know citric acid is a C6 molecule; learn to draw it with the longest carbon chain (shown in red) around three sides of a rectangle with the three -COOH groups pointing out to one side like this. Practice drawing this until you can see it in your sleep.

Now you’ve learned it, you can name it. The longest carbon chain is C5 so the scientific name is based on pentane (C5H12). The C5 chain has -COOH groups at each end, so the core molecule is pentanedioic acid, which has a carboxyl and hydroxyl group on the third carbon atom in the chain. Adding these gives 3-carboxy-3-hydroxypentanedioic acid. That’s the scientific name for citric acid.

If you can learn this name, you can draw citric acid from memory. Awesome. And it’s an easy progression from there to learn the structure of isocitric acid

The only difference is that the hydroxyl group drops down to the second carbon atom in the C5 chain. So the scientific name simply changes to 3-carboxy-2-hydroxypentanedioic acid. The molecule has one carboxyl group on a side chain (the one shown in black) and it’s this one which is released as carbon dioxide along with a hydrogen atom to reduce NAD+, the molecule rearranging itself so that an oxo- group replaces the hydroxyl group on C2

The pentanedioic acid core molecule now has only one substituent; its scientific name is 2-oxopentanedioic acid. Once learned, it’s easy to draw this C5 molecule from memory which is more than can be said for α-ketoglutaric acid. The lower carboxyl group now departs to form carbon dioxide along with a hydrogen atom to reduce NAD+, with a Coenzyme A group taking its place. The resulting molecule is easier to visualize with its C4 chain drawn around three sides of a square like this

Don’t bother with the scientific name of this one, it’s too long. Just call it succinyl-CoA. All the remaining steps of the Krebs cycle retain the C4 chain. Following deesterification the next intermediate, succinic acid, is the simplest structure of all the Krebs cycle intermediates

The scientific name is that of the core molecule, butanedioic acid. Learn this name and you can easily draw succinic acid from memory. The next steps are the removal of two hydrogen atoms from carbon atoms 2 and 3 to form fumaric acid (not shown) followed by hydration to malic acid

whose scientific name is 2-hydroxybutanedioic acid, before a final dehydrogenation to oxaloacetic acid

whose structure is much easier to draw if you can learn its scientific name, 2-oxobutanedioic acid. Here’s a summary of the Krebs cycle intermediate names, ancient and modern:

Citric acid 3-carboxy-3-hydroxypentanedioic acid
Isocitric acid 3-carboxy-2-hydroxypentanedioic acid
α-ketoglutaric acid 2-oxopentanedioic acid
Succinic acid butanedioic acid
Fumaric acid (E)-butenedioic acid
Malic acid 2-hydroxybutanedioic acid
Oxaloacetic acid 2-oxobutanedioic acid

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Suggested further reading

Gaspar Banfalvi, Constructing Energy Maps of Metabolic Cycles (1995)
The pioneering paper whose graphical idea is adopted in this blogpost. Balfalvi’s paper was published in Biochemical Education, now known as Biochemistry and Molecular Biology Education.

S.V. Eswaran, A New Look at the Citric Acid Molecule (1976)
This paper tells the story of Krebs’ belief that if radiolabeled CO2 was assimilated then both the CH2COOH groups in citric acid would be labeled since citric acid is a symmetrical molecule. He was wrong and this paper explains why. It’s worth reading about what was briefly known as the “Ogston Effect”* and the influence A.G. Ogston’s thinking about substrates and planar enzyme surfaces had on the subsequent development of prochirality.

*named after Alexander George Ogston (1911-1996), a British physical chemist.

Nazaret, C et al., Mitochondrial energetic metabolism: A simplified model of TCA cycle with ATP production (2009)
If you’re into modeling the chemical kinetics of metabolic systems on the basis of the Mass Action Law, you might like this paper published in the Journal of Theoretical Biology. The authors describe their model as “very simple and reduced” but even so it’s knee-deep in differential equations. Enjoy.

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P Mander February 2020

Credit: Carbon Engineering

Direct Air Capture (DAC) is the somewhat misleading term that has come into use for chemical fixation processes designed to extract carbon dioxide from the atmosphere. Chemical fixation of atmospheric CO2 is nothing new – organisms capable of photosynthesis are thought to have evolved billions of years ago, while limestone formation from CO2 taken up by the oceans has been occurring for hundreds of millions of years. But mankind hasn’t had a go at it until recently.

The driving force behind the development of DAC is the conviction based on the best available science that current (and rising) atmospheric CO2 levels constitute an existential threat due to global warming and climate change, as well as ocean acidification and marine ecosystem disruption. There is a desire for technologies to augment both naturally-occurring fixation processes and emission-reducing initiatives in order to accelerate the process of bringing the rise in atmospheric CO2 levels to a halt and subsequently to achieve drawdown.

To put this task in perspective, let’s put a few facts and figures into it. Carbon dioxide is a thermodynamically stable gas and the densest component of air, 1.977 kgm-3 at STP compared with 1.293 kgm-3 for air. This does not mean however that CO2 sinks in air and accumulates in the lower atmosphere. Like any gas, CO2 exhibits the phenomenon of diffusion, which is the tendency of a substance to spread uniformly throughout the space available to it. And this is where the challenge of DAC lies. Although there is a lot of CO2 – ca. 3210 gigatonnes in 2018 (ref) – up in the air, the atmosphere is a big place and the concentration of CO2 (currently around 410 ppm) is tiny in the context of extraction. In other words a large amount of air needs to be processed for a modest rate of capture. It is therefore relevant to consider the processes of capture and conversion from a carbon oxidation state perspective, which has a direct bearing on process thermodynamics and economics.

Note: If it proves possible to reduce the partial pressure of CO2 in the atmosphere through DAC, oceanic release of dissolved CO2 should take place according to Henry’s Law to restore equilibrium between oceanic and atmospheric partial pressures. If this were the case, the removal of CO2 from the atmosphere could not be achieved separately from removing CO2 from the vastly bigger ocean reservoir, and the task of stopping the rise of the Keeling curve could not be separated from the rise in oceanic acidification. This would add a whole new dimension to the task of mitigation.

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Oxidation states and conversion energy

Carbon can exist in nine oxidation states, which are conveniently if somewhat abstractly represented by dimensionless numbers ranging from +4 (most oxidized state) to –4 (most reduced state). The lower the number, the more energy is present in carbon’s chemical bonds. So the process of reducing a carbon compound requires energy, the amount of which increases with the degree of reduction.

As the table shows, carbon in carbon dioxide has the highest oxidation state (+4). In terms of conversion energy requirement, it is self-evident that the least demanding option is to capture CO2 without reduction for subsequent release, containment and onward supply to industry. Capture technologies such as aqueous alkanolamine absorption are well-known, but in terms of product use it should be noted that carbon dioxide applications such as synthetic fuel feedstock and carbonated beverages will lead to atmospheric re-release and therefore cannot contribute to CO2 drawdown.

Note that Mother Nature makes use of CO2 capture without reduction in the process of oceanic uptake. The oxidation state of carbon in the carbonate ion that eventually becomes chemically fixed in limestone is still +4.

Turning to carbon dioxide capture with conversion, the oxidation state table shows that there are eight levels of reduction, each possessing its own product outcomes and synthetic opportunities. The formulas shown in the table indicate two strands in the reduction process – the addition of hydrogen and the removal of oxygen (for simplicity I have restricted the examples to combinations solely of C, H and O).

If we ask ourselves the question – What conversion process can we apply such that the energy requirement for reduction of CO2 is minimized, the answer suggests itself: a process that reduces the carbon oxidation state by just one unit, from +4 to +3.

Simple arithmetic suggests this can be achieved by combining two atoms of hydrogen with two molecules of carbon dioxide. And this is borne out in practice, although as we shall see the reducing agent need not necessarily be hydrogen.

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Converting carbon dioxide to oxalic acid

Consider a DAC process in which two moles of carbon dioxide are captured from the atmosphere and converted into one mole of oxalic acid (involving the addition of a mole of hydrogen). Now think about density change. Carbon dioxide is a gas, two moles of which occupy 44.8 liters at STP. Oxalic acid is by contrast a solid, a mole of which occupies 0.0474 liters. So in this conversion process the carbon atoms get packed into a space which is 946 times smaller. Woo. And because oxalic acid is a solid there is no requirement for pressurized containment or its associated cost.

One method by which this capture and conversion process can be realized has been published by researchers at Louisiana State University (ref). The chemical capture unit is built out of four pyridyltriazole chelating units linked by two meta-xylylene groups; complexation with CuCl2 gives a dimeric macrocycle which following cation reduction to Cu+ by sodium ascorbate is able to selectively capture and convert two carbon dioxide molecules into an oxalate ion as shown here

The oxalate ion is released as oxalic acid when treated with dilute mineral acid, regenerating the original copper complex. The reaction conditions are mild, reflecting the minimal reduction of carbon oxidation state from +4 to +3.

As I understand it the essential reaction sequence is as follows:

The oxidation of the ascorbate ion to a radical cation would appear to provide the thermodynamic impulse for this electron transfer sequence. Obviously there are other anions involved here, but the paper didn’t detail them so nor can I. What is evident however is that the energy demands of this process are modest, as one would anticipate for a unit reduction in oxidation state.

This paper was published 5 years ago as exploratory work. The kinetics in particular needed improvement and no doubt this has been addressed in further studies. My point in highlighting this work is to show the advanced level of innovation in facilitating oxidation state reductions by which atmospheric CO2 can be converted to carbon compounds of significant synthetic potential.

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Electrochemical reduction

With an enthalpy of formation ΔHf = -393 kJmol-1, carbon dioxide is a very thermodynamically stable compound. This explains why electrochemical reduction routes using the power of cathodic electrons have been sought to produce a CO2 anion which can enter into protic or aprotic reaction yielding formate or oxalate ions.

In Europe the acronym-intensive SPIRE (Sustainable Process Industry through Resource and Energy Efficiency) Association representing innovative process industries has a project running called OCEAN, which stands for Oxalic acid from CO2 using Electrochemistry At demonstratioN scale. OCEAN aims to develop an electrochemical process for producing high-value C2 chemicals from carbon dioxide via the following sequence:

1) reduce carbon dioxide (C1,+4) to formate (C1,+2)
2) dimerize formate (C1,+2) to oxalate (C2,+3)
3) protonate oxalate (C2,+3) to oxalic acid (C2,+3)
4) reduce oxalic acid (C2,+3) to glycolic acid (C2,+1)

The oxidation state sequence by which oxalic acid (4→2→3) and glycolic acid (4→2→3→1) are obtained from CO2 looks a bit lossy from an energy efficiency perspective, but this is just my impression. The project is running from October 2017 for four years and is EC-financed to the tune of €5.5 m.

Details of the OCEAN project can be found on these links:

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P Mander July 2019