Physics of Energy

Punchline: Oil is a 2-billion-year-old technology. Coal is a 400-million-year-old technology. And they're both about sunlight that decomposers (or anoxia) failed to burn.

Ok Physics of energy question: A quote from The Physics of Climate Change "Respiration is controlled burning. Both breathing and buring combined oxygen with organic compounds to form CO2 Living systems are able to extrace ~30x the energy they would generate by photosynthesis". Can I estimate that from an order of magnitude estimate first principles ?

⏺ Yes — this works out nicely, and it's a great teaching example. The claim is really a ratio of two efficiencies, and you can estimate each from first principles.

Photosynthesis (sunlight → glucose): ~1–2%

Quantum bookkeeping first:

• Need ~8 photons to fix 1 CO₂ (from the Z-scheme — 2 photosystems, 2 photons each, × 2 for NADPH + ATP rebalance)
• Photon at chlorophyll peak (~680 nm): ~1.8 eV
• Energy stored per CO₂ in glucose: ~5 eV (from ΔH_combustion of glucose, 2870 kJ/mol ÷ 6 ≈ 30 eV/glucose ÷ 6 carbons)
• Quantum efficiency: 5 / (8 × 1.8) ≈ 35%

But that's for absorbed photons at the right wavelength. Real-world losses:

• Only ~45% of solar spectrum is PAR (400–700 nm) → ×0.45
• Photorespiration, saturation, reflection, dark respiration → another ×0.1–0.2
• Net: ~1–3% for real plants, up to ~6% for C4 crops

Respiration (glucose → ATP): ~40–50%

• Aerobic respiration: ~30 ATP per glucose
• ATP hydrolysis ≈ 50 kJ/mol under cellular conditions
• Captured work: 30 × 50 = 1500 kJ/mol
• Out of 2870 kJ/mol available → ~52%

Ratio: 40% / 1.5% ≈ 25–30× ✓

So the "30×" is really saying: a plant wastes most of the sunlight it intercepts, but once the energy is locked in glucose, metabolism extracts it with near-heat-engine efficiency. The bottleneck is photon capture, not chemistry.

Nice Physics-of-Energy nugget: this is also why animals can run at ~100 W/m² of body area while plants only produce ~2 W/m² of leaf area — same factor of ~30–50, same reason.

How can I see "Need ~8 photons to fix 1 CO₂ (from the Z-scheme — 2 photosystems, 2 photons each, × 2 for NADPH + ATP rebalance) "

⏺ Good question — this is the cleanest part of the biochemistry to see from a physicist's lens. Think of it as electrons being pumped uphill by two photon kicks in series.

Step 1: What does "fixing 1 CO₂" actually require?

The Calvin cycle (dark reactions) needs:

• 2 NADPH (reducing equivalents)
• 3 ATP

per CO₂ incorporated into sugar. That stoichiometry just comes from balancing the Calvin cycle chemistry — not from photons directly.

Step 2: How many photons per NADPH?

NADPH carries 2 electrons. Those electrons ultimately come from water:

\[2 H_2O \to O_2 + 4 H^+ + 4 e^-\]

The electrons have to be lifted from the redox potential of water (O₂/H₂O ≈ +0.82 V) up to NADP⁺/NADPH (≈ −0.32 V). That's ~1.14 eV uphill per electron — doable in principle with one ~1.8 eV red photon.

But evolution didn't build a single-photon pump. It built a two-stage pump (the Z-scheme):

• PSII absorbs one photon → excited P680* pulls electrons off water (very oxidizing, +1.1 V). Electron drops partway down the chain to ~+0.4 V, dumping energy into a proton gradient.
• PSI absorbs a second photon → excited P700* re-excites the electron up to ~−1.3 V, reducing enough to make NADPH.

So each electron absorbs 2 photons (one per photosystem). NADPH needs 2 electrons → 4 photons per NADPH.

Step 3: Put it together

2 NADPH × 4 photons/NADPH = 8 photons per CO₂ ✓

The ATP requirement (3 per CO₂) is satisfied "for free" by the proton gradient the electrons build on their way down — with some cyclic electron flow around PSI topping up the ATP/NADPH ratio as needed. Hence the "× 2 for NADPH + ATP rebalance" in my earlier shorthand — that was sloppy; the factor of 2 is really the two photosystems in series, not a separate ATP cost.

Physicist's summary:

• 1 electron, 1 photosystem: ~1.8 eV kick
• 2 photosystems in series: ~3.6 eV absorbed per electron
• Useful redox storage: ~1.14 eV per electron
• "Stage efficiency" per electron: ~30%
• 4 electrons needed per CO₂ (to make 2 NADPH) × 2 photons/electron = 8 photons/CO₂

The reason nature uses two pumps instead of one UV photon is that UV is energetic but scarce and destructive; red/blue photons are abundant and gentle, but need to be staged.

Key fact: a eukaryote is essentially an archaeon that swallowed and kept an aerobic bacterium, which became the mitochondrion. (Later, a separate event swallowed a cyanobacterium → chloroplast in plants.) Single endosymbiotic events, never independently re-run in 2 Gyr since — which is why all plants, animals, fungi, and protists share one common eukaryotic ancestor.

Why this matters for the energy story (Nick Lane's argument):

A bacterium does respiration across its single cell membrane. Surface-to-volume scaling means it can't get bigger without suffocating. Every time it wants more ATP capacity, it needs more membrane — but membrane proteins require genes, and those genes have to be copied every division. So bacterial energy per gene hits a hard ceiling.

A eukaryote distributes respiration across hundreds to thousands of mitochondria, each with its own tiny genome (~37 genes) controlling local membrane logistics. The nuclear genome is freed from the energetic tax of copying ATP-making machinery. Result: ~100,000–200,000× more energy available per gene than bacteria.

That's the actual thermodynamic gate to complex life. Multicellularity "appears" around 1 Ga not because evolution was slow, but because:

1. You need O₂ (GOE opens the door ~2.4 Ga)
2. You need mitochondria to budget the energy per gene (eukaryogenesis ~2 Ga)
3. You need enough O₂ for big bodies (NOE ~0.8 Ga)
4. Then animals

Prokaryotes ruled for ~2 Gyr, eukaryotes were unicellular for another ~1 Gyr, then things got weird fast.