# Debunking Jennings

S happens

In 2002 Jennings and coworkers wrote a paper [1] in which they claimed that photosynthesis is more efficient than a Carnot heat engine running between the same temperatures. In fact, their final sentence reads: “Thus, $$1-T/T_r$$ represents a kind of efficiency horizon beyond which negative entropy is produced and the second law is not obeyed. As this is impossible for a heat machine, it serves to underline the difference between photosynthetic photochemistry and a heat machine.”

Even if photosynthetic photochemistry were not a heat machine, and I will come to that, this still presents an enormous problem for the second law, and for thermodynamics. Not being a heat machine does not help as Carnot’s original argument already shows.

In the figure below E is a Carnot engine that takes an amount of heat $$q_h$$ from the (red) high temperature reservoir, converts part of that heat to useful work $w$, and dumps the remaining heat $$q_l$$ in the low temperature (bluegreen) reservoir. The efficiency $$\eta$$ is the amount of work you can get from the heat absorbed at $$T_h$$, and can be written as

$\eta = \frac{w}{q_h} = \frac{q_h-q_l}{q_h} = 1 – \frac{q_l}{q_h}$

Carnot was able to prove that this is also equal to $$1-T_l/T_h$$, which is where the expression in the quote from Jennings’ paper comes from.

You can also run a Carnot engine in reverse, after all it was designed to be a reversible engine. Then it is called a heat pump. P in the picture. This heat pump takes an amount of heat $$q_l’$$ from the cold reservoir, and by performing work $$w’$$ brings it to the high temperature reservoir. The amount of heat $$q_h’$$ dumped is larger than the amount of heat taken from low temperature since the work is added to that: $$q_h’=q_l’+w$$. Both machines E and P are reversible, and Carnot argued that they have to be equally efficient. I repeat his argument here.

Suppose E is more efficient than P, so that $$w>w’$$ and we take $$q_l=q_l’$$. In that case, since $$w=q_h-q_l$$ and $$w’ = q_h’-q_l’$$ we get

$q_h > q_h’$

Running the two engines in parallel therefore implies that we can take an amount of heat equal to $$q_h-q_h’$$ from the high temperature reservoir and completely convert that to work $$w-w’$$. No heat is exchanged with the low temperature reservoir in that case since the heat added by E is taken out again by P. This goes against Kelvin’s statement of the second law, namely that it is impossible in a cyclic process to take heat from a reservoir and completely convert that to work. Kelvin came much later than Carnot, but even Carnot believed this to be impossible. He believed that heat was a conserved quantity and probably thought of the process as similar to taking work from falling water. From his point of view it was like using falling water to obtain work, and subsequently bring the water back to the original level, something nobody had been able to accomplish in Carnot’s time, and indeed even in our time. Numerous attempts notwithstanding. Apparently Carnot was convinced that it could not be done with heat either.

Almost everybody, except apparently Jennings and coworkers, is still convinced such a machine is not possible [2]. This has nothing to do with the fact that these are heat engines. If we replace P by the photosynthetic complex, and let it take a photon from the hot reservoir converting it to work and heat, this process cannot be more efficient that heat pump E either, otherwise we end up with the same situation: heat taken from the hot reservoir (energy of the photon minus the heat $$q_h’$$) is completely converted to work. Carnot came to two conclusions: (1) all reversible machines running between two temperatures have the same efficiency and (2) there can be no machines be more efficient than the reversible machine.

So, If Jennings is correct, it shows that nature has in fact succeeded in what mankind has strived for since time immemorial: get a free lunch. Photosynthesis is then a perpetual motion machine of the second kind. I am fully convinced that if this were the case, the principle would have been applied everywhere, and we would not encounter a single life form which would not use this principle.

But is it true? Let’s look in a little more detail to the photosynthetic complexes and ask the question if it can be viewed as a heat engine, albeit an irreversible one. That turns out to be the case, thus we expect a lower efficiency than that of a Carnot engine, and indeed it has.

Rather than taking the photosynthetic complex of plants, I’ll use a somewhat simpler system, that of photosynthetic purple bacteria to explain what photosynthesis does. In the figure a schematic overview of the process is presented.

The green box is the reaction center (RC) , the blue box another protein complex called cytochrome bc1. Starting point of the cycle is excitation by 870 nm light of a pair of bacteriochlorophylls called P. After that P quickly looses an electron to quinone Q, which picks up a proton from the cytoplasmic side of the membrane. The empty spot at P is quickly filled with a new electron, taken from a small protein called cytochrome c, after which the process more or less repeats itself: excitation, loss of electron, now to QH, picking up a proton so that QH2 is formed, and filling up the electron hole at P. Once QH2 is formed it is released by the RC, which picks up a fresh Q from the so-called quinone pool in the membrane, and the process can repeat itself. The RC  thus converts Q’s in QH2‘s, picking up two protons from the top of the membrane, two electrons from the bottom, and two photons from sunlight. The blue box converts QH2 back to Q, but it does so in a very smart way. When the first QH2 arrives it sends its two protons into the periplasm at the bottom of the membrane, and one electron to a cyt c which returns the electron to P. The second electron goes to the top to the other Q, which can then pick up yet another proton from the top. The Q which is now at the bottom is replaced by a second  QH2 which repeats that process, releasing its two protons and an electron to the bottom, and one electron to the top, to the point where the top Q has now become another QHthat can be released to the quinone pool. A rather briljant strategy that doubles the amount of protons per foton absorbed.

The whole process can thus be summarized as the transport of four protons from the top to the bottom using two 870 nm photons. The electrons run around, when their energy is high, in the RC, they help the Q bind protons, on the QH2 they are transported to the cyt bc1 complex, from there to cyt and back to P. No electrons are used or wasted, they just run around in circles. This is exactly how a heat engine works: the engine takes heat from the high temperature reservoir in the form of a photon, it uses part of that heat to transport protons against the current to another position thus performing work, and dumps the remaining heat in the low temperature reservoir.

We can calculate exactly how much work is done, how much heat is taken from $$T_h$$ and how much heat is dumped at $$T_l$$, and consequently we can also calculate the entropy changes.

The proton is taken from an environment with high pH, and brought to an environment with low pH. At low pH the proton concentration is higher, so in fact we need to bring the proton to a place where there are already many of them, and this costs work. This work can be expressed in the change in chemical potential of the proton, which is

$w = \Delta \mu = -2.3 k_BT\Delta pH = 2.36\times10^{-20}\,{\rm J}$

for a difference of three pH points. The energy of a 870 nm photon is $$2.28\times10^{-19}\,$$J, and consequently the entropy changes are

$\Delta_h S = -\frac{2.28\times10^{-19}}{1100} = -3.8\times10^{-23}\,{\rm J/K}$

for the high temperature reservoir [3], and

$\Delta_l S = \frac{ 2.28\times10^{-19}-2 \times2.36\times10^{-20}}{300} = 5.7\times10^{-22}\,{\rm J/K}$

Therefore the entropy gain in the low temperature reservoir is much larger than the entropy loss in the high temperature reservoir. The process, at an efficiency of  25% does not even come close to Carnot efficiency, which would be 73% for these temperatures.

But this is bacterial photosynthesis, plants can do much better, because in addition to pumping protons over the membrane, photosystem II (PSII)  can split water and move the electrons to a different redox center at much higher (free) energy. This redox center is the copper protein plastocyanin. In order to calculate the work in that case, we can use the known reduction potentials of water and plastocyanin (Pc):

$\frac{1}{2}{\rm O_2} + 2{\rm H^+} + 2{\rm e} \longrightarrow {\rm H_2O} \quad {\cal E}^\circ = 1.23\,{\rm V}$

and

${\rm Pc^{2+}} + {\rm e} \longrightarrow {\rm Pc^+} \quad {\cal E}^\circ = 0.37\,{\rm V}$

so that the free energy difference (the work needed) is

$\Delta G^\circ = 2\times e\times 0.86 = 2.75\times10^{-19}\,{\rm J}$

For this process two 680 nm photons are needed, an amount of energy equal to $$5.84\times10^{-19}\,$$J. The entropy loss in the high temperature compartment is therefore

$\Delta_h S = -\frac{5.84\times10^{-19}}{1100} = -5.31\times 10^{-19}\,{\rm J/K}$

In addition to the oxidation of water and reduction of Pc, four protons are moved across the membrane, so that the entropy gain of the low temperature compartment is

$\Delta_l S =\frac{5.84\times10^{-19}-2.75\times10^{-19} – 4\times9.47\times 10^{-21}}{300} = 9.03\times10^{-22}\,{\rm J/K}$

In other words: the total entropy is still increasing, and the efficiency, at 53% (although better than the bacteria) well below the 73% Carnot predicts. The true numbers can differ slightly since protons and other compounds are not under standard conditions, but this changes the amount of work only in a minor way.

Living systems are pretty smart, but have not yet found a way to circumvent the second law of thermodynamics.

[1] R.C. Jennings, E. Engelmann, F. Garlaschi, A.P. Casazza, and G. Zucchelli, Photosynthesis and negative entropy production, Biochim. Biophys. Acta, 1709, (2005), 251-255

[2] Well, not exactly, we’ll encounter other attempts in due time.

[3] The temperature of 1100 K is estimated as the temperature to be used for light, originally by Duysens who was the first to discuss photosynthetic efficiency. The temperature should probably be taken lower for the photosynthetic bacteria, who live under much more dire circumstances than bright sunlight. This would increase their efficiency somewhat. I’ll devote a page to this issue later.