In the context of renewable energy, hydrogen is usually considered an energy carrier: spare electrical energy is used to split water with the objective of recovering this energy later when the hydrogen is consumed to re-form water and close the cycle. The round-trip efficiency is the ratio of the useful energy that eventually comes out compared with what was put in. A frequent comparison is made with batteries where up to 90% is recoverable. However, the comparison is rather unfair as storing on a grid-electricity scale with batteries can have an enormous impact on world resources (as there is little resource saving with volume increase). Given the advantages of hydrogen on the larger scale, a round-trip efficiency of better than 50% can for some applications be acceptable, particularly if the waste heat can be utilised.
But exactly how efficient is the process? We can look here in detail at the electrolysis segment. The process can be examined in terms of basic physics and chemistry, statistical mechanics or thermodynamics, all giving the same results, but looking at the process in different ways can improve understanding. We can certainly investigate reactions and energy transfer at a molecular level, but what is observable macroscopically is an averaging of the energy distribution functions that give repeatable results consistent with the laws of thermodynamics.
The transition from molecular to macro level is enormous and involves the step up from one molecule to a mole of molecules. One mole is a huge number, 6.022 x 1023: this is approximately the number of coarse grains needed to cover the entire surface of the earth in sand to a depth of 1 m.
We will first look at the energy required to split water in terms of basic molecular parameters. Note that we will need to use average values and these will change with temperature and to a lesser extent pressure (many sources give values without specifying the temperature and pressure, hence some variation in quoted values is expected). The average bond energy of the OH bond in water at normal temperature and pressure (NTP) is 459 kJ mol-1. 494 kJ mol-1 is required to split the OH-H bond then 424 kJ mol-1 to split the O-H bond. Hydrogen gas forms accompanied by a mole of H-H bonds (432 kJ mol-1) as well as half a mole of double-bonded oxygen (0.5 x 495 kJ mol-1). A simple calculation suggests the energy required to split one mole of water by electrolysis is:
Approximation 1
E = (494 + 424 – 432 - 0.5 x 495 ) kJ mol-1 = 238.5 kJ mol-1
However, a full energy calculation must also take into account the bonding in the liquid keeping the molecules partially bound. There is energy transfer relating to this bonding as well. The previous approximation is only appropriate for splitting water vapour, something that is not normally done by electrolysis. In addition, the newly formed gas must move the atmosphere aside to make space for itself – this requires energy pΔV. These two together can be estimated from the latent heat of vaporisation at room temperature, 2500 kJ kg-1, or 45.0 kJ mol-1. Of this, the work done to make space for 1.5 moles of gas (0.036 m3) at normal pressure is 3.7 kJ. We can modify our approximation:
Approximation 2
E = (238.5 + 41.3 + 3.7) kJ mol-1 = 283.5 kJ mol-1
This is close to the accepted value of 285.8 kJ mol-1 (though be aware the dependence on external factors is so great that a value to this precision should never be claimed without stating the conditions, assume NTP if not told otherwise).
The simulation above is based on a classical rather than a quantum mechanical representation of molecules. The reason for is that no one knows how to show a quantum picture! A free water molecule has 9 degrees of freedom (3 x translation, 3 x rotation, 2 x bond vibration, 1 x bending betweeen bonds) and this crude model implements just a few of these. Note that the number in red is energy that has to go in, and in black energy that comes out.
The implication is that 283.5 kJ mol-1 is the amount of electrical energy which needs to be supplied. That is not correct, the closeness of the value to the measured value is just fortuitous! Whilst the basic calculation based on bond energy is useful for illustrative purposes, it neglects the interaction of individual atoms with their surroundings We have assumed the translational, vibrational and rotational energy of the molecules in the liquid is the same as for the gases at that temperature. That is not true - we know that gas molecules move much faster. The gases produced by electrolysis will acquire energy from the liquid to reach the energy level that is consistent for a gas at the electrolysis temperature. Through collisions, the new gas gains the energy required (and it does not have to come from the electrical source). Hence the gas takes energy from the environment. This gain was previously bundled in with the latent heat value and needs be subtracted away to determine how much electrical energy is really needed. Again, an estimate can be made on the basis of the specific heat of the liquid and gas phases with temperature variation estimated using the Nernst equation. The accepted value is that the newly formed gas will absorb 48.7 kJ mol-1 heat from the environment leaving the recognised value of 237.1 kJ mol-1 to come from the electricity source (neglecting all losses).
Note that if the electrolyser is exactly 83% efficient the 17% waste heat produced supplies the energy the gas must take from the environment. A 100% efficient system would need an external heat source otherwise the optimal electrolysis temperature could not be maintained. An ideal fuel cell consuming hydrogen will return the same ratio of heat and electricity – for this reason the process can be assumed to be reversible. The exact efficiency of an electrolyser is subject to debate, but for simplicity it is best to define efficiency with reference to closed cycles only, comparing useful energy in with useful energy out, both possibly a combination of electricity and heat.
The complications at molecular level can be circumvented by treating electrolysis as a thermodynamic process. In terms of thermodynamics, the total energy is referred to as enthalpy (H). The electrical energy that must be supplied is the Gibbs free energy (G). Entropy (S) is dependent on the temperature (T) at which the reaction takes place and relates to the movement of heat. These are connected by the equation:
ΔH = ΔG + TΔS
A further expansion is possible in terms of total internal energy (U), pressure (P) and volume (V):
ΔU = ΔG + TΔS – PΔV
At 298 K, 3.7 kJ mol-1 work is required to make space of the gases produced. The enthalpy we know is 285.8 kJ mol-1 at 298 K and the change in entropy is 163 J K-1 (entropy increases because a gas is more disorganised than a liquid); it means TΔS is 48.7 kJ mol-1. The Gibbs free energy is then 237.1 kJ mol-1 as expected. The change in internal energy (U) is 282.1 kJ mol-1 and this will bundle together bonding, vibrational, kinetic and rotational energy. In a way this is preferable to a low-level analysis at a molecular level because we are working with empirical values and we do not have to concern ourselves with the complexity of the internal energy components of the liquid and gas states, nor do we have to be concerned with the way the molecular binding within molecules is altered when there is the bonding between molecules (specifically hydrogen bonding in liquid water).
Statistical mechanics is the explanation for why such low-level variation and complexity can result in the very simple thermodynamic relations. The idea is that the variation of a parameter when the temperature is constant can be described by a distribution with a shape that is controlled by the random interactions within the equilibrium state. A mole of gas molecules will have a range of velocities that are modified on a molecule-by-molecule basis by collisions (and to a lesser extent photon interactions) without changing the shape of the distribution because of the constraints applied (for example, a closed system) and the huge number of molecules involved.