## Process thermodynamic - Enthalpy

As we have seen, in a Hall-Héroult cell the molten aluminum is produced according to the following reaction:

This reaction occurs at constant temperature T (the bath temperature) and constant pressure p (the atmospheric pressure).

**Enthalpy**

When we speak of enthalpy, we are talking of the 1^{st} law of thermodynamic or, in other words, of the energy conservation principle. In the case of an alumina reduction cell, from the external the only energy we input is electrical energy (W_{El}). This energy is partially used by the cell to produce aluminum while the rest is dissipated as heat. So we can write:

This equation tells us that the electrical energy we supply from the external is used to: 1) increment the internal energy content of the products of the reaction (1) compared to the internal energy of the reactants (term ΔU), 2) give the energy necessary for the CO_{2} bubbles to expand under the atmospheric pressure p (term pΔV), 3) the remaining part of W_{El} is lost in the external as heat (term Q_{D}).

We also know that the term (ΔU + pΔV ) is equal to the enthalpy change ΔH, so we can rewrite the (2) as:

Using thermodynamic tables and the basic reaction (1) we can calculate the enthalpy of the reaction, or, in other terms, the minimum energy required to produce aluminum.

But for a better estimation of the enthalpy we need to take in considerations the following practical aspects:

- We need to consider the energy necessary to heat the reactants from the ambient temperature up to the bath temperature
- The process is not 100% efficient, so we need to take into account the loss in current efficiency

With these considerations, if x is the current efficiency expressed as a fraction (CE = x ∙ 100%), (1-x) is the fraction of aluminum which reoxidizes and the (1) can be rewritten as:

Considering an electrolyte saturated with α alumina, with P_{CO2} = 1 atm, at a temperature of 977°C, inserting the proper figures from the thermodynamic tables we have:

This equation takes into account:

- The energy required to heat the reactants from ambient temperature to 977°C
- A current efficiency lower than 100%

At 100% current efficiency:

The (5) should be further modified if we consider an electrolyte not saturated with alumina (as is the real case) and the fact that usually the alumina delivered to the cell is mostly γ alumina, but these changes introduce only slight modifications to the (5).