Deriving the Cell Equation

Combining Equation 2.15 with the proportionality in Equation 2.16, the mass m and thus also the surface area A of sodium dissolving at a constant electrolyte tempera-ture and composition should follow to first approximation an exponential decay with the same rate constant α,

fIgure 2.2 A globule of substance A floating on a denser substance B. The force balance determines the globule shape and height h_t + h_b. The height is nearly constant for globules whose width substantially exceeds their height.

Sodium 21

dm

dt = −α , m (2.20)

with α to be determined experimentally. Figure 2.3 shows the observed variation of sodium surface area with time from the moment of termination of electrolysis, with the electrolyte at an average temperature of 318°C, and demonstrates that the rate of sodium dissolution is in reasonably good agreement with Equation 2.20.

Most of the small deviation between theory and experiment can be explained by ~ 4°C variation in temperature during the observations, its being difficult to main-tain a constant temperature in the low-thermal conductivity melt during the tran-sition from heating by electrolysis to external heating. In addition, one expects agreement to worsen when time becomes large because the shape of small glob-ules approaches spherical, so that the considerations that led to Equation 2.19 break down, and A becomes proportional to m^2/3 rather than to m. From a fit of the data at these conditions, one obtains a dissolution rate of α = 0.0026 s⁻¹, applicable to Equation 2.20.

The rate of sodium generation at the cathode is by Faraday’s law:

dm

dt = , kI (2.21)

where I is the electrolysis current, and k = M/F; where M is the molar mass of sodium;

and F is Faraday’s constant. Equations 2.20 and 2.21 combine to give dm

dt = − α . kI m (2.22)

200 400 600 800 1000

1 2 3 4

Time, s

Na Mass, g

fIgure 2.3 Variation in the mass of a sodium globule floating on the surface of molten NaOH with time from the moment of termination of electrolysis. The exponential fit provides support for Equation 2.20.

This equation is only valid in Regime 1. Most useful sodium production occurs in Regime 2, when water has reached a steady state distribution in the electrolyte.

Taking into account that in this regime at least half of the generated sodium dis-solves by reaction with water diffusing from the cathode, we arrive at the sodium cell equation:

dm

dt kI kI

= − max( , m)

2 α . (2.23)

This equation has a finite m t( = ∞ to which the mass tends at infinite time. Hence, ) if electrolysis continues indefinitely without removal of sodium, the percentage yield will approach zero, in agreement with the experiment.

Neglecting the processes of Regimes 3, 4a, and 4b, the mass of sodium dissolved in the bath n, plus the amount of bulk sodium m present at any given time, is the amount generated by Faraday’s law minus the amount decomposed by water:

kIt m n

2 = + . (2.24)

We immediately see that a key difference between Equations 2.22 and 2.23 is that in the former case the amount of sodium dissolved in the electrolyte n continuously increases until eventually all current is carried by sodium ions and the cell becomes dead. In the latter case, no sodium concentration is developed in the electrolyte until time t = 1/ α corresponding to yield m^c:

m kI

c= α

2 , (2.25)

which is proportional to the current, and equals half m t( = ∞ , the maximum yield ) that can be obtained from the cell. After this point, n≠ 0 , m starts saturating, while n saturates even faster due to dissolved sodium oxidized at the anode by Processes 2.2b, 2.4, and 2.5.

We thus see that the diffusion of water, rather than being detrimental, is essen-tial to the indefinite operation of the NaOH cell, provided sodium is removed from the cathode at regular intervals, commensurate with 1/α . If this is not done, the amount of bulk sodium starts to saturate, the sodium starts dissolving, and the cell commences functioning in Regimes 3, 4a, and 4b. This explains the low percentage yields reported by von Hevesy [14], where NaOH was electrolyzed for periods of the order of an hour at low currents.

Equation 2.23 does not describe the eventual disappearance of bulk sodium from the cell because it does not take into account that in Regime 4 part of the elemental sodium flux balancing the sodium ion current is carried by sodium already dissolved in the electrolyte rather than sodium generated at the cathode.

Sodium 23 This reduces the effective current I in 2.21, and thus by Equation 2.25, also the maximum amount of sodium m_c the cell can support. Eventually n becomes suffi-ciently large that the ionic sodium current is entirely balanced by the material flux from sodium already dissolved in the electrolyte. This reduces the effective current to zero, and with it m_c.

The variation of the mass of liquid sodium m with electrolysis time, Figure 2.4, is in good agreement with Equation 2.23. The mass m was estimated by applying Equation 2.16 to time-lapse images captured during electrolysis. Because the cur-rent I varies with time during the experiment, kt in Equation 2.23 is replaced by the charge Q. The graphs show that initially m varies linearly with a charge passed through the cell, and starts saturating at large times. The agreement with theory is good despite the dissolution rate α used in the theory being measured under static conditions in the absence of material and electric currents.

200 400 600 800 1000

3

2

1

Time, s

Na Mass, g

200 400 600 800 1000

4 3 2 1

Time, s

Na Mass, g

fIgure 2.4 Variation of mass of liquid sodium m generated in the electrolysis of molten NaOH as a function of charge passed through the cell. The solid line represents data calcu-lated from the cell Equation 2.23, and shows good agreement with experimental results, in particular demonstrating that the amount of sodium starts to saturate at large times.