Goodenough-JB primer: the "glass battery" and where its energy comes from

Document studied: M. H. Braga, N. S. Grundish, A. J. Murchison and J. B. Goodenough, "Alternative strategy for a safe rechargeable battery", Energy & Environmental Science (2016), doi:10.1039/c6ee02888h. The full text layer was read. Every figure page (Figs. 1 to 9, pp.3 to 7) was rendered and read as an image, the later ones in a deep read requested by quality engineering (QE) [read; read: page image p.3 to p.7]. Verdicts follow arbiter.md, including its "QE revision" section.

Source gap (repeated from Step 1). This is the only document in the folder. The three earlier papers that hold the evidence for the glass's conductivity, its wetting by lithium and its "ferroelectric dipoles" were not provided and not read: Braga et al. 2014 (J. Mater. Chem. A), Braga et al. 2016 (Energy Environ. Sci. 9, 948) and Braga et al. 2017 (J. Electrochem. Soc. 164, A1). No raw data and no peer-reviewed replication or critique was read either. So anything below about the glass itself (claims 1 and 2, and the dipole part of claim 3) rests on the paper's citations plus [web] or [memory] context. The energy-budget and Nernst arguments (claims 3 to 5) use only the paper's own numbers and standard physics. Where a conclusion depends on [memory] alone, it says so.

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1. The idea in one paragraph

John B. Goodenough co-invented the cathode used in almost every lithium-ion battery and shared the 2019 Nobel Prize in Chemistry [memory, high]. In this paper he and Maria Helena Braga propose a new kind of solid-state battery. A thin sheet of glass made of lithium (or sodium), oxygen and chlorine sits between a lithium-metal anode and a copper/carbon current collector. They say the glass conducts lithium ions very well, does not react with lithium, and lets lithium plate and strip without forming needle-like "dendrites". The unusual part is the cathode. In an ordinary battery the cathode is a material that soaks up lithium, and the chemical energy of that reaction is the battery's energy. Here they say discharge simply moves lithium metal from the anode and plates it as lithium metal onto the copper. A small amount of sulfur, ferrocene or manganese dioxide, which they say is not used up, "sets" the voltage. So capacity would be limited only by how much lithium you start with. They explain the cell voltage with two electrical capacitors in series rather than with ordinary electrode chemistry. Their coin cells did deliver large charge over weeks and lit LEDs. The puzzle this primer works through is simple: if lithium metal ends up as lithium metal, where does the energy come from?

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2. What you need to know first

Charge, current and capacity. Current I (amperes, A) is charge per second; one coulomb (C) is one ampere for one second. Battery people count charge in milliampere-hours: 1 mAh = 10⁻³ A × 3600 s = 3.6 C. "Specific capacity" is charge per gram of material (mAh/g). Energy is charge times voltage: delivered energy = ∫ V dQ. If V is roughly constant, energy ≈ Q × V̄, where V̄ is the average voltage. Read: any intro physics text (Halliday, Resnick and Walker, Fundamentals of Physics, circuits chapters).

Faraday's law of electrolysis. Every atom that gives up one electron moves a charge e = 1.602×10⁻¹⁹ C. A mole of electrons carries the Faraday constant F = N_A·e = 96 485 C/mol. So a gram of a metal of molar mass M (g/mol) that gives n electrons per atom can deliver n·F/M coulombs. For lithium (M = 6.94 g/mol, n = 1) that is 13 900 C/g = 3862 mAh/g. Read: Atkins, Physical Chemistry, electrochemistry chapter; Bard and Faulkner, Electrochemical Methods.

Gibbs free energy and cell voltage. At constant temperature and pressure the maximum useful (non-expansion) work a reaction can do is −ΔG, where ΔG is the Gibbs free energy change. In a battery that work is electrical: ΔG = −nFE, where E is the open-circuit cell voltage and n the number of electrons transferred per formula unit. No free-energy change, no voltage, no energy. Read: Schroeder, An Introduction to Thermal Physics (chapter on free energy and chemical thermodynamics).

Electrode potentials and the Nernst equation. Each electrode reaction (a "couple", such as Li⁺ + e⁻ ⇌ Li) has a potential measured against a reference, often the standard hydrogen electrode (SHE). Lithium is about −3.04 V vs SHE [memory, high]. Cell voltage is the difference between the two electrodes' potentials. When concentrations differ from standard, the Nernst equation adds a term (RT/nF)·ln(activity ratio); "activity" is the effective concentration. Read: Bard and Faulkner, Electrochemical Methods, early chapters on potentials and thermodynamics.

Fermi level. In a metal, electrons fill energy states up to the Fermi level E_F. Connect two metals and electrons flow until their Fermi levels (strictly, their electrochemical potentials) match. In battery language, the voltage of a cell is the difference in electron electrochemical potential between the two terminals divided by e. The work function Φ is the energy needed to pull an electron from the Fermi level out of the metal; different metals have different Φ. When two different metals touch, a little charge flows once, until the Fermi levels line up, leaving a small fixed potential step between their surfaces called the contact (Volta) potential, ΔΦ/e. That step is a one-time equilibrium, not a source of current: around any closed loop of metals the contact potentials add to zero, so a voltmeter cannot read it and it cannot drive current [memory, high]. Read: Kittel, Introduction to Solid State Physics (free-electron chapter); Griffiths does not cover this.

Capacitors, series combination and energy. A capacitor holds charge +Q and −Q on two conductors; V = Q/C, with capacitance C in farads (F). Two capacitors in series carry the same Q, and 1/C_eq = 1/C_A + 1/C_C. The energy stored is U = ½QV = Q²/(2C). A parallel-plate capacitor has C = ε_rε₀A/d, where A is plate area, d the gap, ε₀ = 8.854×10⁻¹² F/m and ε_r the relative permittivity (dielectric constant) of the material between. Read: Griffiths, Introduction to Electrodynamics, chapter 2 (capacitors) and chapter 4 (fields in matter, dielectrics).

Electric double layer (EDL). Where an electrode meets an electrolyte, ions pile up a nanometre or so from the surface, opposite to charge on the metal. That thin layer is a capacitor with a tiny gap, so it has large capacitance per area: typically 10 to 100 µF/cm² [memory, high]. Supercapacitors use this on carbon with enormous surface area. Read: Bard and Faulkner, Electrochemical Methods (double-layer chapter).

Ion conduction in solids. In a solid electrolyte, ions hop between sites over an energy barrier ΔH_m (the "motional enthalpy"). Conductivity follows an Arrhenius law: σT = A·exp(−ΔH_m / k_BT), where k_B = 8.617×10⁻⁵ eV/K is Boltzmann's constant. At 300 K, k_BT = 0.0259 eV, so a barrier of 0.25 eV suppresses hopping by e⁻⁹·⁷ ≈ 6×10⁻⁵, a barrier of 0.06 eV by only e⁻²·³ ≈ 0.1. Read: Kittel (point defects / ionic conductivity); Huggins, Advanced Batteries: Materials Science Aspects.

Band gap vs electrochemical stability window. A band gap E_g is the energy between filled and empty electron states of a solid (in eV). An electrolyte's electrochemical stability window is the voltage range (in V) over which it is not reduced or oxidised. Goodenough's own well-known picture (Goodenough and Kim, Chem. Mater. 2010 [memory, high]) relates the two: an electrolyte is stable if its lowest empty level lies above the anode's Fermi level and its highest filled level lies below the cathode's. But a large gap is necessary, not sufficient. Real stability is set by whether a chemical reaction with lithium (e.g. making Li₂O or LiCl) lowers the free energy [memory, high].

SEI and dendrites. Lithium metal reduces most electrolytes and forms a thin solid-electrolyte interphase (SEI). A good SEI stops further reaction. Lithium also tends to plate unevenly as dendrites, which can short a cell. Both are central problems for lithium-metal batteries [memory, high].

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3. The textbook view

3.1 A battery is a controlled chemical reaction

A cell has an anode (oxidised on discharge), a cathode (reduced on discharge), an electrolyte that passes ions but not electrons, and an external circuit that passes electrons. For a lithium–sulfur (Li–S) cell:

• Anode: Li → Li⁺ + e⁻
• Cathode: S + 2Li⁺ + 2e⁻ → Li₂S (written in the paper as 16Li + S₈ → 8Li₂S [read])
• Net: 2Li + S → Li₂S, with ΔG < 0.

The open-circuit voltage is

E = −ΔG / (nF)

where E is in volts, ΔG in J/mol, n the number of electrons per formula unit, F = 96 485 C/mol. Conventional Li–S cells discharge on plateaus around 2.3–2.4 V and about 2.1 V vs Li/Li⁺ [memory, high].

3.2 Nernst equation

For one electrode couple Ox + ne⁻ ⇌ Red,

E = E° − (RT/nF) ln(a_Red / a_Ox)

with E° the standard potential, R = 8.314 J/(mol·K), T temperature in kelvin, and a the activities. At 298 K, RT/F = 25.7 mV, and (RT/F)·ln 10 = 59.2 mV per factor of ten. If the same couple is on both sides of a cell (a "concentration cell"), the E° terms cancel:

E_cell = (RT/nF) ln(a₁/a₂).

Only a difference in activity can then give a voltage, and it grows only logarithmically.

3.3 Capacity and energy

Theoretical specific capacity: q = nF/(3.6·M) mAh/g. For Li, q = 96485/(3.6 × 6.94) = 3862 mAh/g. For sulfur going to Li₂S (n = 2 per S, M = 32.06), q = 1672 mAh/g (the paper's figure of about 1675 is the same to rounding). Energy delivered = ∫V dQ.

3.4 Where capacitors fit

Every electrode/electrolyte interface has a double-layer capacitance. It explains small, fast changes: the brief voltage spike when current switches on, or the slow drift of an open-circuit reading as interfaces settle. It cannot store much: Q = CV with C ≈ 10⁻⁵–10⁻⁴ F on a coin-cell electrode is microcoulombs to millicoulombs, not the hundreds of coulombs a battery delivers [memory, high on EDL values]. The voltage of a battery comes from ΔG of the reaction; the double layer is where that voltage drops physically, not where it comes from.

3.5 Solid electrolytes

Good crystalline lithium conductors reach about 10⁻² S/cm at room temperature with barriers near 0.2–0.3 eV; for example Li₁₀GeP₂S₁₂ is about 1.2×10⁻² S/cm with about 0.25 eV [memory, medium, not web-verified]. Crystalline Li₃OCl "antiperovskite" is generally reported near 10⁻³ S/cm or lower with a barrier near 0.3 eV [memory, medium, not web-verified].

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4. Where this researcher departs

Fair starting points first. Several of the ideas are real physics:

• Glassy and amorphous conductors can conduct ions well, and Li₃OCl-type materials are a genuine research family [memory, high].
• Good wetting of a solid electrolyte by lithium does lower interface resistance, and pressure helps [memory, medium].
• The band-gap picture of electrolyte stability is Goodenough's own standard teaching tool [memory, high].
• Double layers really are capacitors in series with each other, and Eqs. (1)–(2) are correct algebra for two series capacitors [read; checked].
• Fermi levels really do set cell voltage in the sense that V = (difference of electron electrochemical potentials)/e.

The departures, side by side:

The exact point of departure is the sentence: "the sulfur acts as a redox center determining the voltage of the cell at which electrons from the anode reduce the Li⁺ at the electrolyte/cathode interface to plate lithium rather than reducing the sulfur" [read]. Accept it, and the net reaction is

Li (metal, anode side) → Li (metal, cathode side),

which is the same substance in the same state. Then ΔG ≈ 0, so E ≈ 0 (Section 3.1), and a catalyst-like center that is not consumed cannot change an equilibrium voltage (catalysts change rates, not ΔG). The capacitor model is introduced to explain the voltage anyway. Section 5 tests whether it can.

A note on signs. The paper writes μ_A = −3.05 V and μ_C = −0.40 V vs SHE and plots E_F = −μ in Fig. 4 [read: page image p.5]. The minus signs are lost in the PDF text layer, and earlier drafts of this review quoted them as positive. Only their difference, 2.65 V, is used below, and that is unchanged. μ_A matches the textbook Li⁺/Li value of about −3.04 V vs SHE [memory, high].

Three internal points the paper does not reconcile:

1. Fixed Q or fixed potentials? The paper fixes the electrode potentials (μ_A and μ_C above) and also says Q is fixed while V changes as C changes. If Q is whatever equalises fixed potentials, then Q = C_eq·|μ_A − μ_C| and V = |μ_A − μ_C| = 2.65 V regardless of C. If Q is fixed, V moves with C but the potentials cannot also be fixed. Both cannot hold (arbiter: internally inconsistent). The fitted numbers make it worse: Fig. 4 says C_C ≪ C_A, but going from 2.65 V to the aged 1.6 V would need C_C about 0.43 C_A at the start, not much smaller [arbiter, QE revision, calc].
2. The 2.65 V coincidence, softened. |−3.05 − (−0.40)| = 2.65 V, the same as the reported V_max ≈ 2.65 V [read]. Earlier drafts took this as proof that the cell is simply the ordinary Li/S couple (sulfur being reduced). That is too strong. The paper gives no method for μ_C, and Fig. 4 itself lists V_cell ≈ 2.65 V for a fresh cell, so μ_C may have been set from the same cells, which makes the match partly circular. Also, conventional Li–S cells discharge at about 2.3–2.4 V and 2.1 V (Section 3.1), so a 2.65–2.68 V start does not match them cleanly either. "Ordinary sulfur reduction" stays a reasonable but unproven reading. The refutations in Section 5 do not depend on it.
3. A Fermi-level difference is a contact potential, not a battery. The paper says the voltage is set by the Fermi-level difference between the anode and the cathode current collector, and that with no redox center the cell reads about 3.5 V [read p.2, p.7]. A Fermi-level (work-function) difference between two different metals is exactly the contact potential of Section 2. Rough numbers [memory, medium]: Φ(Li) ≈ 2.9 eV and Φ(Cu) ≈ 4.6 eV, so ΔΦ/e ≈ 1.7 V. When the two are connected, charge Q = C·ΔΦ/e flows once and stops. Treat the 0.06 mm glass between 1.5 cm² electrodes as an ordinary capacitor with ε_r = 10 (assumed, a typical glass): C = 10 × 8.854×10⁻¹² × 1.5×10⁻⁴ / 6×10⁻⁵ = 2.2×10⁻¹⁰ F, and Q = 2.2×10⁻¹⁰ × 1.7 ≈ 3.8×10⁻¹⁰ C. The cell delivered 129 C, about 3.4×10¹¹ times more (example.py, Step 6). A voltage that comes from equalising Fermi levels cannot drive a sustained current, because nothing is consumed to keep restoring the difference. A battery needs a chemical reaction whose free energy is released; an unconsumed catalyst or collector cannot supply it [arbiter, QE revision, item 7].

The paper itself also notes that the cell voltage "was limited to avoid electrostatic energy storage at a high voltage, as has been demonstrated can occur (ref. 2)" [read p.4]. So the authors treat capacitor-like storage as a separate effect they tried to avoid. This does not rescue the capacitor model: Section 5 shows a capacitor cannot hold the 129 C anyway.

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5. Worked example: where did 315 joules come from?

Everything here is reproduced by example.py (run: .venv/bin/python review/Goodenough-JB/example.py). Paper inputs [read]: 1.99 mg sulfur; Fig. 1 curve; 90.1 % of Li capacity; 0.06 mm membrane; 30 mA/g. The Fig. 1 curve was digitised by eye from the rendered page (start of the discharge plateau 2.68 V, knee 2.34 V at about 850 % of sulfur capacity, end about 1.62 V at about 1075 %) [read: page image p.3], so voltages are good to about ±0.03 V. Strictly, the trace starts near 1.65 V, the aged open-circuit voltage, and jumps almost vertically to 2.68 V once discharge current flows [read: page image p.3, p.6]. The jump carries almost no charge, so it does not change the energy integral; but it is an anomaly in its own right (see Section 7). Two values are assumed because the paper does not state them: electrode area A = 1.5 cm² (typical coin cell) and double-layer capacitance 20 µF/cm².

Step 1: capacities (Faraday). - Li: 1 × 96485 / 6.94 / 3.6 = 3862 mAh/g. - S → Li₂S: 2 × 96485 / 32.06 / 3.6 = 1672 mAh/g.

Step 2: charge delivered. - Sulfur alone could deliver 1.99 mg × 1.672 mAh/mg = 3.33 mAh. - Fig. 1a ends at about 1075 % of that: Q = 10.75 × 3.33 = 35.8 mAh = 129 C. The cell gave about 10.8 times what its sulfur can give. (A critic's blog reads the same, about 35.9 mAh [web: https://lacey.se/2017/03/29/on-the-skepticism-surrounding-the-goodenough-battery].) - "90.1 % of Li capacity" then implies a Li anode of 35.8 / (0.901 × 3862) = 10.3 mg (an inference; the paper gives no Li mass). - Rate check: 30 mA per gram of sulfur × 1.99 mg = 59.7 µA; 35.8 mAh / 59.7 µA = 600 h = 25 days, close to the stated 28 days. (Assumes "30 mA/g" is per gram of sulfur; the paper does not say.)

Step 3: energy, and the paper's 10.5 Wh/g. - Integrating V dQ over the digitised curve gives 315 J (88 mWh), mean voltage 2.45 V. - Per gram of Li: 315 J / 3600 / 0.0103 g = 8.5 Wh/g for the whole curve. Above 2.34 V only: 7.0 Wh/g. - The paper says 10.5 Wh/g (whole curve) and 8.5 Wh/g (above 2.34 V) [read]. - To get 10.5 Wh/g from 90.1 % of 3.862 Ah/g = 3.48 Ah/g needs an average of 10.5 / 3.48 = 3.02 V. The curve never exceeds 2.68 V. Even a flat 2.68 V would give only 9.3 Wh/g. - So both headline numbers are about 22 % higher than the paper's own plot supports. (Oddly, the paper's "reversible" 8.5 Wh/g equals the whole-curve value computed here; why is unknown.) The arbiter's estimate, using a rough 2.4 V average, was 8.4 Wh/g: the same conclusion. - Remember also that "per gram of Li" is not a cell energy density. Good Li-ion cells give about 0.25–0.30 Wh per gram of whole cell [memory, high]; these numbers are not comparable.

Step 4: could two series capacitors supply it? - To hold Q = 129 C at 2.65 V needs C = Q/V = 48.6 F. - A typical double layer: 20 µF/cm² × 1.5 cm² = 30 µF. Short by a factor of 1.6 million. Even 100 µF/cm² leaves a factor of about 300 000. - Suppose instead the whole 0.06 mm glass were the dielectric of a parallel-plate capacitor: ε_r = C·d/(ε₀A) = 48.6 × 6×10⁻⁵ / (8.854×10⁻¹² × 1.5×10⁻⁴) = 2.2×10¹². The highest-permittivity real materials reach roughly 10⁴–10⁶ [memory, medium]. - Even granting a 48.6 F capacitor, the energy it holds when charged to that Q and V is U = ½QV = ½ × 129 × 2.65 = 171 J, only 54 % of the 315 J delivered. - And a capacitor's voltage falls in proportion to the charge removed (V = Q/C). Fig. 1 shows a slow slope from 2.68 to 2.34 V while 80 % of the charge flows, not a straight fall to zero. - Changing the assumed area by a factor of 100 either way does not rescue the model.

Step 5: Nernst for lithium on both sides. If both electrodes are lithium metal, E = (RT/F) ln(a₁/a₂). For 2.65 V: ln(a₁/a₂) = 2.65 / 0.0257 = 103, i.e. a₁/a₂ = 10⁴⁴·⁸. Two pieces of lithium metal have essentially equal activity (both are pure metal, activity 1). This is why plating Li on Cu in an ordinary Li‖Cu test cell happens near 0 V vs Li/Li⁺ [memory, high].

Step 6: how much charge can a contact potential move? Using the Section 4 numbers (ΔΦ/e ≈ 1.7 V [memory, medium]; ordinary glass ε_r = 10, assumed): C = 2.2×10⁻¹⁰ F, Q = 3.8×10⁻¹⁰ C, once. The cell delivered 129 C, about 3.4×10¹¹ times more. A Fermi-level difference cannot be the energy source.

What the example shows. The cell delivered real charge at real voltage: about 129 C at about 2.45 V. The paper's mechanism (Li moved to Li, an unconsumed redox center, voltage from capacitors) cannot account for it: no free-energy source, a capacitor a million times too small, and too little stored energy even if it were big enough. So the energy must come from some chemical reaction the paper did not identify. Candidates include reduction of the sulfur after all (possible, though the 2.65 V match is partly circular; Section 4), reactions of lithium with the glass, with water or hydroxide in it, with the carbon, or with air entering through the "imperfect seal" the paper mentions [read]. The membrane also contains residues that react with lithium metal: the glass is held in recycled paper (cellulose, rich in –OH groups) or fiberglass, cast from an ethanol slurry and heated only to above 130 °C [read p.3]; residual ethanol, water and cellulose –OH all react with Li [memory, high]. Each is a hypothesis, not a demonstrated finding. Only a mass balance (Section 7) would decide.

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6. Evidence table

How it was received: the paper appeared in a mainstream Royal Society of Chemistry journal and drew skepticism about missing data and a first-law (energy-conservation) problem; it "remains controversial after several follow-up works" [web: https://en.wikipedia.org/wiki/John_B._Goodenough, secondary]. The reviewers found no evidence of blocked publication. A formal comment in Energy Environ. Sci. arguing a first-law problem, with a reply by Braga and Goodenough, is recalled [memory, medium], but the attempt to verify it on the web failed, so its authors, date and arguments are not given here. This is a "published, unreplicated, thermodynamically challenged, mass balance never done" case, not "dismissed without testing". The authors' reputation is not evidence either way.

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7. What a learner should take away

Solid. - The cells did deliver measurable charge (about 129 C) at about 2.4–2.7 V for weeks, and lit LEDs [read]. That observation stands unless shown to be a measurement error, and nobody has shown that. - Series-capacitor algebra and the band-gap picture of electrolyte stability are correct physics in their proper place. - Low-barrier glassy ion conductors and Li-wetting solid electrolytes are legitimate research directions.

Wrong, and why the mistakes are instructive. - Voltage without a reaction. Moving lithium from lithium to lithium has ΔG ≈ 0, so E ≈ 0. Whenever a device seems to deliver energy with "nothing consumed", find the ΔG. This is the most useful habit this project teaches. - Confusing where a voltage appears with where it comes from. The double layer is where the potential drops, but the energy comes from the reaction. A capacitor big enough to hold this charge would need about 49 F, and even then it would hold only about half the energy delivered. - Band gap ≠ stability window. A gap measured in eV and a window measured in V are related but not the same, and chemical reactions with Li can happen across a wide gap. - Bookkeeping. The 10.5 Wh/g needs a 3.0 V average from a curve that peaks at 2.68 V. Always integrate the plot yourself. - Instrument limits. EDS cannot see lithium, so it cannot show lithium was (or was not) plated. - Contact potential is not a power source. Two metals with different work functions show a small potential step, but it moves a tiny charge once (about 4×10⁻¹⁰ C here) and then stops. Any "voltage from Fermi levels with nothing consumed" claim fails the same way. - Read the test protocol, not just the cycle count. Cycling at fixed current for fixed times cannot show capacity fade, and cycles that use a quarter of the cathode's own capacity cannot show excess capacity. "Hundreds of cycles" says little until you know what each cycle asked of the cell. - Fitted curves are not tests. Fig. 5 fits the voltage-vs-time data with exponentials whose time constants are free parameters. A model with as many knobs as data points will fit; it is evidence only if it predicts something new.

Genuinely open. - Where the roughly 36 mAh actually came from: sulfur reduction, the glass, water or hydroxide, carbon, the leaky seal, or something unrecognised. - Whether the glass conducts as claimed, and what it really is (the 2014–2017 papers and the Graz paper were not read here). - Why the Fig. 1 voltage jumps about 1 V above the open-circuit voltage (from about 1.65 V to 2.68 V) once discharge current flows. In an ordinary cell the voltage under load is below open circuit [memory, high]. The paper says current lowers C_C; ordinary candidates (a resistive surface film broken down by current, a contact or instrument effect) are untested hypotheses. - What is reduced at about 2 V vs Na in the Na–ferrocene cell. Ferrocene's usual reaction is its oxidation, Fc → Fc⁺, near 3.1 V vs Na [memory, medium], about 1 V above where this cell works, and "ferrocene was not oxidized" says nothing about a discharge, which is a reduction. The charge–discharge gap also widens from about 0.15–0.4 V to about 0.7 V by cycle 200 [read: page image p.6], a degradation sign the paper does not mention. - The decisive test is cheap: a full mass balance on a discharged Li–S cell (measure metallic lithium on the cathode side by reacting it with water and collecting the hydrogen, or by ⁷Li NMR), plus control cells with no redox center and a Cu‖Cu dummy, all from a lab outside the authors' group.

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8. Check yourself

1. A cell has lithium metal on both electrodes and the same electrolyte throughout. Using ΔG = −nFE, what open-circuit voltage do you expect, and why?
2. Using the numbers in Section 5, how many coulombs would a 1.5 cm² double layer at 50 µF/cm² hold at 2.65 V? What fraction of the 129 C delivered is that?
3. A battery delivers 3.48 Ah per gram of Li and its voltage falls steadily from 2.68 V to 2.34 V, then drops quickly to 1.6 V. Without integrating, why can it not reach 10.5 Wh per gram of Li?
4. The paper fixes the electrode potentials so that their difference is 2.65 V and also says Q is constant while V = Q/C_eq changes with C_C. Show that if Q is chosen to equalise the fixed potentials, V does not depend on C_eq.
5. A glass has a 0.06 eV hopping barrier. By roughly what factor does its conductivity fall between 25 °C and −30 °C (ignore the 1/T prefactor)?

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9. Further reading

Mainstream. - A. J. Bard and L. R. Faulkner, Electrochemical Methods: Fundamentals and Applications (Wiley). Cell thermodynamics, Nernst, double layer. - P. W. Atkins and J. de Paula, Physical Chemistry (Oxford). Electrochemistry chapters. - D. J. Griffiths, Introduction to Electrodynamics, chapters 2 and 4. Capacitors and dielectrics. - D. V. Schroeder, An Introduction to Thermal Physics. Gibbs free energy. - R. A. Huggins, Advanced Batteries: Materials Science Aspects (Springer). Battery thermodynamics and solid electrolytes. - J. B. Goodenough and Y. Kim, "Challenges for rechargeable Li batteries", Chem. Mater. 22, 587 (2010) [memory, high]. Goodenough's own statement of the band-gap stability picture. - M. Lacey, "On the skepticism surrounding the 'Goodenough battery'" (blog, 2017; one critic, not peer reviewed): https://lacey.se/2017/03/29/on-the-skepticism-surrounding-the-goodenough-battery - J. B. Goodenough and Y. Kim (above) and Kittel (Section 2) for Fermi levels; for contact potentials and work functions, any solid-state text (Ashcroft and Mermin, Solid State Physics, chapter on surface effects) [memory, high]. - Comment and reply on this paper in Energy Environ. Sci. (about 2018): recalled [memory, medium], not verified; look them up on pubs.rsc.org under doi:10.1039/c6ee02888h. - Hanghofer et al., Chem. Mater. 30, 8134 (2018), on Li₃OCl and Li₂OHCl synthesis and air sensitivity: https://pubs.acs.org/doi/10.1021/acs.chemmater.8b02568 (not read here; access failed).

The researchers' own work. - Braga, Grundish, Murchison, Goodenough, "Alternative strategy for a safe rechargeable battery", Energy Environ. Sci. (2016), doi:10.1039/c6ee02888h (the document reviewed here). - Braga et al., J. Mater. Chem. A (2014), Li₃ClO-type glasses (cited as ref. 1; not read). - Braga et al., Energy Environ. Sci. 9, 948 (2016), glass electrolytes and symmetric cells (ref. 2; not read). - Braga et al., J. Electrochem. Soc. 164, A1 (2017), electric dipoles in the Na glass (ref. 3; not read).

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Answers

1. 0 V. The net reaction Li → Li has the same substance in the same state at start and end, so ΔG = 0 and E = −ΔG/nF = 0. Any small reading would come from surface or strain effects, of the order of millivolts to tens of millivolts, and could not be sustained.
2. C = 50 µF/cm² × 1.5 cm² = 75 µF; Q = CV = 75×10⁻⁶ × 2.65 = 2.0×10⁻⁴ C. That is 2.0×10⁻⁴ / 129 ≈ 1.5×10⁻⁶, about a millionth of the charge delivered.
3. Energy per gram = (Ah/g) × (average V). The average voltage is below the maximum, 2.68 V, so the energy is below 3.48 × 2.68 = 9.3 Wh/g. Reaching 10.5 needs an average of 3.02 V.
4. If Q is the charge that makes the voltage across the series pair equal the fixed difference Δμ/e = 2.65 V, then Q = C_eq × 2.65 V. Then V = Q/C_eq = 2.65 V for any C_eq: changing C changes Q, not V. The paper's claim that V changes with C at fixed Q needs Q to be fixed independently, which contradicts fixing the potentials.
5. k_B = 8.617×10⁻⁵ eV/K. 1/k_BT at 298 K is 38.9 /eV; at 243 K it is 47.8 /eV. Ratio = exp[0.06 × (47.8 − 38.9)] = exp(0.53) ≈ 1.7. So conductivity falls by only about 40 % from 25 °C to −30 °C. That is why the authors say the low barrier promises good low-temperature operation, if the barrier is real.