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Schola Gregoriana Pragensis (English: The Gregorian School of Prague) is an a cappella male voice choir from the Czech Republic, founded in 1987 by David Eben. Their core repertoire consists of Gregorian chant, Bohemian plainchant, and early polyphony, but they also perform modern works including some composed for them.

True sandbox
$$ \eta = A \times \ln \displaystyle \left (\frac {i} {i_0} \right)$$ =Lithium batteries copy= https://www.coursehero.com/file/pa0ffb/recently-in-hybrid-electric-vehicles-the-batterys-relatively-low-energy-and/

My References
Bergveld, H.J., Kruijt W.S. and Notten P.H.L. (2002) Battery Management Systems: Design by Modelling, Springer, pp107-108,113. http://www.springer.com/gp/book/9781402008320

Dhameja, S. (2001) Electric Vehicle Battery Systems, Newnes, p12. https://www.elsevier.com/books/electric-vehicle-battery-systems/dhameja/978-0-7506-9916-7

Electrochemistry
The participants in the electrochemical reactions in a lithium-ion battery are the negative and positive electrodes with the electrolyte providing a conductive medium for lithium ions to move between the electrodes.

Both electrodes allow lithium ions to move in and out of their interiors. During insertion (or intercalation) ions move into the electrode. During the reverse process, extraction (or deintercalation), ions move back out. When a lithium-ion based cell is discharging, the positive lithium ion moves from the negative electrode (usually graphite = "$$\mathrm{C_6}$$" below) and enters the positive electrode (lithium containing compound). When the cell is charging, the reverse occurs.

Useful work is performed when electrons flow through a closed external circuit. The following equations show one example of the chemistry.

The cathode (marked +) half-reaction is within a substrate of lithium-doped cobalt oxide as follows:

$$\mathrm{CoO_2}+\mathrm{Li^+}+\mathrm{e^-}\leftrightarrows\mathrm{LiCoO_2}$$

The anode (marked -) half reaction is: $$\mathrm{LiC_6}\leftrightarrows \mathrm{C_6}+\mathrm{Li^+}+\mathrm{e^-}$$

The full reaction being: $$\mathrm{LiC_6}+\mathrm{CoO_2}\leftrightarrows \mathrm{C_6}+\mathrm{LiCoO_2}$$

The overall reaction has its limits. Overdischarge supersaturates lithium cobalt oxide, leading to the production of lithium oxide, possibly by the following irreversible reaction:

$$\mathrm{Li^+} + \mathrm{e^-} + \mathrm{LiCoO_2} \rightarrow \mathrm{Li_2O} + \mathrm{CoO}$$

Overcharge up to 5.2 volts leads to the synthesis of cobalt(IV) oxide, as evidenced by x-ray diffraction: $$ \mathrm{LiCoO_2} \rightarrow \mathrm{Li^+} + \mathrm{CoO_2} +\mathrm{e^-} $$

In a lithium-ion battery the lithium ions are transported to and from the positive or negative electrodes by oxidizing the transition metal, cobalt (Co), in from  to  during charge, and reduced from  to  during discharge. The cobalt electrode reaction is only reversible for x < 0.5, limiting the depth of discharge allowable. This chemistry was used in the Li-ion cells developed by Sony in 1990.

The cell's energy is equal to the voltage times the charge. Each gram of lithium represents Faraday's constant/6.941 or 13,901 coulombs. At 3 V, this gives 41.7 kJ per gram of lithium, or 11.6 kWh per kg. This is a bit more than the heat of combustion of gasoline, but does not consider the other materials that go into a lithium battery and that make lithium batteries many times heavier per unit of energy.

Electrolytes
The cell voltages given in the Electrochemistry section are larger than the potential at which aqueous solutions will electrolyze.

Liquid electrolytes in lithium-ion batteries consist of lithium salts, such as lithium hexafluorophosphate, lithium tetrafluoroborate or lithium perchlorate in an organic solvent, such as ethylene carbonate, dimethyl carbonate, and diethyl carbonate. A liquid electrolyte acts as a conductive pathway for the movement of cations passing from the negative to the positive electrodes during discharge. Typical conductivities of liquid electrolyte at room temperature (20 C) are in the range of 10 mS/cm, increasing by approximately 30–40% at 40 C and decreasing slightly at 0 C.

The combination of linear and cyclic carbonates (e.g., ethylene carbonate (EC) and dimethyl carbonate (DMC)) offers high conductivity and SEI-forming ability. A mixture of a high ionic conductivity and low viscosity carbonate solvents is needed, because the two properties are mutually exclusive in a single material.

Organic solvents easily decompose on the negative electrodes during charge. When appropriate organic solvents are used as the electrolyte, the solvent decomposes on initial charging and forms a solid layer called the solid electrolyte interphase (SEI), which is electrically insulating yet provides significant ionic conductivity. The interphase prevents further decomposition of the electrolyte after the second charge. For example, ethylene carbonate is decomposed at a relatively high voltage, 0.7 V vs. lithium, and forms a dense and stable interface.

Composite electrolytes based on POE (poly(oxyethylene)) provide a relatively stable interface. It can be either solid (high molecular weight) and be applied in dry Li-polymer cells, or liquid (low molecular weight) and be applied in regular Li-ion cells.

Room temperature ionic liquids (RTILs) are another approach to limiting the flammability and volatility of organic electrolytes.