Question $4(40$ points$)$

One type of Li$-$ion battery includes a graphite negative electrode and manganese dioxide spinel

positive electrode, described by the following reactions $($written in the discharge direction$)$:

$L{i}_x{C}_{6}xL{i}^{+}+x{e}^{--}+C_6(negative)$

$xL{i}^{+}+x{e}^{--}+M{n}_2{O}_{4}L{i}_{x}M{n}_2{O}_4(positive)$

The electrolyte consists of an organic solvent that contains a binary $LiP{F}_6$ lithium salt, where the

anion is $P{F}_6.$ Assume a one$-$dimensional cell with the cathode located at $x=0$ and the anode

located at $x=L.$ The following properties are known:

$D_{L{i}^{+}}=1\mathrm{.}8{10}^{-10}{m}^2{s}^{-1}$

$D_{P{F}_6-}=2\mathrm{.}6{10}^{-10}{m}^2{s}^{-1}$

$t_{+}=0\mathrm{.}4$

For our purposes here, we assume that the electrodes are flat surfaces, and that they are separated

by an electrolyte$-$containing separator that is $25m$ thick. The initial concentration of $LiP{F}_6$ in

the electrolyte is $1\mathrm{.}0M.$ The parameters given in the question are for transport in the separator.

a$)$ In which direction does the current in solution flow during discharge? Is this positive or

negative relative to the $x-$direction? $(10$ points$)$

b$)$ Where is the concentration highest, at $x=0$ or $x=L?(10$ points$)$

c$)$ What is the concentration difference across the separator if a cell with a $2cm5cm$

electrode is operated at a current of $10mA?$ Is this difference significant? $(15$ points$)$

d$)$ Briefly describe how you would calculate the cell potential for a constant current

discharge of this cell if the current is known and the concentration variation is known.

The potential that would be measured is between the current collectors of the cathode and

anode during discharge. You do not need to include all the equations that you would use.

The important thing is that you know and are able to identify the factors that contribute to

the measured cell potential, and that you know the process by which you might determine

their values. $(5$ points$)$