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Problem 1. Sodium-ion battery Sodium-ion batteries are a class of electrochemical systems that are being researched as alternatives to lithium-ion batteries in certain energy storage applications. In these batteries, sodium ions are the mobile and electrochemically active species. a) Sodium ion battery cathodes researched today are typically transition metal oxides. Examples of such cathodes include NaTiO2, NaMnO2, Nao.7VO2. Write down the battery charging reactions for these materials. What chemical transitions define the potential of these reactions? Compute the theoretical gravimetric capacity (in mAh/g) of these materials. b) Alloys of various materials with sodium are among the most-researched materials for use as sodium-ion baHittery anodes. Examples of materials (in the charged state) include NaisSn4, Na3P, and Na3Sb. Write down the battery charging reactions for these materials. What chemical transition defines the potential of these reactions? Compute the theoretical gravimetric capacity (in mAh/g) of these materials. c) The theoretical capacity of an entire battery is not equal to the capacity of the individual cathode or anode materials. Develop an analytical expression for the theoretical capacity (in mAh/g) of an entire, perfectly balanced battery, given the theoretical capacities of the anode and cathode materials. Ignore any mass added to a battery by the electrolyte, separators, current collectors, etc. What combination of anode and cathode materials from parts a) and b) will give a battery with the highest theoretical capacity? What is the value of this capacity? d) The theoretical capacity of a Li-based batteries can go up to 700 mAh/g. How does this value compare to the theoretical capacity of a Na-ion battery? When would it be advantageou to use a Na-ion battery over a Li-based battery? Problem 3. Nitrogen Cycle Nitrogen-containing compounds are key to life on earth, and one of the most important chemical reactions is the conversion of dinitrogen to ammonia. When conducted thermochemically, this reaction is known as the Haber-Bosch reaction: In this problem, we will explore the thermodynamics of this reaction - you may need to refresh yourself on chemical thermodynamics, which will be helpful as we derive the electrochemical analogs of familiar expressions in class during the next few lectures. a) Assuming a stoichiometric feed of nitrogen and hydrogen, what is the equilibrium conversion of nitrogen to ammonia at 25°C and 1 atm (i.e., what fraction of nitrogen will be converted to products)? Why is the Haber-Bosch reaction not operated near 25°C and 1 atm? What pressure is necessary to convert 50% of the input nitrogen to ammonia at the realistic operating temperature of 450°C (assume constant enthalpy of reaction)? b) Unfortunately, one of the problems with the Haber-Bosch reaction is the hydrogen feed; the production of high purity hydrogen produces a lot of carbon dioxide. An intriguing idea would be to convert nitrogen to ammonia using water as a hydrogen source. Why is this reaction impossible at reasonable temperatures, under 1000°C? You can assume the water is always a gas since it will be for the majority of the temperature range 25°C- 1000 °C. c) Now, instead of reacting nitrogen to ammonia in a typical chemical reactor, you decide to use electrochemistry. Write out the half reactions in acidic electrolyte for the cathode and anode in the following two cases: (i) a nitrogen and hydrogen feed; (ii) a nitrogen and water feed. What do you notice about the cathode half reactions with the different feeds? d) *Repeat Part C for basic conditions. e) For now, we are only interested in the overall reaction. Given a feed of nitrogen and water, what is the equilibrium potential at 25°C? What other electrochemical reaction occurs near this potential? (hint: it involves one of the reactants) Why might this be a problem? f) *Given a feed of nitrogen and hydrogen, what is the equilibrium potential at 25°C? What does your result mean, physically, in particular when compared to the result from Part E?

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