Question

The sodium-sulfur battery has molybdenum electrodes with anode and cathode compartments separated by \(\beta\) -alumina, a ceramic through which sodium ions can pass. Because the battery operates at temperatures above \(300^{\circ} \mathrm{C}\), all the reactants and products are present in a molten solution. The cell voltage is about \(2.0 \mathrm{~V}\). (a) What is the cell reaction if the shorthand notation is \(\mathrm{Mo}(s)\left|\mathrm{Na}(s o l n), \mathrm{Na}^{+}(s o l n) \| \mathrm{S}(s o l n), \mathrm{S}^{2-}(s o l n)\right| \mathrm{Mo}(s) ?\) (b) How many kilograms of sodium are consumed when a \(25 \mathrm{~kW}\) sodium- sulfur battery produces current for \(32 \mathrm{~min} ?\)

Short answer

(a) \( 2\mathrm{Na} + \mathrm{S} \rightarrow 2\mathrm{Na}^+ + \mathrm{S}^{2-} \). (b) 5.724 kg of sodium are consumed.

Step-by-step solution

Understand the Shorthand Notation

The shorthand notation for the cell is given as \( \mathrm{Mo}(s)|\mathrm{Na}(soln), \mathrm{Na}^{+}(soln) \| \mathrm{S}(soln), \mathrm{S}^{2-}(soln)| \mathrm{Mo}(s) \). This indicates the oxidation and reduction happening in the battery. \( \mathrm{Na} \) is oxidized to \( \mathrm{Na}^+ \) at the anode, and \( \mathrm{S} \) is reduced to \( \mathrm{S}^{2-} \) at the cathode. The electrodes are \( \mathrm{Mo} \), which do not participate in the reaction.

Write the Half-Reactions

At the anode, sodium \( (\mathrm{Na}) \) loses an electron to form sodium ions \( (\mathrm{Na}^+) \): \[ \mathrm{Na} \rightarrow \mathrm{Na}^+ + e^- \]At the cathode, sulfur \( (\mathrm{S}) \) gains electrons to form sulfide ions \( (\mathrm{S}^{2-}) \): \[ \mathrm{S} + 2e^- \rightarrow \mathrm{S}^{2-} \]

Write the Net Cell Reaction

To balance the electrons, we combine the half-reactions. Two moles of sodium react with one mole of sulfur.Combined Reaction: \[ 2\mathrm{Na} + \mathrm{S} \rightarrow 2\mathrm{Na}^+ + \mathrm{S}^{2-} \]

Understand Power Consumption

The battery is generating \( 25 \text{ kW} \) of power, equivalent to \( 25000 \text{ W} \). Power is the product of voltage and current, \( P = V \times I \). We must find the current \( I \) generated by the battery: \( I = \frac{P}{V} = \frac{25000}{2} = 12500 \text{ A} \).

Calculate Electric Charge from Current

Convert the operation time from minutes to seconds: \( 32 \text{ min} = 1920 \text{ s} \). Using \( Q = I \times t \), calculate the total charge \( Q \): \( Q = 12500 \text{ A} \times 1920 \text{ s} = 24000000 \text{ C} \).

Convert Charge to Moles of Electrons

Using Faraday's constant \( F = 96485 \text{ C/mol} \), convert charge to moles of electrons:\[ n = \frac{Q}{F} = \frac{24000000}{96485} \approx 248.8 \text{ mol} \].

Determine Moles of Sodium Consumed

From the balanced equation \( 2\mathrm{Na} \rightarrow 2\mathrm{Na}^+ + 2e^- \), each mole of sodium consumes one mole of electrons. Therefore, 248.8 moles of sodium are consumed.

Convert Moles of Sodium to Mass

The molar mass of sodium is \( 22.99 \text{ g/mol} \). Convert moles to mass:\[ \text{Mass of Na} = 248.8 \text{ mol} \times 22.99 \text{ g/mol} = 5724.212 \text{ g} = 5.724 \text{ kg} \].

Key concepts

Sodium-sulfur battery

The sodium-sulfur battery is a type of rechargeable battery known for its high-energy density. It operates at elevated temperatures, typically above 300°C, where the sodium and sulfur are in a molten state. The battery is designed with molybdenum electrodes, which do not actively participate in the electrochemical reaction. Instead, they serve as a solid support structure.

The key feature of the sodium-sulfur battery is the ceramic electrolyte, specifically β-alumina, which allows sodium ions (Na⁺) to pass through while separating the two compartments of the battery. This setup ensures the movement of ions while preventing the mixing of reactants, which is crucial for the battery's efficiency and longevity.

• Anode: Molten sodium (Na)
• Cathode: Molten sulfur (S)
• Electrolyte: β-alumina
• Voltage: Typically 2.0 V

This design provides an effective way to store and deliver large amounts of energy, making the sodium-sulfur battery suitable for power grid applications and electric vehicles.

Cell reaction

Understanding the cell reaction in an electrochemical cell is key to grasping how energy is produced. In a sodium-sulfur battery, the cell reaction refers to the chemical changes occurring at both the anode and cathode.

The shorthand notation illustrates the flow of electrons and ions between these two electrodes. At the anode, molten sodium (Na) gives up an electron, turning into sodium ions (March: Na⁺), a process known as oxidation. Meanwhile, at the cathode, sulfur elements and sodium ions participate in a redox reaction, where sulfur gains electrons to form sulfide ions (S^2-).

The overall cell reaction is balanced as follows:\[ 2\mathrm{Na} + \mathrm{S} \rightarrow 2\mathrm{Na}^+ + \mathrm{S}^{2-} \]This reaction depicts the transformation of sodium and sulfur into their ionized states and manages the electron equilibrium necessary for generating electrical energy.

Electrochemical cells

Electrochemical cells are fundamental components of batteries, converting chemical energy into electrical energy. They consist of two electrodes immersed in electrolyte solutions, where oxidation-reduction reactions occur.

There are two main types of electrochemical cells:

• Galvanic (Voltaic) Cells: These cells discharge energy and are used in batteries to power devices.
• Electrolytic Cells: These require external energy to drive reactions, often used in industrial processes.

In the case of the sodium-sulfur battery, it functions as a galvanic cell during discharge, where sodium's oxidation and sulfur's reduction naturally generate electricity.

This cell operates by maintaining an ionic flow within the electrolyte while preventing direct mixing of the reactants, crucial for maintaining a steady voltage and lasting battery performance.

Molar mass calculation

Molar mass calculation is essential when determining how much of a substance is involved in a chemical reaction. It relates the mass of a substance to its amount in moles.

For any element or compound, the molar mass is calculated by adding up the atomic masses of its constituent atoms. For example, the molar mass of sodium (Na) is 22.99 g/mol.

In the sodium-sulfur battery exercise, computing the mass of sodium consumed involves several steps:

• First, calculate the total charge (in coulombs) from the current and time the battery operates.
• Next, use Faraday's constant to convert this charge into moles of electrons.
• According to the balanced reaction, relate the moles of electrons to moles of sodium.
• Finally, multiply the moles of sodium by its molar mass to get the total mass in grams.

This serves not just to understand the specific application within the battery, but also as a general method for stoichiometric calculations in chemistry.