Problem 104 - Chemistry and Chemical Reactivity

Question

Sodium metal is produced by electrolysis of molten sodium chloride. The cell operates at \(7.0 \mathrm{V}\) with a current of \(25 \times 10^{3} \mathrm{A}\) (a) What mass of sodium can be produced in 1 hour? (b) How many kilowatt-hours of electricity are used to produce \(1.00 \mathrm{kg}\) of sodium metal \((1 \mathrm{kWh}=\) \(\left.3.6 \times 10^{6} \mathrm{J}\right) ?\)

Step-by-Step Solution

Answer

(a) 21.47 kg of sodium (b) 0.82 kWh of electricity

Step 1: Calculate Faraday's Constant

Faraday's constant is the charge of one mole of electrons, which we find by multiplying Avogadro's number by the charge of one electron: \[ F = N_A imes e = 6.022 imes 10^{23} ext{ mol}^{-1} imes 1.602 imes 10^{-19} ext{ C} = 96485 ext{ C/mol} \]

Step 2: Calculate the Moles of Electrons

Calculate the charge passed through the system in an hour. Find the moles of electrons transferred. The electric current is given, hence: \[ Q = I imes t = 25000 ext{ A} imes 3600 ext{ s} = 90000000 ext{ C} \] Then, calculate the number of moles of electrons: \[ n_e = \frac{90000000 ext{ C}}{96485 ext{ C/mol}} \approx 933.88 ext{ mol} \]

Step 3: Determine Moles of Sodium

The overall reaction in the electrolysis is: \[ 2 ext{NaCl} ightarrow 2 ext{Na} + ext{Cl}_2 \] For one mole of sodium to form, one mole of electrons is needed. So, the moles of sodium produced are equal to the moles of electrons consumed: \[ n_{ ext{Na}} = n_e = 933.88 ext{ mol} \]

Step 4: Calculate Mass of Sodium Produced

The molar mass of sodium is approximately \(22.99 \text{ g/mol}\). Therefore, convert moles of sodium to grams: \[ m_{ ext{Na}} = 933.88 ext{ mol} \times 22.99 \text{ g/mol} = 21467.53 \text{ g} \approx 21.47 \text{ kg} \]

Step 5: Calculate Energy for 1 kg of Sodium

Using the potential difference, calculate the energy: \[ E = V \times Q = 7.0 \text{ V} \times \frac{96485 \text{ C/mol} \times 43.50 \text{ mol}}{1} = 2935437.5 \text{ J} \approx 2.94 \text{ MJ}\]

Step 6: Convert Joules to kWh

Convert the energy from joules to kWh:\[ 1 ext{ kWh} = 3.6 \times 10^6 \text{ J} \] Thus, the energy required is: \[ \frac{2935437.5 \text{ J}}{3.6 \times 10^6 \text{ J/kWh}} \approx 0.82 \text{ kWh} \]

Key Concepts

Faraday's constant

Faraday's constant is a fundamental constant used in electrolysis and electrochemistry. It represents the total electric charge carried by one mole of electrons. To calculate Faraday's constant, we multiply Avogadro's number, which is approximately \(6.022 \times 10^{23} \text{ mol}^{-1}\), by the elementary charge of an electron, about \(1.602 \times 10^{-19} \text{ C}\). Hence, Faraday's constant is approximately \(96485 \text{ C/mol}\). This value helps us understand how much charge is needed to drive reactions involving moles of electrons in electrochemical cells, like those used in the production of sodium metal.

Moles of electrons

Moles of electrons are a vital component in understanding electrochemical reactions. The number of moles of electrons transferred during electrolysis relates directly to the quantity of substance being converted. By calculating the total charge transferred using the formula \(Q = I \times t\), where \(I\) is the current in amperes and \(t\) is the time in seconds, we can find the moles of electrons. Given \(Q\), we then use Faraday's constant to determine the moles:

• \(n_e = \frac{Q}{F}\)

This equation illustrates that the more current or longer time applied, the more moles of electrons are transferred, driving the formation of chemical products.

Sodium production

Sodium production through electrolysis involves the conversion of sodium chloride into sodium metal and chlorine gas, as per the equation:

• \(2 \text{NaCl} \rightarrow 2 \text{Na} + \text{Cl}_2\)

Each mole of sodium chloride produces one mole of sodium and releases one mole of chlorine gas. The process requires the transfer of one mole of electrons for each mole of sodium produced. Given the equivalence in moles between sodium and electrons, if \(933.88 \text{ moles}\) of electrons are transferred, the same number of moles of sodium, \(933.88 \text{ moles}\), would be formed. This fundamental understanding links the quantities of reactants to products via their electron exchange.

Energy calculation

Calculating the energy involved in electrolysis requires an understanding of the basic relationship between voltage, charge, and energy. The energy \(E\) used in producing a certain mass of sodium, measured in joules, can be calculated by the formula:

• \(E = V \times Q\)

Here, \(V\) is the potential difference, and \(Q\) is the total charge that has been used. For the production of sodium, the energy required reveals the efficiency and costs involved in the industrial process. For instance, using \(7.0 \text{ V}\) and the charge accrued over the reaction time, we convert that into energy consumed in megajoules (MJ) or kilowatt-hours (kWh) for practical and economic considerations.

Electrochemical reactions

Electrochemical reactions are the foundation of processes like sodium production by electrolysis. They involve redox reactions, where oxidation and reduction occur simultaneously. In electrolysis, an external voltage source drives the non-spontaneous reaction. Sodium ions \(\text{Na}^+\) are reduced to sodium metal at the cathode, while chloride ions \(\text{Cl}^-\) are oxidized to chlorine gas at the anode.

• Reduction: \(\text{Na}^+ + e^- \rightarrow \text{Na}\)
• Oxidation: \(2 \text{Cl}^- \rightarrow \text{Cl}_2 + 2e^-\)

Understanding these reactions helps in controlling and optimizing the production of materials like sodium and highlights the link between electron flow and chemical change.