Problem 73 - Chemistry The Molecular Science

Question

In a Downs cell, molten \(\mathrm{NaCl}\) is electrolyzed to sodium metal and chlorine gas. $$2 \mathrm{NaCl}(\ell) \longrightarrow 2 \mathrm{Na}(\ell)+\mathrm{Cl}_{2}(\mathrm{~g})$$ \(\Delta_{\mathrm{r}} H^{\circ}\) and \(\Delta_{\mathrm{r}} S^{\circ}\) for the reaction are \(+820 \mathrm{~kJ} / \mathrm{mol}\) and \(+180 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1},\) respectively. (a) Calculate \(\Delta_{\mathrm{r}} G^{\circ}\) at \(600 .{ }^{\circ} \mathrm{C},\) the electrolysis temperature. (b) Calculate the voltage required for the electrolysis.

Step-by-Step Solution

Answer

(a) \( \Delta_{\mathrm{r}} G^{\circ} = 662833 \mathrm{~J/mol} \). (b) Required voltage is 3.43 V.

Step 1: Convert Temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin. The formula to convert Celsius to Kelvin is:\[ T(K) = T(°C) + 273.15 \]For 600 °C, the temperature in Kelvin is:\[ T(K) = 600 + 273.15 = 873.15 \]

Step 2: Use Gibbs Free Energy Formula

Use the Gibbs Free Energy equation to calculate \( \Delta_{\mathrm{r}} G^{\circ} \):\[ \Delta_{\mathrm{r}} G^{\circ} = \Delta_{\mathrm{r}} H^{\circ} - T \Delta_{\mathrm{r}} S^{\circ} \]Given:\( \Delta_{\mathrm{r}} H^{\circ} = +820 \mathrm{~kJ} / \mathrm{mol} = 820000 \mathrm{~J} / \mathrm{mol} \) (since we need unit consistency)\( \Delta_{\mathrm{r}} S^{\circ} = +180 \mathrm{~J} / \mathrm{K} / \mathrm{mol} \)\( T = 873.15 \mathrm{~K} \)Plug in the values:\[ \Delta_{\mathrm{r}} G^{\circ} = 820000 \mathrm{~J/mol} - (873.15 \mathrm{~K} \times 180 \mathrm{~J/K/mol}) \]\[ = 820000 - 157167 = 662833 \mathrm{~J/mol} \]

Step 3: Convert Gibbs Free Energy to Voltage

To find the voltage required for electrolysis, use the relation:\[ \Delta_{\mathrm{r}} G^{\circ} = -nFE \]where \( n = 2 \) (moles of electrons transferred per formula unit of \( \mathrm{NaCl} \)), \( F = 96485 \mathrm{~C/mol} \) (Faraday's constant), and \( E \) is the cell voltage.Rearrange to solve for \( E \):\[ E = -\frac{\Delta_{\mathrm{r}} G^{\circ}}{nF} \]Substitute the values:\[ E = -\frac{662833 \mathrm{~J/mol}}{2 \times 96485 \mathrm{~C/mol}} \]\[ = -\frac{662833}{192970} \]\[ \approx -3.43 \mathrm{~V} \]

Step 4: Interpret the Results

Since we calculated a negative value of \( E \), it indicates that the reaction is non-spontaneous and thus external energy is required to drive the reaction during electrolysis. The minimum required voltage will be positive 3.43 V for electrolysis to occur.

Key Concepts

Downs Cell

A Downs Cell is an industrial apparatus used for the electrolysis of molten sodium chloride (\( \text{NaCl} \)) to produce sodium metal and chlorine gas. This cell is specially designed to handle high temperatures because the process involves electrolyzing molten \( \text{NaCl} \). Here's what happens inside a Downs Cell:

• Molten sodium chloride is used because the ions are free to move, allowing electric current to flow.
• When electrical energy is supplied, \( \text{Na}^+ \) ions move towards the cathode, where they gain electrons and form sodium metal.
• Simultaneously, \( \text{Cl}^- \) ions move towards the anode, where they lose electrons and form chlorine gas.

This process facilitates the separation of sodium and chlorine in their elemental form, which are both valuable resources in various industries. Downs Cell is a significant method for producing sodium metal efficiently.

Gibbs Free Energy

Gibbs Free Energy (\( \Delta_{ ext{r}} G^{\circ} \)) is a measure of the maximum reversible work that a thermodynamic system can perform at constant temperature and pressure.It helps to determine whether a reaction is spontaneous. In the context of a Downs Cell, it's crucial because:

• If \( \Delta_{ ext{r}} G^{\circ} \) is negative, the reaction can occur spontaneously without additional energy.
• If positive, as in the case of a Downs Cell at high temperatures, \( \Delta_{ ext{r}} G^{\circ} \) indicates that external energy is needed for the reaction to proceed.

The formula used in calculations is:\[\Delta_{ ext{r}} G^{\circ} = \Delta_{ ext{r}} H^{\circ} - T \Delta_{ ext{r}} S^{\circ}\]Here, \( \Delta_{ ext{r}} H^{\circ} \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta_{ ext{r}} S^{\circ} \) is the change in entropy.For the given reaction, calculating \( \Delta_{ ext{r}} G^{\circ} \) at 600°C helps to determine how much electrical energy we need to supply to make the electrolysis happen in a Downs Cell.

Electrolysis

Electrolysis is a chemical process wherein electric current passes through a substance, causing it to break down into its components. For the reaction in a Downs Cell, electrolysis is key in the transformation of \( \text{NaCl} \) into sodium metal and chlorine gas. During electrolysis:

• The cathode attracts cations, where reduction occurs. In the Downs Cell, sodium ions get reduced to form sodium metal.
• The anode attracts anions, where oxidation happens. Here, chloride ions are oxidized to release chlorine gas.

Electrolysis fundamentally requires electrical energy to drive non-spontaneous reactions. This energy input ensures that the reaction can proceed, even when natural conditions would not favor it. In the case of a Downs Cell, the required voltage calculated tells us exactly how much energy it takes to make \( \text{Na} \) and \( \text{Cl}_2 \) separate from molten \( \text{NaCl} \). This process is vital for industrial production, especially when chemicals cannot be obtained through simpler reactions.