Question

Calculate how much energy \((\mathrm{kWh})\) is required to prepare a ton of sodium in a typical Downs cell operating at \(25,000 \mathrm{~A}\) and \(7.0 \mathrm{~V}\).

Short answer

The energy required is 630 kWh.

Step-by-step solution

Determine Energy Formula

We need to calculate energy in kilowatt-hours (kWh). The general formula for electrical energy is given by \( E = V \cdot I \cdot t \), where \( E \) is energy in watt-hours, \( V \) is voltage, \( I \) is current, and \( t \) is time in hours.

Calculate the Energy for One Second

First, let's convert the energy formula to calculate the energy per second. We substitute the given values of voltage and current into the formula: \[ E_{ ext{second}} = 7.0 ext{ V} \times 25,000 ext{ A} \times 1 ext{ s} = 175,000 ext{ J/s} = 175,000 ext{ W} \] This value is in watts, as joules per second is the definition of a watt.

Calculate Energy per Hour

To find out the energy consumed in one hour, we multiply the energy per second by the number of seconds in an hour: \[ E_{ ext{hour}} = 175,000 ext{ W} \times 3600 ext{ s/hour} = 630,000,000 ext{ J/hr} = 630,000 ext{ Wh} \].

Convert Energy to Kilowatt-hours

Now, convert watt-hours to kilowatt-hours since the question asks for energy in kWh: \[ E_{ ext{kWh}} = \frac{630,000}{1000} ext{ kWh} = 630 ext{ kWh} \].

Key concepts

Downs Cell

A Downs cell is an electrochemical cell used specifically for producing alkali metals like sodium. It is a type of electrolytic cell, meaning it uses an external source of electricity to drive a chemical reaction that wouldn't occur spontaneously.
This cell is vital for industrial applications, especially in the production of sodium, which is widely used in chemical industries.

• In a Downs cell, molten sodium chloride ( NaCl ) is electrolyzed to obtain sodium ( Na ) and chlorine gas ( Cl_2 ).
• The cell uses a high current (often around 25,000 Amperes) and a voltage of about 7 Volts to facilitate this reaction.
• The large current is necessary to overcome the chemical bonds in NaCl , allowing sodium and chloride ions to separate and form elemental sodium and chlorine gas.

The Downs cell operates at high temperatures because NaCl is in a molten state to allow the ions to move freely and the reactions to occur effectively. It showcases the importance of understanding how electrical energy is used in chemical manufacturing.

Kilowatt-Hours

Kilowatt-hours ( ext{kWh} ) are a unit of energy commonly used in the context of electricity consumption. It is crucial to understand this unit when calculating energy usage in devices and systems like the Downs cell.
One kilowatt-hour is equivalent to the amount of energy a device consuming 1,000 watts ( 1 ext{ kW} ) would use if it were running for one hour.

• This makes ext{kWh} a practical measure for energy usage, as it directly relates to how electricity is billed in households and industries.
• When calculating energy consumption in kWh, we use the formula E = P imes t , where E is energy in kWh, P is power in kilowatts, and t is time in hours.
• In the case of a Downs cell, this conversion is necessary to express the energy consumption in a more usable format like ext{kWh} from joules or watt-hours.

Energy conversion from smaller units like watt-hours to kilowatt-hours helps standardize measurements and makes comparisons between systems more straightforward.

Electricity Consumption

Electricity consumption refers to the amount of electrical energy used by a device, system, or facility over a specific period.
For systems like a Downs cell that operate at high amperage and voltage, understanding electricity consumption is vital.

• Electricity consumption can be quantified using the formula E = V imes I imes t , where V is voltage, I is current, and t is time. This provides energy in watt-hours.
• In industrial processes, knowing electricity consumption helps optimize processes, reduce costs, and manage energy resources efficiently.
• By converting this consumption into kilowatt-hours, companies can better understand their electricity usage over time and control operational expenses.

Calculating electricity consumption accurately is crucial for maintaining efficiency in production scales like those involving Downs cells, allowing for better resource management.