Problem 84 - Chemistry: The Central Science

Question

The galvanizing of iron sheet can be carried out electrolytically using a bath containing a zinc sulfate solution. The sheet is made the cathode, and a graphite anode is used. Calculate the cost of the electricity required to deposit a 0.49-mm layer of zinc on both sides of an iron sheet \(2.0 \mathrm{~m}\) wide and \(80 \mathrm{~m}\) long if the current is \(30 \mathrm{~A}\), the voltage is \(3.5 \mathrm{~V}\), and the energy efficiency of the process is \(90 \%\). Assume the cost of electricity is \(\$ 0.082\) per kilowatt hour. The density of zinc is \(7.1 \mathrm{~g} / \mathrm{cm}^{3}\).

Step-by-Step Solution

Answer

The cost of the electricity required to deposit a 0.49-mm layer of zinc on both sides of an iron sheet 2.0 m wide and 80 m long is approximately $1,060.19.

Step 1: 1. Calculating the volume of zinc required

As we need to deposit a 0.49-mm layer of zinc on both sides of an iron sheet with the dimensions of 2.0 m wide and 80 m long, let's first calculate the volume of zinc required for one side. Area of the iron sheet = width × length = 2.0 m × 80 m = 160 m² We'll now convert the thickness of the zinc layer to meters for ease of calculation: 0.49 mm = 0.00049 m Now, let's calculate the volume of the zinc layer for one side: Volume of zinc (one side) = Area × thickness = 160 m² × 0.00049 m = 0.0784 m³ As we need to deposit zinc on both sides of the sheet, we'll multiply the volume by 2 to obtain the total volume of zinc required: Total volume of zinc = 0.0784 m³ × 2 = 0.1568 m³

Step 2: 2. Calculate the mass of the deposited zinc

We are given the density of zinc, which is 7.1 g/cm³. First, we'll convert the density to kg/m³ for ease of calculation: Density of zinc (kg/m³) = 7.1 g/cm³ × (1000 kg/1 g) × (1 m³/1,000,000 cm³) = 7100 kg/m³ Now, using the volume of zinc calculated in step 1, we can find the mass of the deposited zinc: Mass of zinc = density × volume = 7100 kg/m³ × 0.1568 m³ ≈ 1113.28 kg

Step 3: 3. Calculate the charge required to deposit the zinc

To calculate the charge required to deposit the zinc, we will use Faraday's Law of Electrolysis: \[Q = \frac{m \times F}{M}\] Where Q is the charge in Coulombs, m is the mass of zinc in kg, F is Faraday's constant (96485 C/mol), and M is the molar mass of zinc (65.38 g/mol). First, let's convert the mass of zinc to grams: 1113.28 kg × 1000 g/kg = 1,113,280 g Now, we can calculate the required charge (Q): Q = (1,113,280 g) × (96485 C/mol) / (65.38 g/mol) ≈ 1.658 × 10⁸ C

Step 4: 4. Calculate the energy used in the process

We are given the amount of electricity used by the process and the energy efficiency of the process (90%). We can calculate the energy consumed in kilowatt-hours using the formula: Energy (kWh) = (Voltage × Current × Time × Efficiency) / 3600 First, let's calculate the time required for the process: Time (seconds) = Charge (Coulombs) / Current (Amperes) Time = 1.658 × 10⁸ C / 30 A ≈ 5.527 × 10⁶ s Now we can calculate the energy consumed: Energy (kWh) = (35 V × 30 A × 5.527 × 10⁶ s × 0.9) / 3600 ≈ 1.293 × 10⁴ kWh

Step 5: 5. Calculate the cost of the electricity required

Finally, we are given the cost of electricity, which is $0.082 per kWh. We can now calculate the cost of the electricity required to deposit the zinc: Cost = Energy (kWh) × Price per kWh Cost = 1.293 × 10⁴ kWh × \(0.082/kWh ≈ \)1,060.19 Therefore, the cost of the electricity required to deposit a 0.49-mm layer of zinc on both sides of an iron sheet 2.0 m wide and 80 m long is approximately $1,060.19.

Key Concepts

Faraday's Law of Electrolysis

Electrolytic galvanization, a process you might encounter in electrochemistry, often involves the application of Faraday's Law of Electrolysis. This fundamental law helps us understand how electric charge relates to chemical change during electrolysis.

Essentially, Faraday's Law states that the amount of chemical change is directly proportional to the total electric charge that flows through an electrolyte. This means that by knowing the quantity of electricity used (charge in Coulombs), we can determine the amount of substance that has been oxidized or reduced at each electrode during electrolysis.

In practical terms, to deposit a certain amount of metal onto a surface, like zinc on iron for galvanization, we need to know the equivalent charge required. The formula derived from Faraday's Law is: \[ Q = \frac{m \times F}{M} \] where:

• \( Q \) is the charge in Coulombs,
• \( m \) is the mass of the metal to be deposited in grams,
• \( F \) is Faraday’s constant (96485 C/mol), and
• \( M \) is the molar mass of the metal (in this case, zinc, which is 65.38 g/mol).

With these variables, one can calculate the exact charge needed to deposit a layer of metal, which then translates to the electrical energy required for the electrolytic process.

Electrochemistry

Electrochemistry, the study of the relationship between chemical changes and electrical energy, becomes quintessential when working with electrolytic galvanization. This branch of chemistry explores how electrical currents drive chemical reactions and vice versa. The process relies on an electric current to induce a redox reaction which would not spontaneously occur in an electrolytic cell.

In the context of galvanization, the iron sheet acts as the cathode where zinc ions from the zinc sulfate solution are reduced and deposited as a solid metal layer. The graphite anode is where oxidation occurs. The current and voltage required for this process are crucial parameters, as they dictate the rate of zinc deposition and, by extension, the efficiency and cost of galvanization.

The energy efficiency of the electrolytic galvanization process is also vital. It reflects how much of the electrical energy is actually used for the deposition of zinc. In our scenario, an efficiency of 90% means that 10% of the energy is lost and not used for galvanization, which is common in real-world applications due to heat loss and other factors.

Stoichiometry

Stoichiometry, a central concept in chemistry and particularly critical in electrochemistry, involves the quantitative relationships between the reactants and products in a chemical reaction. When applying stoichiometry to electrolytic galvanization, it helps in determining the precise amount of zinc that needs to be deposited to achieve a desired thickness of the protective layer.

By calculating the volume of the zinc layer needed, using the area and thickness of the iron sheet, and the density of zinc, stoichiometry allows us to determine the mass of zinc that must be deposited. With this mass and knowing the molar mass of zinc, one can use stoichiometry to calculate the number of moles and then apply Faraday’s Law to find the charge necessary for the galvanization process.

Ultimately, stoichiometry bridges the gap between the macroscopic level of mass and volume and the microscopic level of moles and Faraday's Law, culminating in the practical assessment of the cost of electricity needed for the electrolytic deposition of zinc.