Question

A construction company is installing an iron culvert (a long cylindrical tube) that is \(40.0 \mathrm{~m}\) long with a radius of \(0.900 \mathrm{~m}\). To prevent corrosion, the culvert must be galvanized. This process is carried out by first passing an iron sheet of appropriate dimensions through an electrolytic cell containing \(\mathrm{Zn}^{2+}\) ions, using graphite as the anode and the iron sheet as the cathode. If the voltage is \(3.26 \mathrm{~V}\), what is the cost of electricity for depositing a layer \(0.200 \mathrm{~mm}\) thick if the efficiency of the process is 95 percent? The electricity rate is \(\$ 0.12\) per kilowatt hour \((\mathrm{kWh})\), where \(1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s}\) and the density of \(\mathrm{Zn}\) is \(7.14 \mathrm{~g} / \mathrm{cm}^{3}\).

Short answer

The cost of electricity is $0.11.

Step-by-step solution

Compute the Surface Area of the Culvert

The culvert is a long cylindrical tube. The formula for the surface area of a cylinder without the ends is given by \(A = 2 \pi r h\), where \(r\) is the radius and \(h\) is the height (or length) of the cylinder. Here, \(r = 0.900\, \mathrm{m}\) and \(h = 40.0\, \mathrm{m}\). Compute the surface area:\[A = 2 \pi (0.900) (40.0) \approx 226.19\, \mathrm{m}^2\]

Convert Thickness to Meters and Calculate Volume of Zinc Layer

The thickness of the zinc layer is given as \(0.200 \mathrm{~mm}\). Convert this into meters, \(0.200 \mathrm{~mm} = 0.0002 \mathrm{~m}\). The volume of the zinc layer is the surface area times the thickness:\[V = 226.19 \times 0.0002 = 0.045238\, \mathrm{m}^3\]

Convert Volume to Mass of Zinc

Now convert the volume to mass using the density of zinc \(7.14 \mathrm{~g} / \mathrm{cm}^{3}\). Remember to convert this to \(\mathrm{kg} / \mathrm{m}^{3}\), noting that \(1\, \mathrm{g} / \mathrm{cm}^3 = 1000\, \mathrm{kg} / \mathrm{m}^3\), so \(7.14\, \mathrm{g}/\mathrm{cm}^{3} = 7140\, \mathrm{kg}/\mathrm{m}^{3}\):\[m = 0.045238 \times 7140 = 323.79732\, \mathrm{kg}\]

Compute the Energy Required for Galvanization

Calculate the total energy needed using the formula \(E = \frac{m}{e \cdot \eta} \cdot \Delta z\), where \(e =\) the electrochemical equivalent of Zn (found using the molar mass and Faraday's constant), \(\eta = 0.95\) (efficiency), and \(\Delta z =\) the ionization of Zn, which is 2 (since Zn loses 2 electrons to form \(\mathrm{Zn}^{2+}\)). First calculate \(e\) using:\[\text{Molar mass of Zn} = 65.38\, \mathrm{g/mol},\]\[\text{Faraday's constant} = 96,485\, \mathrm{C/mol}\]\[e = \frac{65.38}{96485 \times 2} = 0.000338\, \mathrm{kg/C}\]Then, calculate energy required:\[E = \frac{323.79732}{0.000338 \times 0.95} = 995328\, \mathrm{C}\]

Calculate Total Cost of Electricity

Convert the charge to kilowatt-hours:\[\text{Energy in Joules} = Q \times \text{Voltage} = 995328 \times 3.26 \approx 3247582\, \mathrm{J}\]Convert to kWh:\[\text{kWh} = \frac{3247582}{3.6 \times 10^6} = 0.902\, \mathrm{kWh}\]Finally, calculate the cost:\[\text{Cost} = 0.902 \times 0.12 = 0.108\, \$\]

Key concepts

Electrolytic cell

In a galvanization process, an electrolytic cell is crucial to depositing a protective layer of zinc onto the iron surface. The electrolytic cell consists of two main components - the anode, which is made of graphite, and the cathode, which is the iron sheet being galvanized. Through this cell, an electric current is passed, facilitating the movement of zinc ions ( \( \text{Zn}^{2+} \)) from the solution to the iron surface.
This process involves applying a voltage, in our case, \( 3.26 \, \text{V} \), to drive the reduction reaction that plates zinc onto the iron.

• The iron sheet serves as the cathode, accepting electrons to attract zinc ions.
• The graphite anode helps in completing the electrical circuit by allowing electrons to flow away.
• Zinc ions gain electrons and get deposited as solid zinc on the iron surface.

During galvanization, maintaining an efficient electrolytic cell process is vital to minimize energy waste and maximize zinc adherence to the culvert surface.

Corrosion prevention

Corrosion prevention is essential for maintaining the integrity of metal structures over time. One effective method is through galvanization, where iron structures, such as culverts, are coated with a layer of zinc. This zinc layer acts as a protective barrier, preventing corrosive substances from reaching the iron.
Here are several ways in which zinc prevents corrosion:

• Barrier Protection: Zinc provides a physical shield that keeps moisture and oxygen from reaching the iron surface.
• Cathodic Protection: Zinc is more reactive than iron, which means it will corrode preferentially. This sacrificial action preserves the underlying iron.
• Reduced Contact with Corrosive Agents: By covering the iron with zinc, the area exposed to potentially corrosive elements is minimized.

Overall, galvanization extends the life of metal structures while providing a maintenance-free solution against rust.

Zinc plating

Zinc plating is the application of a protective zinc coating applied to iron, using an electrolytic process. This plating forms a corrosion-resistant barrier and enhances the metal's appearance. During the plating process, several factors are considered:

• Thickness of Zinc Layer: For our culvert, a layer of \(0.200 \, \text{mm} \) is applied. The thickness is crucial for ensuring adequate protection and varies based on environmental exposure.
• Plating Efficiency: The efficiency of the plating process impacts the energy consumption and cost. A 95% efficiency indicates minimal energy loss, optimizing resource use.
• Surface Uniformity: A smooth and even zinc layer prevents weaknesses that could compromise protection.

Zinc plating through galvanization remains a widely adopted method due to its cost-effectiveness and durability in protecting metal surfaces.

Surface area calculation

Calculating the surface area of a structure is fundamental in determining the amount of zinc required for galvanization. For a cylindrical object like the culvert, the surface area is essential for assessing both material needs and costs.
In the given exercise, the surface area \( A \) of the culvert without the ends is calculated using the formula: \[ A = 2 \pi r h \]where \( r \) is the radius and \( h \) is the height (or length). For our culvert, \( r = 0.900 \, \text{m} \) and \( h = 40.0 \, \text{m} \).
This results in a surface area of approximately \( 226.19 \, \text{m}^2 \). Knowing the surface area helps us calculate the volume of zinc required, which directly impacts the cost and energy calculation in the galvanization process.