Problem 73 - Fundamentals of Physics

Question

A coil of current-carrying Nichrome wire is immersed in a liquid. (Nichrome is a nickel-chromium-iron alloy commonly used in heating elements.) When the potential difference across the coil is \(12 \mathrm{~V}\) and the current through the coil is \(5.2 \mathrm{~A}\), the liquid evaporates at the steady rate of \(21 \mathrm{mg} / \mathrm{s} .\) Calculate the heat of vaporization of the liquid

Step-by-Step Solution

Answer

The heat of vaporization of the liquid is approximately 2971.43 J/g.

Step 1: Calculate the Power Dissipated by the Coil

To calculate the power dissipated by the Nichrome coil, we use the formula for electrical power: \( P = IV \), where \( I \) is the current and \( V \) is the voltage across the coil. Substitute the given values: \( I = 5.2 \, \text{A} \) and \( V = 12 \, \text{V} \). So, \( P = 5.2 \times 12 = 62.4 \, \text{W} \).

Step 2: Determine the Energy Provided per Second

The power calculated in Step 1, \( 62.4 \, \text{W} \), is the energy provided per second for the evaporation process since 1 Watt is equivalent to 1 Joule per second.

Step 3: Calculate the Mass Evaporated per Second

The problem states that the liquid evaporates at a steady rate of \( 21 \, \text{mg/s} \), which is \( 0.021 \, \text{g/s} \) when converted to grams per second.

Step 4: Calculate the Heat of Vaporization

The heat of vaporization \( L \) is defined as the amount of energy required to convert 1 g of liquid into vapor without a temperature change. Using the power provided (energy per second) and the mass evaporated per second, we have: \( L = \frac{\text{Energy per second}}{\text{Mass evaporated per second}} \). Thus, \( L = \frac{62.4}{0.021} = 2971.43 \, \text{J/g} \).

Key Concepts

Nichrome wire

Nichrome wire is a special type of wire used in heating applications. It is an alloy composed mainly of nickel, chromium, and iron. One of the main reasons Nichrome is used in situations that require controlled heat, like the exercise, is because of its high resistance to oxidation and its ability to withstand high temperatures without melting.

• Durability: Nichrome can maintain its toughness and strength, even at high temperatures.
• Resistance: It has a relatively high electrical resistance, which leads to the generation of heat as an electric current passes through it.
• Stability: It doesn’t corrode or oxidize easily, making it reliable for long-term use in heating elements.

The wire heats up when an electrical current flows through. In the exercise, this heat is transferred to a liquid, causing it to evaporate.

Electrical power

Electrical power is fundamental in understanding how the heating of the Nichrome wire occurs. It is the rate at which electrical energy is transferred or converted.
In the exercise, the electrical power (\( P \)) is calculated using the formula \( P = IV \), where \( I \) is the current in amperes and \( V \) is the voltage in volts.

• Current is a flow of electric charge, and it is given as \( 5.2 \) A in this example.
• Voltage is the potential difference that pushes the current through the wire, here it is \( 12 \) V.
• Power is the energy used per second, calculated here as \( 62.4 \) W, or \( 62.4 \) Joules per second.

This concept is integral because it shows how much energy is available for the evaporation process, directly affecting the rate at which the liquid turns into vapor.

Evaporation process

Evaporation is the process where liquid turns into vapor. In the exercise, this occurs thanks to the heat produced by the Nichrome wire. The temperature increase causes molecules in the liquid to gain energy, break free from the surface, and become gas.
A steady rate of \( 21 \) mg/s of the liquid is evaporated, which translates to \( 0.021 \) g/s.

• Temperature doesn’t change significantly in this process because the energy is used to change the state, not increase temperature.
• Steady rates indicate a balance of energy in the system.
• This process showcases how energy can change a substance’s state from liquid to gas.

Understanding evaporation is crucial to calculating the heat of vaporization, as it relates to how much energy is needed for this transition.

Energy conversion

Energy conversion is the process of changing energy from one form to another. In this example, electrical energy is converted into thermal energy by the Nichrome wire, which then drives the evaporation of the liquid.

• Electrical energy: The initial form of energy provided to the Nichrome wire through current and voltage.
• Thermal energy: Produced by the resistance in Nichrome, which heats up the wire.
• Phase change energy: The thermal energy is used to change the state of the liquid from liquid to gas.

This conversion is essential for the evaporation process because it determines how much of the wire's electrical input is available to convert the liquid state without significant temperature change. Calculating the heat of vaporization shows the efficiency of this energy conversion, emphasizing the energy needed to vaporize each gram of the liquid.