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A pot of water is boiling on a stove under one atmosphere of pressure. Assume that heat enters the pot only through its bottom, which is copper and rests on a heating element. In a certain time, a mass \(m\) of water boils away. (a) What is the temperature of the boiling water and does it change during this time? (b) What determines the amount of heat needed to boil the water? (c) Is the temperature of the heating element in contact with the pot greater than, smaller than, or equal to \(100{ }^{\circ} \mathrm{C} ?\) Explain. Problem In two minutes, the mass of water boiled away is \(m=0.45 \mathrm{~kg} .\) The radius of the pot bottom is \(R=6.5 \mathrm{~cm}\) and the thickness is \(L=2.0 \mathrm{~mm}\). What is the temperature \(T_{\mathrm{E}}\) of the heating element in contact with the pot? Verify that your answer is consistent with your answers to the Concept Questions.

Step by step solution

01

Understanding Boiling Point

The boiling point of water at one atmosphere of pressure is 100°C (373 K). Since the system is at one atmosphere and we assume no impurities or pressure change, the temperature remains constant at 100°C during boiling.

02

Determining Heat Needed to Boil Water

The heat required to boil away a mass of water, \( m \), is given by the formula \( Q = mL_v \), where \( L_v \) is the latent heat of vaporization for water, approximately \( 2260 \text{ kJ/kg} \). For \( m = 0.45 \text{ kg} \), the heat required \( Q = 0.45 \times 2260 = 1017 \text{ kJ} \).

03

Understanding the Role of Copper Pot

The temperature of the heating element must be higher than 100°C to account for heat transfer through the copper bottom. This temperature gradient is necessary for heat flow, facilitated by the thermal conductivity of copper.

04

Apply Heat Conduction through the Pot

Using Fourier's law of heat conduction, \( q = k \cdot A \cdot \frac{(T_E - T_W)}{L} \), where \( q = \frac{Q}{t} \) is the heat per unit time, \( k \) is the thermal conductivity of copper (approximately \( 400 \text{ W/mK} \)), \( A = \pi R^2 \) is the area, \( T_E \) is the temperature of the element, and \( T_W = 100°C \).

05

Calculating Heat Flux

Calculate the heat flux, \( q \), using the time \( t = 120 \text{ s} \) and the heat \( Q = 1017 \text{ kJ} = 1017000 \text{ J} \). \( q = \frac{1017000}{120} = 8475 \text{ W} \).

06

Calculating Temperature of Heating Element

Set up the equation \( 8475 = 400 \cdot \pi \cdot (0.065)^2 \cdot \frac{(T_E - 100)}{0.002} \). Solve for \( T_E \).

07

Solve for Temperature T_E

Simplify the equation to find \( T_E \). First, calculate the area \( A \): \( A = \pi \times (0.065)^2 = 0.01327 \text{ m}^2 \). Then, rearrange and solve: \( T_E = \frac{8475 \times 0.002}{400 \times 0.01327} + 100 \approx 103.3°C \).

08

Verify Consistency with Concept Questions

Confirm that the temperature of the heating element greater than 100°C is consistent with the requirement for heat flow from the element to the boiling water.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Boiling Point of Water

When water is heated, it reaches a specific temperature known as the boiling point, where it transitions from liquid to vapor. Under one atmosphere of pressure, the boiling point of water is precisely 100°C (212°F or 373 Kelvin). This is a key characteristic of water under normal atmospheric conditions.

During boiling, the temperature of the water does not rise beyond this point as long as the pressure remains constant. The heat energy added to the water at this stage is used to transform the water into vapor rather than increase the temperature.

To facilitate effective boiling, it is essential that the heat source maintains at least this temperature, ensuring the water continues to convert to steam until all heat energy transfers.

Heat Conduction

Heat conduction is the process through which heat transfers from a high temperature area to a low temperature area. In this scenario involving a boiling pot of water, heat flows from a heating element, through the pot's bottom, and into the water.

In this context, the bottom of the pot plays a critical role and is made of a highly conductive material like copper. Copper efficiently facilitates the transfer of heat due to its high thermal conductivity, approximately 400 W/mK.

This heat conduction process is described by Fourier's law, which states, \[q = k \cdot A \cdot \frac{(T_E - T_W)}{L} \]where:

• \( q \): heat per unit time
• \( k \): thermal conductivity of copper
• \( A \): area of the pot's bottom
• \( T_E \): temperature of the heating element
• \( T_W \): temperature of the boiling water
• \( L \): thickness of the pot bottom

This equation shows that the temperature of the heating element must be greater than that of the boiling water to ensure continuous heat flow.

Latent Heat of Vaporization

The latent heat of vaporization is the amount of energy required to change a unit mass of a liquid into vapor without a change in temperature. For water, this is a significant amount, approximately 2260 kJ/kg.

Understanding this concept is essential in boiling, as all the heat energy going into the water is used to break intermolecular bonds during the phase change from liquid to vapor. This requirement is independent of any temperature change.

To determine how much heat is needed to vaporize a certain mass of water, we use the formula:\[ Q = mL_v \]where:

• \( Q \): total heat required
• \( m \): mass of the water
• \( L_v \): latent heat of vaporization

Thus, for boiling away 0.45 kg of water, knowing these values helps calculate that approximately 1017 kJ of heat energy is necessary, underlining the energy-intensive nature of the boiling process.