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Ch 19: Work, Heat, and the First Law of Thermodynamics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 19: Work, Heat, and the First Law of ThermodynamicsProblem 40
Chapter 19, Problem 40

The burner on an electric stove has a power output of 2.0 kW. A 750 g stainless steel teakettle is filled with 20°C water and placed on the already hot burner. If it takes 3.0 min for the water to reach a boil, what volume of water, in cm3, was in the kettle? Stainless steel is mostly iron, so you can assume its specific heat is that of iron.

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Step 1: Identify the given values and constants. The power output of the burner is 2.0 kW (2000 W), the mass of the stainless steel kettle is 750 g (0.750 kg), the initial temperature of the water is 20°C, the time to reach boiling is 3.0 minutes (180 seconds), and the final temperature of the water is 100°C. The specific heat capacity of water is approximately 4,186 J/(kg·°C), and the specific heat capacity of iron (for the kettle) is approximately 450 J/(kg·°C).
Step 2: Write the energy balance equation. The total energy provided by the burner is used to heat both the water and the kettle. The energy provided by the burner is given by \( Q_{total} = P \cdot t \), where \( P \) is the power and \( t \) is the time. The energy required to heat the water is \( Q_{water} = m_{water} \cdot c_{water} \cdot \Delta T \), and the energy required to heat the kettle is \( Q_{kettle} = m_{kettle} \cdot c_{kettle} \cdot \Delta T \).
Step 3: Express the total energy equation. Combine the energy terms: \( Q_{total} = Q_{water} + Q_{kettle} \). Substituting the expressions for \( Q_{water} \) and \( Q_{kettle} \), we get \( P \cdot t = m_{water} \cdot c_{water} \cdot \Delta T + m_{kettle} \cdot c_{kettle} \cdot \Delta T \).
Step 4: Solve for the mass of the water. Rearrange the equation to isolate \( m_{water} \): \( m_{water} = \frac{P \cdot t - m_{kettle} \cdot c_{kettle} \cdot \Delta T}{c_{water} \cdot \Delta T} \). Substitute the known values: \( P = 2000 \ \text{W} \), \( t = 180 \ \text{s} \), \( m_{kettle} = 0.750 \ \text{kg} \), \( c_{kettle} = 450 \ \text{J/(kg·°C)} \), \( c_{water} = 4186 \ \text{J/(kg·°C)} \), and \( \Delta T = 100 - 20 = 80 \ \text{°C} \).
Step 5: Convert the mass of the water to volume. Use the density of water, which is approximately \( 1 \ \text{g/cm}^3 \) or \( 1000 \ \text{kg/m}^3 \), to find the volume. The volume \( V \) is given by \( V = \frac{m_{water}}{\text{density}} \). Substitute the calculated mass of the water into this equation to find the volume in \( \text{cm}^3 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Power and Energy Transfer

Power is the rate at which energy is transferred or converted. In this context, the electric stove's power output of 2.0 kW indicates how much energy it provides to heat the water over time. The energy transferred can be calculated using the formula: Energy = Power × Time, which helps determine how much energy is available to heat the water in the kettle.
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Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. For stainless steel, which is assumed to have a specific heat similar to iron, this property is crucial for calculating how much energy is needed to heat the kettle and the water. The formula Q = mcΔT, where Q is the heat energy, m is mass, c is specific heat, and ΔT is the change in temperature, is used to find the energy needed to reach boiling.
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Volume and Density Relationship

The volume of a substance can be determined using its mass and density, expressed as Volume = Mass / Density. In this problem, knowing the mass of water (750 g) allows us to find its volume, given that the density of water is approximately 1 g/cm³. This relationship is essential for converting the mass of water into a volume measurement, which is the final goal of the question.
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Related Practice
Textbook Question

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