Problem 34 - University Physics with Modern Physics

Question

In an effort to stay awake for an all-night study session, a student makes a cup of coffee by first placing a \(200-\mathrm{W}\) electric immersion heater in 0.320 \(\mathrm{kg}\) of water. (a) How much heat must be added to the water to raise its temperature from \(20.0^{\circ} \mathrm{C}\) to \(80.0^{\circ} \mathrm{C}\) ? (b) How much time is required? Assume that all of the heater's power goes into heating the water.

Step-by-Step Solution

Answer

80,371.2 J of heat is required, and it takes approximately 402 seconds.

Step 1: Understand the Problem

We need to find how much heat is required to raise the temperature of water from 20°C to 80°C. Then, we calculate the time needed using the power of the heater.

Step 2: Use the Heat Formula

The heat required to change the temperature of a substance can be calculated using the formula: \[ q = mc\Delta T \] where: - \( q \) is the heat added,- \( m \) is the mass of the water (0.320 kg),- \( c \) is the specific heat capacity of water (4,186 J/kg°C),- \( \Delta T \) is the change in temperature (80°C - 20°C = 60°C).

Step 3: Calculate the Heat Required

Substitute the known values into the formula: \[ q = (0.320 \, \text{kg})(4186 \, \text{J/kg°C})(60 \, \text{°C}) \]Calculate the result.\[ q = 80,371.2 \, \text{J} \] So, 80,371.2 Joules of heat are needed.

Step 4: Find Time Using Power

We have the heater's power (200 W) and the total energy required (80,371.2 J). Time can be found using the formula: \[ t = \frac{q}{P} \]where:- \( t \) is the time,- \( q \) is the energy in Joules,- \( P \) is the power in Watts (200 W).

Step 5: Calculate the Time Required

Substitute the known values into the formula: \[ t = \frac{80,371.2 \, \text{J}}{200 \, \text{W}} \]Calculate the result.\[ t = 401.856 \, \text{seconds} \] Thus, it takes approximately 402 seconds.

Key Concepts

Understanding Specific Heat Capacity

Specific heat capacity is a fundamental concept in physics that describes how much heat energy is required to change the temperature of a substance. It is defined as the amount of heat needed to raise the temperature of 1 kilogram of a substance by 1 degree Celsius (or 1 Kelvin).
In mathematical terms, this is represented by the symbol \( c \) and is included in the heat formula \( q = mc\Delta T \).

• \( q \) represents the total heat energy added (measured in Joules).
• \( m \) is the mass of the substance (in kilograms).
• \( c \) is the specific heat capacity (in J/kg°C).
• \( \Delta T \) is the change in temperature (in °C).

This formula helps us calculate how much heat to add to a known mass to achieve a desired temperature change, assuming no heat loss to the surroundings.
In our example, the specific heat capacity of water is used because the problem involves heating water. It tells us how efficiently water can absorb heat, which is crucial for understanding thermal processes.

Electric Power in Heat Transfer

Electric power is a measure of the rate at which energy is transferred or converted. It is expressed in watts (W), where 1 watt equates to 1 joule per second.
In the context of heating, power tells us how quickly a device like an electric immersion heater can transfer energy to a substance such as water.
The relationship is captured in the equation \( P = \frac{q}{t} \), where:

• \( P \) is the power in watts.
• \( q \) represents the energy in joules.
• \( t \) is the time in seconds.

By rearranging the formula to \( t = \frac{q}{P} \), we can calculate how long it will take for the heater to supply enough energy to achieve the target temperature.
This calculation assumes that all the power is used for heating, with no losses, ensuring the accuracy of the time estimate. Understanding electric power is essential, as it connects the physical operation of heating with the energy requirements of the process.

Approach to Problem Solving in Physics

Problem-solving in physics follows a structured approach to break down complex issues into manageable parts. Often, it involves:

• Clearly understanding the problem and identifying what is being asked.
• Using known formulas and variables to express the problem mathematically.
• Substituting the given values into equations to find unknown variables.
• Performing calculations carefully, paying attention to units.

In solving this exercise, we first determined the amount of heat needed using the heat equation \( q = mc\Delta T \). Next, we utilized the power relationship \( t = \frac{q}{P} \) to find the time required by the heater to add the calculated heat to the water.
This step-by-step methodology not only provides the solution but also ensures a deep understanding of the physical concepts involved.