Problem 33 - University Physics with Modern Physics

Question

An aluminum tea kettle with mass 1.50 \(\mathrm{kg}\) and containing 1.80 \(\mathrm{kg}\) of water is placed on a stove. If no heat is lost to the surroundings, how much heat must be added to raise the temperature from \(20.0^{\circ} \mathrm{C}\) to \(85.0^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The total heat required is 615.6 kJ.

Step 1: Identify Given Information

We have an aluminum kettle with a mass of 1.50 kg and water inside it with a mass of 1.80 kg. We need to raise the temperature from 20.0°C to 85.0°C.

Step 2: Determine Specific Heat Capacities

The specific heat capacity of aluminum is 0.900 J/g°C, and for water, it is 4.186 J/g°C.

Step 3: Convert Mass to Grams

Convert the mass from kilograms to grams for consistency with the specific heat capacity units: 1.50 kg of aluminum = 1500 g 1.80 kg of water = 1800 g.

Step 4: Calculate Heat Required for Aluminum

Use the formula for heat: \[ q = m \, \times \, c \, \times \, \Delta T \]where: - \( q \) is the heat added,- \( m \) is the mass,- \( c \) is the specific heat capacity,- \( \Delta T \) is the change in temperature.For aluminum:\[ q_{\text{aluminum}} = 1500 \, \text{g} \, \times \, 0.900 \, \text{J/g°C} \, \times \, (85.0°C - 20.0°C) \]\[ q_{\text{aluminum}} = 1500 \, \text{g} \, \times \, 0.900 \, \text{J/g°C} \, \times \, 65.0°C \]Calculate the result.

Step 5: Calculate Heat Required for Water

Apply the same formula for water:\[ q_{\text{water}} = 1800 \, \text{g} \, \times \, 4.186 \, \text{J/g°C} \, \times \, (85.0°C - 20.0°C) \]\[ q_{\text{water}} = 1800 \, \text{g} \, \times \, 4.186 \, \text{J/g°C} \, \times \, 65.0°C \]Calculate the result.

Step 6: Calculate Total Heat Required

Add the heat required for the aluminum and the water to find the total heat required:\[ q_{\text{total}} = q_{\text{aluminum}} + q_{\text{water}} \] Substitute in the values from the previous steps and calculate the total heat.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a fascinating property of materials. It tells us how much heat energy is needed to raise the temperature of one gram of a substance by one degree Celsius. In this exercise, we deal with two different substances: aluminum and water.
Aluminum has a specific heat capacity of 0.900 J/g°C, while water holds a much higher value of 4.186 J/g°C. This means that water requires more heat energy than aluminum for the same mass and temperature change.
Understanding specific heat capacity helps us determine how different substances will react to heat. It allows us to calculate precisely how much energy is needed to reach a desired temperature change.

Thermal Energy Calculation

In thermal energy calculations, we use a straightforward formula to find the amount of heat energy required:

• \( q = m \times c \times \Delta T \)

Here, \( q \) represents the heat added, \( m \) is the mass in grams, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change.
In the problem at hand, we calculate the heat required for both the aluminum kettle and the water inside. For example, to heat the aluminum, we use its mass (1500 g), its specific heat capacity (0.900 J/g°C), and the temperature change (65°C).
Applying these values gives us an estimate of the energy needed, which we can similarly do for the water. This straightforward process enables clear assessments of the energy demands involved in heating different substances.

Temperature Change

Temperature change is at the heart of any thermal energy problem. It represents the difference between the initial and final temperatures of the substances in question.
In this scenario, the temperature change \( \Delta T \) is from 20.0°C to 85.0°C. To find \( \Delta T \), we simply subtract the initial temperature from the final temperature:

• \( \Delta T = 85.0°C - 20.0°C = 65.0°C \)

This result tells us how much each substance's temperature needs to rise. Whether working with metals like aluminum, or liquids such as water, knowing the temperature change is crucial for calculating the total thermal energy required. These calculations rely on the specific properties of each material, emphasizing how important it is to understand those variations across different substances.