Problem 26 - 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

The heat required is 80,371.2 J, and the time required is about 402 seconds.

Step 1: Identify the Given Information

We are given that the power of the heater is 200 W, the mass of the water is 0.320 kg, the initial temperature is 20.0 °C, and the final temperature is 80.0 °C. The specific heat capacity of water is approximately 4,186 J/(kg·°C).

Step 2: Calculate the Heat Required

To calculate the heat \(Q\) required, use the formula: \[ Q = m \, c \, \Delta T \]where \(m\) is the mass of the water, \(c\) is the specific heat capacity, and \(\Delta T\) is the change in temperature. Substitute the given values:\[ Q = 0.320 \, \text{kg} \times 4,186 \, \text{J/(kg·°C)} \times (80.0 \, °C - 20.0 \, °C) \]\[ Q = 0.320 \times 4,186 \times 60 \]\[ Q = 80,371.2 \, \text{J} \]

Step 3: Calculate the Time Required

To find the time \(t\) required, use the formula: \[ Q = P \, t \]where \(P\) is the power of the heater. Rearrange the formula to solve for time: \[ t = \frac{Q}{P} \]Substitute the values calculated previously:\[ t = \frac{80,371.2 \, \text{J}}{200 \, \text{W}} \]\[ t = 401.856 \, \text{s} \]Thus, it takes approximately 402 seconds.

Key Concepts

Heat Transfer

Understanding heat transfer is crucial when dealing with temperature changes in thermodynamics. It is the movement of thermal energy from a substance at a higher temperature to one at a lower temperature.
Basically, heat transfer occurs in three main ways: conduction, convection, and radiation. Conduction involves direct transfer through contact, like when the heater warms the water. Convection occurs in fluids, where warmer areas rise and cooler areas sink, creating a circulation pattern. Radiation involves the transfer of energy through electromagnetic waves, like sunlight warming the Earth.
The fundamental principle here is that heat will flow from the hotter material to the cooler one until thermal equilibrium is reached. In our exercise, the heat supplied by the heater is transferred to the water, raising its temperature.

Specific Heat Capacity

The concept of specific heat capacity is essential when calculating how much heat energy is required to change a substance's temperature. It is defined as the amount of heat needed to raise the temperature of 1 kilogram of a substance by 1 degree Celsius.
The formula to calculate heat using specific heat capacity is:

• \( Q = m \, c \, \Delta T \)

Here,

• \( Q \) is the heat energy in joules
• \( m \) is the mass in kilograms
• \( c \) is the specific heat capacity in joules per kilogram per degree Celsius
• \( \Delta T \) is the change in temperature

For water, which has a specific heat capacity of approximately 4,186 J/(kg·°C), the calculations show the importance of this property in heating applications. It signifies that water can absorb a lot of heat before its temperature rises significantly, making it an effective coolant.

Energy Conversion

Energy conversion is a fundamental concept in thermodynamics, describing how energy changes from one form to another. In many practical situations, mechanical, electrical, or chemical energy is converted into thermal energy, as in the example of the electric immersion heater.
This conversion process is vital in everyday appliances, where electrical energy is consumed to produce heat for cooking, heating, or industrial processes. The formula

• \( Q = P \, t \)

is used to calculate the time required to achieve the desired heat output after energy conversion. Here,

• \( Q \) represents the heat energy,
• \( P \) represents the power in watts (the rate of energy conversion),
• \( t \) represents time in seconds.

In our exercise, the electrical heater converts 200 watts of electrical power into heat energy, effectively warming the water by transferring and converting energy to meet the set thermal criteria.