Problem 57 - University Physics with Modern Physics

Question

A refrigerator door is opened and room-temperature air \(\left(20.0^{\circ} \mathrm{C}\right)\) fills the \(1.50-\mathrm{m}^{3}\) compartment. A \(10.0-\mathrm{kg}\) turkey, also at room temperature, is placed in the refrigerator and the door is closed. The density of air is 1.20 \(\mathrm{kg} / \mathrm{m}^{3}\) and its specific heat is 1020 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) . Assume the specific heat of a turkey, like that of a human, is 3480 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) . How much heat must the refrigerator remove from its compartment to bring the air and the turkey to thermal equilibrium at a temperature of \(5.00^{\circ} \mathrm{C}\) ? Assume no heat exchange with the surrounding environment.

Step-by-Step Solution

Answer

The refrigerator must remove 549,540 J of heat to reach thermal equilibrium.

Step 1: Calculate the Mass of Air

Using the density formula \( \text{density} = \frac{\text{mass}}{\text{volume}} \), we find the mass of the air in the refrigerator. Given the density of air is 1.20 \( \text{kg/m}^3 \) and the volume is 1.50 \( \text{m}^3 \), the mass of the air is: \[\text{mass of air} = (1.20 \, \text{kg/m}^3) \times (1.50 \, \text{m}^3) = 1.80 \, \text{kg}\]

Step 2: Calculate Heat Removed from Air

The heat removed from the air, \( Q_\text{air} \), is calculated using: \[Q = m \cdot c \cdot \Delta T\] where \( m \) is the mass of the air, \( c \) is the specific heat capacity of air (1020 \( \text{J/kg} \cdot \text{K}\)), and \( \Delta T \) is the change in temperature (from 20.0°C to 5.0°C, so \( \Delta T = 20.0 - 5.0 = 15.0 \) K). Calculating: \[Q_\text{air} = 1.80 \, \text{kg} \times 1020 \, \text{J/kg} \cdot \text{K} \times 15.0 \, \text{K} = 27540 \, \text{J}\]

Step 3: Calculate Heat Removed from Turkey

The heat removed from the turkey, \( Q_\text{turkey} \), uses the same formula: \[Q = m \cdot c \cdot \Delta T\] The mass of the turkey is 10.0 kg, the specific heat capacity is 3480 \( \text{J/kg} \cdot \text{K}\), and \( \Delta T = 15.0 \) K. Calculating: \[Q_\text{turkey} = 10.0 \, \text{kg} \times 3480 \, \text{J/kg} \cdot \text{K} \times 15.0 \, \text{K} = 522000 \, \text{J}\]

Step 4: Calculate Total Heat Removed

The total heat \( Q_\text{total} \) that the refrigerator must remove is the sum of the heat removed from the air and the turkey. \[Q_\text{total} = Q_\text{air} + Q_\text{turkey} = 27540 \, \text{J} + 522000 \, \text{J} = 549540 \, \text{J}\]

Key Concepts

Heat Transfer

Heat transfer is the movement of thermal energy from one substance or material to another. In our exercise, the refrigerator needs to transfer heat out of its compartment to cool everything inside. When you open the fridge, warm room air replaces the cold air inside. Once the refrigerator door is closed, the refrigerator's job is to remove this heat, bringing the contents to a cooler temperature, in this case, from 20°C to 5°C.

Heat transfer can occur in three ways: conduction, convection, and radiation. In our scenario, conduction is the most relevant. This involves direct contact where heat moves from a warmer area to a cooler one. As the refrigerator operates, its cooling system transfers heat from the contents inside, including the air and the turkey, to the outside environment. However, in this problem, we're simplifying to just consider the energy change needed without external heat exchange.

The energy required for the refrigerator to remove heat is derived from the temperature change required and the specific heat capacity of the substances being cooled. You'll calculate the energy based on the differences in initial and final temperatures for both the contents of the fridge itself and the turkey placed inside.

Specific Heat Capacity

Specific heat capacity is a measure of how much heat energy is needed to change the temperature of a specific quantity (usually 1 kg) of a substance by 1 degree Celsius (or 1 Kelvin). Different materials require different amounts of energy to change temperature.

In the example, the specific heat capacity differs between the air and the turkey:

• For air, the specific heat capacity is 1020 J/kg·K. This means that 1020 Joules of energy will change the temperature of 1 kg of air by 1°C.
• For the turkey, it is 3480 J/kg·K, meaning more energy is needed as the turkey has higher specific heat requirements.

The use of specific heat capacity helps us calculate exactly how much energy needs to be extracted from each item in the refrigerator to reach the desired temperature drop.

By multiplying specific heat capacity with the mass of the substance and the temperature change (\( \Delta T \)), we derive the amount of heat energy, \( Q \), to be removed. This is calculated using the formula:\[ Q = m \times c \times \Delta T \] where \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.

Thermal Equilibrium

Thermal equilibrium occurs when two or more substances within a closed system reach the same temperature, and no net heat flows between them. This means that all parts of the system are at the same thermal state and will not transfer energy to one another.

In the case of our refrigerator problem, achieving thermal equilibrium is a key goal. Initially, when the turkey (at 20°C) is placed in the refrigerator with the air (also at 20°C), they need to reach a new common temperature of 5°C. This involves removing enough heat from both air and turkey until neither gains nor loses additional heat energy again. The system stabilizes when both reach the thermal equilibrium at the cooler temperature.

The concept of thermal equilibrium is crucial in thermodynamics because it signifies a state where energy distribution within a system has become balanced. Once thermal equilibrium is achieved, the refrigerator's work, in terms of using its cooling system to facilitate heat transfer, is effectively complete.