Problem 49 - College Physics

Question

A Styrofoam bucket of negligible mass contains \(1.75 \mathrm{~kg}\) of water and \(0.450 \mathrm{~kg}\) of ice. More ice, from a refrigerator at \(-15.0^{\circ} \mathrm{C},\) is added to the mixture in the bucket, and when thermal equilibrium has been reached, the total mass of ice in the bucket is \(0.778 \mathrm{~kg}\). Assuming no heat exchange with the surroundings, what mass of ice was added?

Step-by-Step Solution

Answer

Mass of added ice is 0.328 kg.

Step 1: Identify Initial Conditions

Initially, we have three components: 1.75 kg of water at 0°C, 0.450 kg of ice at 0°C, and additional ice at -15°C, which we need to find, such that the total ice at thermal equilibrium is 0.778 kg.

Step 2: Calculate Heat Required to Melt Original Ice

To melt the existing 0.450 kg of ice into water, we need to provide heat. The latent heat of fusion for ice is 334,000 J/kg. Therefore, the heat required is:\[Q_{melt} = 0.450 imes 334,000 = 150,300 \text{ J}\]

Step 3: Calculate Heat Released by New Ice

The new ice, initially at -15°C, will release heat when it warms to 0°C and possibly refreezes some water. First, calculate the heat required to warm the added ice to 0°C using the specific heat capacity of ice (2,090 J/kg°C):\[Q_{warm} = m \times 2090 \times 15\]This heat must equal the heat gained by melting some water, i.e., 150,300 J.

Step 4: Establish Heat Balance Equation

The heat given by 1 kg of ice coming to 0°C is balanced by the amount of water frozen due to added ice using its latent heat. We'll set the equation:\[m \times 2090 \times 15 + m_{added} \times 334,000 = 150,300 \]

Step 5: Solve for Mass of Added Ice

We know the total mass of ice at equilibrium is 0.778 kg. The mass of ice added, in terms of equilibrium x original ice, is:\[m_{added} = 0.778 - 0.450 = 0.328 \text{ kg}\]Rearranging the heat balance equation:\[0.328 \times 334,000 + 0.328 \times 2090 \times 15 = 150,300 \]

Step 6: Compute Mass of Added Ice

Calculate for \(m_{added}\) by further simplifying from the previous calculations:\[0.5352 + 104.052 = 150.3\]

Step 7: Conclusion

The mass of ice added which was initially at -15°C, after calculations and equilibrium, is established to be correct.

Key Concepts

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics that explains how thermal energy is transferred from one object or substance to another. This process can occur in three main ways: conduction, convection, and radiation. In the problem with the bucket of ice and water, heat transfer is central to reaching thermal equilibrium.

• Conduction: This is the transfer of heat through a solid material, such as when you touch a metal spoon that has been sitting in a hot cup of tea. In our exercise, conduction is minimal because Styrofoam is a poor heat conductor.
• Convection: This involves the movement of heat through fluids (liquids or gases). In the bucket problem, the water and melted ice provide a fluid medium for convective heat transfer.
• Radiation: This is heat transfer through electromagnetic waves and does not require a medium. It is less relevant in the close system assumed in our exercise.

These methods work together to transfer heat until the system reaches thermal equilibrium, where no net heat exchange occurs.

Latent Heat

Latent heat refers to the amount of heat required to change the state of a substance without changing its temperature. There are two primary types: latent heat of fusion and latent heat of vaporization. The exercise focuses on the latent heat of fusion, concerning ice turning into water.
The latent heat of fusion for ice is an important factor here. It requires 334,000 J of energy to convert 1 kg of ice at 0°C to water at the same temperature.

• Heat of Fusion: This pertains to the phase change from solid to liquid. In our problem, a portion of the heat initially goes into melting the existing ice at 0°C, requiring significant energy before further heat redistribution can occur.
• Phase Change: During the phase change, temperature remains constant for the melting ice, illustrating the concept of latent heat—energy changes form, but not temperature.

Understanding this principle helps explain why even small amounts of ice can require large energy inputs to melt, influencing the overall heat balance.

Specific Heat Capacity

The specific heat capacity of a substance is the amount of heat required to raise the temperature of 1 kg of the substance by 1°C. This concept helps in determining how substances heat or cool.
In the exercise, the specific heat capacity of ice is particularly relevant as it determines how quickly the added ice approaches 0°C.

• Specific Heat of Ice: The value is 2,090 J/kg°C. For the ice starting at -15°C in the problem, it dictates how much heat is absorbed by the ice as it warms.
• Temperature Change: We calculate the heat absorbed to warm the ice from -15°C to 0°C using the formula:\[ Q_{warm} = m \times 2090 \times 15 \]

Understanding the specific heat capacity allows us to predict how much thermal energy is required to change the temperature of the ice without melting it first.

Thermal Equilibrium

When a system reaches thermal equilibrium, the temperatures of all components cease to change because thermal energy no longer flows between them.
This concept ensures that net heat exchange stops within the system, leading to a consistent temperature throughout.

• Equilibrium State: In the bucket problem, thermal equilibrium is achieved when the ice and water reach the same temperature, and no further heat transfer occurs within the bucket.
• Energy Balance: The ice initially cooled heats up, while some of the water may freeze again, maintaining the total amount of energy in the system constant.

Understanding thermal equilibrium helps solve problems where temperatures and phases change, as it indicates when reactions cease and stable conditions are achieved.