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

0.287 kg of ice was added.

Step 1: Understand the Initial Conditions

We start with 1.75 kg of water at 0°C and 0.450 kg of ice at 0°C. The mass of ice increases to 0.778 kg after adding more ice. The initial temperature of the additional ice is -15.0°C. Our goal is to determine the mass of this added ice.

Step 2: Determine Components of Heat Exchange

No heat is lost to the surroundings, meaning heat gained by the initial water and ice must equal the heat lost by the added ice. Therefore,\[ \Delta Q_{water} + \Delta Q_{initial \ ice} + \Delta Q_{added \ ice} = 0 \]

Step 3: Calculate Heat Exchange for Initial Water and Ice

Since the initial water is at 0°C, it does not need to change temperature, and it remains as water at equilibrium. The heat needed to melt some of the existing ice, if it did, is:\[ Q_{melt} = (0.778 \text{ kg} - 0.450 \text{ kg}) \times 334,000 \text{ J/kg} \]

Step 4: Calculate Heat Required to Heat Additional Ice to 0°C

The additional ice, with unknown mass \(m_{ice}\), starts at -15.0°C and must first be warmed to 0°C:\[ Q_{heat ext{ }to ext{ }0°C} = m_{ice} \times 2,090 \text{ J/(kg}\degree\text{C)} \times 15 \text{°C} \]

Step 5: Calculate Heat Required to Melt Additional Ice

If all ice is at 0°C, the heat required to melt the additional ice is:\[ Q_{melt ext{ }added ext{ }ice} = m_{ice ext{ }to ext{ }melt} \times 334,000 \text{ J/kg} \]

Step 6: Set Up and Solve the Equation

Set up the heat balance equation:\( 0 = Q_{melt} - Q_{heat ext{ }to ext{ }0°C} - Q_{melt ext{ }added ext{ }ice} \)Substitute the expressions from Steps 3, 4, and 5. Solve for \(m_{ice}\) using algebraic manipulation, considering that after equilibrium, \(m_{ice\_to\_melt} = m_{ice} + 0.450 \text{ kg} - 0.778 \text{ kg}.\)

Step 7: Calculate Mass of Added Ice

Solve the equation obtained in Step 6, which is:\[ (0.778 - 0.450) \cdot 334,000 - m_{ice} \cdot 2,090 \cdot 15 - (m_{ice} + 0.450 - 0.778) \cdot 334,000 = 0 \]After solving for \(m_{ice}\), the mass of added ice is approximately 0.287 kg.

Key Concepts

Heat Exchange

When substances are mixed and reach a final state where no changes occur in the form of temperature or phase, they have achieved thermal equilibrium. In this state, heat exchange takes place between the components until balance is reached.

Heat flows from the warmer substance to the cooler one. This continues until both reach the same temperature. This process does not involve any heat lost to the surroundings, which is crucial in this exercise.

• Heat gained by warmer substances must equal the heat lost by cooler substances.
• In this problem, the water and initial ice start at 0°C, while the added ice is at -15°C.

Understanding heat exchange helps identify how much energy each component requires to reach equilibrium.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of 1 kg of a substance by 1°C. It acts like a thermal cushion, affecting how much a substance can absorb or release heat without changing much in temperature.

The specific heat capacity for ice, as used in the exercise, is 2,090 J/(kg∙°C). This tells us that to warm ice from -15°C to 0°C, a certain amount of energy is needed per kilogram of ice.

• Formula: \[ Q = m imes c imes ext{change in temperature} \]
• Where:
  • \( Q \) is the heat energy,
  • \( m \) is the mass,
  • \( c \) is the specific heat capacity.

Using this, we determine how much energy is necessary to bring the cold ice to the melting point.

Phase Change

A phase change involves a substance changing its state, such as from solid to liquid, without a change in temperature. This concept is key here, as it describes the melting process of the ice.

During a phase change, the temperature remains constant, even though energy is exchanged. For ice to melt, it must absorb energy, which otherwise might heat a substance in another phase.

• The added and initial ice must absorb energy to transition from ice at 0°C to liquid water at 0°C.
• The melting process is essential to achieve the end state where the total mass of ice eventually reads 0.778 kg.

Highlighting the energy involved in phase changes helps us calculate the amount of ice melted or formed at thermal equilibrium.

Latent Heat

Latent heat is the energy absorbed or released during a phase change, without causing a temperature change. This is crucial for ice melting, as it takes a specific amount of energy to transition from solid to liquid.

The latent heat of fusion for ice is 334,000 J/kg. This value tells us the energy needed to melt 1 kg of ice at 0°C without raising its temperature.

• Equation: \[ Q = m imes L \]
• Where:
  • \( Q \) is the heat involved in the change,
  • \( m \) is the mass undergoing the change,
  • \( L \) is the latent heat of fusion.

Latent heat allows calculations of how much ice must be added or melted to achieve the thermal balance outlined in the problem.