Problem 41 - University Physics with Modern Physics

Question

A 6.00-kg piece of solid copper metal at an initial temperature \(T\) is placed with 2.00 kg of ice that is initially at -20.0\(^\circ\)C. The ice is in an insulated container of negligible mass and no heat is exchanged with the surroundings. After thermal equilibrium is reached, there is 1.20 kg of ice and 0.80 kg of liquid water. What was the initial temperature of the piece of copper?

Step-by-Step Solution

Answer

The initial temperature of the copper was approximately 151.9°C.

Step 1: Understand the System

You have a 6 kg piece of copper initially at temperature \( T \), 2 kg of ice starting at -20.0\(^\circ\)C in an insulated container. At thermal equilibrium, 1.20 kg of ice and 0.80 kg of liquid water remain. We need to find the initial temperature of the copper.

Step 2: Calculate Heat Needed by Ice to Reach 0°C

Ice needs to first be warmed from -20.0\(^\circ\)C to 0°C. Use the formula \( q = m \, c_i \, \Delta T \) where \( m = 2.00 \text{ kg} \), \( c_i = 2.09 \text{ J/g°C} \) (specific heat capacity of ice), and \( \Delta T = 20\, \text{°C} \). It follows:\[q_1 = 2,000 \times 2.09 \times 20 = 83,600 \, \text{J}.\]

Step 3: Calculate Heat Needed to Melt Ice

Next, a portion of the ice (0.80 kg) needs to melt. Use \( q = m \, L_f \) where \( L_f = 334,000 \text{ J/kg} \) is the latent heat of fusion. Thus:\[q_2 = 0.80 \times 334,000 = 267,200 \, \text{J}.\]

Step 4: Total Heat Needed by Ice

Add the heat required to raise the ice temperature to 0°C and to melt the ice:\[q_{\text{total}} = 83,600 + 267,200 = 350,800 \, \text{J}.\]

Step 5: Heat Lost by Copper Metal

This heat is taken in by the copper, which cools from temperature \( T \) to 0\(^\circ\)C. Using \( q = m \, c_c \, \Delta T \), where \( m = 6 \text{ kg} \), \( c_c = 385 \text{ J/kg°C} \):\[q = 6 \times 385 \times (T - 0) = 2,310 \, T.\]

Step 6: Equate Heats to Find Initial Temperature

Since no heat is exchanged with surroundings, heat lost by copper equals heat gained by ice:\[2,310 \, T = 350,800.\]Solve for \( T \):\[T = \frac{350,800}{2,310} \approx 151.9 \, \text{°C}.\]

Step 7: Verify Solution

Ensure no errors by recalculating steps if needed. Check assumptions, like negligible container mass, are valid. Ensures a balanced heat exchange with no external losses.

Key Concepts

Heat Transfer

Heat transfer is the process of thermal energy moving from a hotter object to a cooler one. In this exercise, we examine heat transfer between a piece of copper and ice in an insulated container. This is pivotal, as it defines how the initial thermal energy from the hotter copper is used to warm and melt the ice.

• The first phase involves the ice absorbing heat to rise from -20.0°C to 0°C.
• The second involves the phase change from solid ice to liquid water, known as melting.

These processes showcase the two main types of heat transfer: sensible heat (which changes temperature) and latent heat (which changes state). It's crucial to consider both in calculations to understand the overall heat transfer in the system.

Specific Heat Capacity

Specific heat capacity is a property that defines how much energy it takes to raise the temperature of a unit mass of a substance by one degree Celsius. Each material has a unique specific heat capacity. In this scenario:

• The specific heat capacity of ice is 2.09 J/g°C, showing that it requires 2.09 Joules to raise 1 gram of ice by 1°C.
• Copper has a specific heat capacity of 385 J/kg°C, indicating that each kilogram requires 385 Joules for the same temperature change.

Understanding these differences is essential. It helps us calculate how much heat is exchanged when one substance cools down or heats up. In this exercise, specific heat capacity is key to determining how much the temperature of copper and ice change during the process.

Latent Heat

Latent heat describes the energy absorbed or released during a phase change, without any change in temperature. For example, when ice melts to water, it absorbs heat but does not change in temperature. Here:

• The latent heat of fusion for ice is 334,000 J/kg, meaning that for 1 kilogram of ice to melt completely, 334,000 Joules of energy must be infused.
• This phase is crucial in the problem as 0.80 kg of ice melts, absorbing a significant amount of energy, impacting the copper's initial temperature.

Understanding latent heat is vital because it accounts for the energy needed even after the ice has reached 0°C but has yet to turn into water. This concept helps explain why a lot of heat is used in melting before raising the liquid water's temperature.

Thermal Equilibrium

Thermal equilibrium describes the state achieved when two objects in contact do not exchange any further heat, meaning they have reached the same temperature. In this problem:

• After the heat exchange between copper and ice-water, the system reaches thermal equilibrium.
• At this point, the temperature of both copper and the ice-water mixture is the same, halting further heat flow.

Achieving thermal equilibrium is foundational for the process. It allows us to equate the heat lost by copper with the heat gained by the ice. It ensures a clear understanding of how temperatures adjust and eventually stabilize in isolated systems, enabling accurate calculations of initial conditions like the initial temperature of copper in this exercise.