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Chapter 17: Problem 41

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?

Short Answer

Expert verified
The initial temperature of the copper was approximately 151.9掳C.

Step by step solution

01

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.
02

Calculate Heat Needed by Ice to Reach 0掳C

Ice needs to first be warmed from -20.0\(^\circ\)C to 0掳C. Use the formula \( q = m \, c_i \, \Delta T \) where \( m = 2.00 \text{ kg} \), \( c_i = 2.09 \text{ J/g掳C} \) (specific heat capacity of ice), and \( \Delta T = 20\, \text{掳C} \). It follows:\[q_1 = 2,000 \times 2.09 \times 20 = 83,600 \, \text{J}.\]
03

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:\[q_2 = 0.80 \times 334,000 = 267,200 \, \text{J}.\]
04

Total Heat Needed by Ice

Add the heat required to raise the ice temperature to 0掳C and to melt the ice:\[q_{\text{total}} = 83,600 + 267,200 = 350,800 \, \text{J}.\]
05

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 \( m = 6 \text{ kg} \), \( c_c = 385 \text{ J/kg掳C} \):\[q = 6 \times 385 \times (T - 0) = 2,310 \, T.\]
06

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 \( T \):\[T = \frac{350,800}{2,310} \approx 151.9 \, \text{掳C}.\]
07

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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掳C to 0掳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掳C, showing that it requires 2.09 Joules to raise 1 gram of ice by 1掳C.
  • Copper has a specific heat capacity of 385 J/kg掳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掳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.

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Most popular questions from this chapter

The emissivity of tungsten is 0.350. A tungsten sphere with radius 1.50 cm is suspended within a large evacuated enclosure whose walls are at 290.0 K. What power input is required to maintain the sphere at 3000.0 K if heat conduction along the supports is ignored?

A laboratory technician drops a 0.0850-kg sample of unknown solid material, at 100.0\(^\circ\)C, into a calorimeter. The calorimeter can, initially at 19.0\(^\circ\)C, is made of 0.150 kg of copper and contains 0.200 kg of water. The final temperature of the calorimeter can and contents is 26.1\(^\circ\)C. Compute the specific heat of the sample.

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BIO Temperatures in Biomedicine. (a) Normal body temperature. The average normal body temperature measured in the mouth is 310 K. What would Celsius and Fahrenheit thermometers read for this temperature? (b) Elevated body temperature. During very vigorous exercise, the body鈥檚 temperature can go as high as 40\(^\circ\)C. What would Kelvin and Fahrenheit thermometers read for this temperature? (c) Temperature difference in the body. The surface temperature of the body is normally about 7 C\(^\circ\) lower than the internal temperature. Express this temperature difference in kelvins and in Fahrenheit degrees. (d) Blood storage. Blood stored at 4.0\(^\circ\)C lasts safely for about 3 weeks, whereas blood stored at -160\(^\circ\)C lasts for 5 years. Express both temperatures on the Fahrenheit and Kelvin scales. (e) Heat stroke. If the body鈥檚 temperature is above 105\(^\circ\)F for a prolonged period, heat stroke can result. Express this temperature on the Celsius and Kelvin scales.
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