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Ch. 19 - Heat and the First Law of Thermodynamics
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 19 - Heat and the First Law of ThermodynamicsProblem 77
Chapter 19, Problem 77

What will be the final result when equal masses of ice at 0°C and steam at 100°C are mixed together?

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Step 1: Identify the key processes involved. When ice at 0°C and steam at 100°C are mixed, the ice will absorb heat to melt into water (latent heat of fusion), and the steam will release heat to condense into water (latent heat of vaporization). After these phase changes, the resulting water masses will exchange heat until thermal equilibrium is reached.
Step 2: Write the heat transfer equations for each process. For the ice melting, the heat absorbed is given by \( Q_{\text{ice}} = m L_f \), where \( m \) is the mass of the ice and \( L_f \) is the latent heat of fusion of ice. For the steam condensing, the heat released is \( Q_{\text{steam}} = m L_v \), where \( L_v \) is the latent heat of vaporization of steam.
Step 3: Account for the temperature changes after the phase changes. If the ice melts completely, the resulting water at 0°C may absorb additional heat to warm up. The heat absorbed by this water is \( Q_{\text{water, ice}} = m c_w \Delta T \), where \( c_w \) is the specific heat capacity of water and \( \Delta T \) is the temperature change. Similarly, if the steam condenses completely, the resulting water at 100°C may release heat to cool down. The heat released by this water is \( Q_{\text{water, steam}} = m c_w \Delta T \).
Step 4: Apply the principle of conservation of energy. The total heat gained by the ice and its resulting water must equal the total heat lost by the steam and its resulting water. Set up the equation: \( Q_{\text{ice}} + Q_{\text{water, ice}} = Q_{\text{steam}} + Q_{\text{water, steam}} \).
Step 5: Solve the equation step by step. Substitute the known values for \( L_f \), \( L_v \), \( c_w \), and the masses of ice and steam. Determine whether the ice melts completely, the steam condenses completely, or if both reach equilibrium as water at an intermediate temperature. Simplify the equation to find the final equilibrium temperature or state of the system.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Phase Change

Phase change refers to the transition of matter from one state to another, such as solid to liquid (melting) or liquid to gas (vaporization). In this scenario, ice at 0°C will undergo melting to become water, while steam at 100°C will condense to form water. Understanding these phase changes is crucial for predicting the final state of the system.
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Heat Transfer

Heat transfer is the process of thermal energy moving from one object or substance to another due to a temperature difference. In this case, heat will flow from the steam (higher temperature) to the ice (lower temperature) until thermal equilibrium is reached. This concept is essential for determining how much ice melts and how much steam condenses.
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Latent Heat

Latent heat is the amount of heat required to change the phase of a substance without changing its temperature. For instance, the latent heat of fusion is the energy needed to melt ice, while the latent heat of vaporization is the energy required to condense steam. These values are critical for calculating the energy exchanges that occur when equal masses of ice and steam are mixed.
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Related Practice
Textbook Question

The temperature within the Earth’s crust increases about 1.0 C° for each 30 m of depth. The thermal conductivity of the crust is 0.80 J/s C°. Determine the heat transferred from the interior to the surface for the entire Earth in 1.0 h.

Textbook Question

(a) Estimate the total power radiated into space by the Sun, assuming it to be a perfect emitter at T = 5500 K. The Sun’s radius is 7.0 x 10⁸ m.

(b) From this, determine the power per unit area arriving at the Earth, 1.5 x 10¹¹ m away (Fig. 19–37).

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Textbook Question

A microwave oven is used to heat 250 g of water. On its maximum setting, the oven can raise the temperature of the liquid water from 20°C to 100°C in 1 min 45 s ( = 105 s).

(a) At what rate does the oven put energy into the liquid water?

(b) If the power input from the oven to the water remains constant, determine how many grams of water will boil away if the oven is operated for 2 min (rather than just 1 min 45 s).

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Textbook Question

A ceramic teapot (e = 0.70) and a shiny metal one (e = 0.10) each hold 0.55 L of tea at 85°C. (a) Estimate the rate of heat loss from each, and (b) estimate the temperature drop after 30 min for each. Consider only radiation, and assume the surroundings are at 20°C.

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Textbook Question

A copper rod and an aluminum rod of the same length and cross-sectional area are attached end to end (Fig. 19–35). The copper end is placed in a furnace maintained at a constant temperature of 205°C. The aluminum end is placed in an ice bath held at a constant temperature of 0.0°C. Calculate the temperature at the point where the two rods are joined.

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