Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 17: Temperature and Heat

Problem 55

Chapter 17, Problem 55

A vessel whose walls are thermally insulated contains $2.40$ kg of water and $0.450$ kg of ice, all at $0.0$ °C. The outlet of a tube leading from a boiler in which water is boiling at atmospheric pressure is inserted into the water. How many grams of steam must condense inside the vessel (also at atmospheric pressure) to raise the temperature of the system to $28.0$ °C? You can ignore the heat transferred to the container.

Verified step by step guidance

First, understand the problem: We need to find out how much steam must condense to raise the temperature of the water and ice mixture from 0.0°C to 28.0°C. The steam will release heat as it condenses and cools down to 28.0°C, which will be absorbed by the water and ice.

Calculate the heat required to melt the ice. Use the formula:

Q = m L

_f, where

m

is the mass of the ice and

L

_f is the latent heat of fusion for ice (334,000 J/kg).

Calculate the heat required to raise the temperature of the melted ice and the water from 0.0°C to 28.0°C. Use the formula:

Q = m c ΔT

, where

m

is the total mass of water (melted ice + initial water),

c

is the specific heat capacity of water (4,186 J/kg°C), and

ΔT

is the change in temperature (28.0°C - 0.0°C).

Calculate the heat released by the steam as it condenses and cools to 28.0°C. First, use the formula for the heat released during condensation:

Q = m L

_v, where

m

is the mass of the steam and

L

_v is the latent heat of vaporization for water (2,260,000 J/kg). Then, calculate the heat released as the condensed water cools to 28.0°C using:

Q = m c ΔT

, where

ΔT

is the change in temperature (100.0°C - 28.0°C).

Set up the equation where the total heat absorbed by the ice and water equals the total heat released by the steam. Solve for the mass of the steam:

m

_steam=Q_totalL_v+cΔT. Convert the mass from kilograms to grams.

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Key Concepts

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

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. It is crucial for calculating the heat needed to raise the temperature of the water and ice in the vessel. For water, this value is typically 4.18 J/g°C, which helps determine the total energy required for the temperature change.

Latent Heat of Fusion and Vaporization

Latent heat refers to the energy absorbed or released during a phase change without changing temperature. The latent heat of fusion is needed to melt the ice, while the latent heat of vaporization is required for steam condensation. These values are essential for calculating the energy exchanges during the melting of ice and condensation of steam, respectively.

Heat Transfer and Energy Conservation

In thermally insulated systems, the principle of energy conservation states that the total energy remains constant. The heat from the condensing steam must equal the energy needed to melt the ice and raise the temperature of the water. Understanding this concept allows us to set up an equation balancing the energy input from steam with the energy required for the temperature increase.

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Related Practice

Textbook Question

A laboratory technician drops a $0.0850$ -kg sample of unknown solid material, at $100.0$ °C, into a calorimeter. The calorimeter can, initially at $19.0$ °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 $26.1$ °C. Compute the specific heat of the sample.

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Textbook Question

Suppose that the rod in Fig. $17.24$ a is made of copper, is $45.0$ cm long, and has a cross-sectional area of $1.25$ cm² . Let $T H equals 100.0$ °C and $T C equals 0.0$ °C. What is the final steady-state temperature gradient along the rod?

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Textbook Question

A carpenter builds an exterior house wall with a layer of wood $3.0$ cm thick on the outside and a layer of Styrofoam insulation $2.2$ cm thick on the inside wall surface. The wood has $k = 0.080 W / m \cdot K$ , and the Styrofoam has $k = 0.027 W / m \cdot K$ . The interior surface temperature is $19.0$ °C, and the exterior surface temperature is $negative 10.0$ °C. What is the temperature at the plane where the wood meets the Styrofoam?

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Textbook Question

A $4.00$ -kg silver ingot is taken from a furnace, where its temperature is $750.0$ °C, and placed on a large block of ice at $0.0$ °C. Assuming that all the heat given up by the silver is used to melt the ice, how much ice is melted?

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Textbook Question

Two rods, one made of brass and the other made of copper, are joined end to end. The length of the brass section is $0.300$ m and the length of the copper section is $0.800$ m. Each segment has cross-sectional area $0.00500$ m². The free end of the brass segment is in boiling water and the free end of the copper segment is in an ice–water mixture, in both cases under normal atmospheric pressure. The sides of the rods are insulated so there is no heat loss to the surroundings. What mass of ice is melted in $5.00$ min by the heat conducted by the composite rod?

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Textbook Question

An insulated beaker with negligible mass contains $0.250$ kg of water at $75.0$ °C. How many kilograms of ice at $negative 20.0$ °C must be dropped into the water to make the final temperature of the system $40.0$ °C?

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