   Chapter 11, Problem 73AP

Chapter
Section
Textbook Problem

At time t = 0, a vessel contains a mixture of 10. kg of water and an unknown mass of ice in equilibrium at 0°C. The temperature of the mixture is measured over a period of an hour, with the following results: During the first 50. min. the mixture remains at 0CC; from 50. min to 60. min, the temperature increases steadily from 0°C to 2.0°C Neglecting the heat capacity of the vessel, determine the mass of ice that was initially placed in it. Assume a constant power input to the container.

To determine
The amount of ice initially placed inside the vessel.

Explanation

Given info:

For the first 50 the temperature of the mixture remains at 0°C . This indicates that the heat energy supplied by the constant power source melts the ice present in the container.

Formula to calculate the rate of energy supplied by the source is,

P=Q1Δt1

• P is the power input of the constant power source,
• Q1 is the heat supplied by the source for all the ice to melt,
• Δt1 is the time duration till all ice melt,

The formula for the heat supplied is,

Q1=miceLf

• Lf is the latent heat of fusion of ice,
• mice is the mass of ice,

Use miceLf for Q1 in P=Q1Δt1 to rewrite P.

P=miceLfΔt1

For the next 10 minute, the power input by the source increases the temperature of the cold water from the ice and the water already present in the container.

Formula to calculate the rate of energy supplied by the source to increase the temperature is,

P=Q2Δt2

• Q2 is the heat supplied by the source to raise the temperature of the mixture,
• Δt2 is the time duration taking by the source to raise the temperature of the mixture,

Formula to calculate the energy supplied by the source to raise the temperature of the mixture is,

miceLfΔt1=(mwater+mice)cwater(TfTi)Δt2miceLfΔt1=mwatercwater(TfTi)Δt2+micecwater(TfTi)Δt2mice[LfΔt1cwater(TfTi)Δ</

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