   Chapter 11, Problem 35P

Chapter
Section
Textbook Problem

A 40.-g block of ice is cooled to −78°C and is then added to 560 g of water in an 80.-g copper calorimeter at a temperature of 25°C. Determine the final temperature of the system consisting of the ice, water, and calorimeter. (If not all the ice melts, determine how much ice is left.) Remember that the ice must first warm to 0°C, melt, and then continue warming as water. (The specific heat of ice is 0.500 cal/g · °C = 2.090J/kg · °C.)

To determine
The final temperature of the system consisting of the ice, water and calorimeter.

Explanation

Given Info: Mass of ice block is 40 g, initial temperature of ice block is 78°C ,  mass of water is 560 g, mass of copper calorimeter is 80 g, initial temperature of water and calorimeter is 25°C , and Specific heat of ice is 2090J/kg°C

There are two possibilities; the ice can melt completely or partially. This can be identified only with the final temperature of the mixture. If the final temperature is greater than 0°C then all the ice melts else only partially melts. To find the final temperature let us assume that the ice melts completely.

Heat energy lost be ice is equal to heat energy gained by water and calorimeter.

Formula to calculate the heat required to raise the temperature of ice 0°C is,

Qice=micecice(Tf,iceTi,ice) (1)

• Qice is the heat required to raise the temperature of ice to 0°C ,
• mice is the mass of the ice,
• Ti,ice is the initial temperature of ice,
• Tf,ice is the final temperature of ice,

Formula to calculate the heat required to melt the ice to cold water is,

Qmelt=miceLf (2)

• Qmelt is the energy required to melt the ice to cold water,
• mice is the mass of the ice,
• Lf is the latent heat of fusion of ice,

Formula to calculate the heat required to raise the temperature of cold water.

Qice-water=micecwater(TfTi,ice-water) (3)

• Qice-water is the heat required to raise the temperature of ice water.
• cwater is the specific heat of water,
• Ti,ice-water is the initial temperature of cold water from ice,
• Tf is the final temperature of the mixture.

Formula to calculate the heat lost by the water is,

Qwater=mwatercwater(TfTi,water) (4)

• Qwater is the heat lost be the water,
• cwater is the specific heat of water,
• Ti,water is the initial temperature of cold water from ice,

Formula to calculate the heat lost by the calorimeter is,

Qcal=mcalccu(TfTi,cal) (5)

• Qcal is the heat lost by the calorimeter,
• mcal is the mass of the calorimeter,
• ccu is the specific heat of  the copper,
• Ti,cal is the initial temperature of the calorimeter,

Heat gained by water and calorimeter is equal to the heat lost by the ice.

Qwater+Qcal=(Qice+Qmelt+Qice-water) (6)

Substitute equation (1), (2), (3), (4), (5) in equation (6) and rewrite in terms of Tf .

{[mwatercwater(TfTi,water)+mcalccu(TfTi,cal)]=[micecice(Tf,iceTi,ice)+miceLf+micecwater(TfTi,ice-water)]}

Substitute 40 g for mice , 2090J/kg°C for cice , 0°C for Tf,ice , 78°C for Ti,ice , 3

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