An insulated beaker with negligible mass contains 0.2500.250 kg of water at 75.075.0°C. How many kilograms of ice at −20.0-20.0°C must be dropped into the water to make the final temperature of the system 40.040.0°C?

Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 17: Temperature and Heat

Problem 53

Chapter 17, Problem 53

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?

Verified step by step guidance

Identify the heat transfer processes involved: The water will lose heat as it cools down to the final temperature, and the ice will gain heat as it warms up to 0°C, melts, and then the resulting water warms up to the final temperature.

Calculate the heat lost by the water as it cools from 75.0°C to 40.0°C using the formula:

Q = m c (T

_i-T_f), where

m

is the mass of the water,

c

is the specific heat capacity of water,

T

_i is the initial temperature, and

T

_f is the final temperature.

Calculate the heat required to warm the ice from -20.0°C to 0°C using the formula:

Q = m c (T

_f-T_i), where

c

is the specific heat capacity of ice.

Calculate the heat required to melt the ice using the formula:

Q = m L

, where

L

is the latent heat of fusion for ice.

Calculate the heat required to warm the melted ice (now water) from 0°C to 40.0°C using the formula:

Q = m c (T

_f-T_i), where

c

is the specific heat capacity of water. Set the total heat gained by the ice equal to the total heat lost by the water and solve for the mass of the ice.

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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 exchange between the water and ice, as it determines how much energy is needed to change their temperatures.

Latent Heat of Fusion

Latent heat of fusion is the heat required to change a substance from solid to liquid at its melting point without changing its temperature. This concept is essential for determining the energy needed to melt the ice, which must be accounted for in the heat balance of the system.

Heat Transfer and Equilibrium

Heat transfer involves the exchange of thermal energy between substances, leading to a change in temperature until thermal equilibrium is reached. Understanding this concept is key to solving the problem, as it involves calculating the heat lost by water and gained by ice to reach the final temperature of 40.0°C.

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Overview of Heat Transfer

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