Ch 19: Work, Heat, and the First Law of Thermodynamics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 19: Work, Heat, and the First Law of Thermodynamics

Problem 21a

Chapter 19, Problem 21a

A 750 g aluminum pan is removed from the stove and plunged into a sink filled with 10.0 L of water at 20.0°C . The water temperature quickly rises to 24.0°C. What was the initial temperature of the pan in °C and in °F?

Verified step by step guidance

Step 1: Identify the principle of conservation of energy. The heat lost by the aluminum pan is equal to the heat gained by the water. Use the formula for heat transfer:

Q = mcΔT

, where

Q

is the heat,

m

is the mass,

c

is the specific heat capacity, and

ΔT

is the change in temperature.

Step 2: Write the equation for energy conservation:

Q_{pan} = Q_{water}

. Substitute the heat transfer formula for both the pan and the water:

m_{pan}c_{pan}(T_{initial, pan} - T_{final}) = m_{water}c_{water}(T_{final} - T_{initial, water})

Step 3: Convert the mass of the aluminum pan to kilograms (750 g = 0.750 kg) and the volume of water to mass (10.0 L of water = 10.0 kg, assuming the density of water is 1 kg/L). Use the specific heat capacities:

c_{pan} = 900 \, \(\text{J/kg°C}\)

for aluminum and

c_{water} = 4186 \, \(\text{J/kg°C}\)

for water.

Step 4: Rearrange the equation to solve for the initial temperature of the pan:

T_{initial, pan} = T_{final} + \(\frac{m_{water}\)c_{water}(T_{final} - T_{initial, water})}{m_{pan}c_{pan}}

. Substitute the known values:

T_{final} = 24.0 \, \(\text{°C}\)

T_{initial, water} = 20.0 \, \(\text{°C}\)

m_{pan} = 0.750 \, \(\text{kg}\)

m_{water} = 10.0 \, \(\text{kg}\)

c_{pan} = 900 \, \(\text{J/kg°C}\)

, and

c_{water} = 4186 \, \(\text{J/kg°C}\)

Step 5: Once the initial temperature of the pan in °C is calculated, convert it to °F using the formula:

T_{F} = \(\frac{9}{5}\)T_{C} + 32

. This will give the initial temperature of the pan in Fahrenheit.

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Key Concepts

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

Heat Transfer

Heat transfer is the process by which thermal energy moves from one object to another due to a temperature difference. In this scenario, the aluminum pan, being hotter than the water, transfers heat to the water until thermal equilibrium is reached. This concept is crucial for understanding how the temperature of the water increases as it absorbs heat from the pan.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. Each material has a unique specific heat capacity, which influences how much its temperature changes when heat is added or removed. For aluminum and water, knowing their specific heat capacities allows us to calculate the initial temperature of the pan based on the heat exchanged.

Thermal Equilibrium

Thermal equilibrium occurs when two objects in contact reach the same temperature, resulting in no net heat transfer between them. In this problem, the aluminum pan and the water reach thermal equilibrium at a temperature of 24.0°C. Understanding this concept is essential for solving the problem, as it allows us to set up equations based on the heat lost by the pan equaling the heat gained by the water.

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