Physics 001.001.014 Problems Involving Heat
Alignment
Learning Intentions
By the end of the lesson, students will be able to:
- Identify when to use
, , and thermal equilibrium energy balances. - Solve multi-step heating and cooling problems involving temperature change and phase change.
- Apply conservation of energy to thermal equilibrium problems.
- Explain assumptions used in calorimetry-style thermal equilibrium problems.
Success Criteria
By the end of the lesson, students have successfully:
- Selected the correct equation for each stage of a thermal process.
- Converted mass from grams to kilograms before substituting into equations.
- Used
for temperature change without phase change. - Used
for phase change at constant temperature. - Applied
for an isolated system. - Solved for an unknown final temperature, mass, or energy transfer.
Syllabus Reference
- Unit 1: Thermal, Nuclear and Electrical Physics
- Topic 1: Heating Processes
- Solve problems involving specific heat capacity using
. - Solve problems involving specific latent heat using
. - Describe thermal equilibrium in terms of temperature and average kinetic energy.
- Solve problems involving specific heat capacity, specific latent heat and thermal equilibrium.
Phenomenon
A student places ice cubes into a warm drink. The drink cools down, the ice warms up, and some or all of the ice melts. Eventually, the drink and melted ice reach the same final temperature.
The key question is:
Why can we calculate the final temperature of the drink using energy conservation?
Key Idea
Thermal energy transfers from the hotter substance to the colder substance until thermal equilibrium is reached.
If the system is treated as isolated, then:
For multi-step problems, energy may be used to:
- warm or cool a substance:
- melt or freeze a substance:
- reach a shared final temperature:
Concept
Specific heat capacity describes the energy required to change the temperature of
Specific latent heat describes the energy required to change the state of
Thermal equilibrium occurs when two or more systems in thermal contact reach the same temperature and no net thermal energy transfer occurs.
Convention
The key conventions associated with this concept are below.
-
Use SI units:
- mass in kilograms,
- temperature change in
or - energy in joules,
- specific heat capacity in
- specific latent heat in
- mass in kilograms,
-
Temperature change is calculated using:
- In thermal equilibrium problems, choose a system boundary.
For an isolated system:
or
- Common values:
- water:
- ice:
- steam:
- latent heat of fusion of ice:
- latent heat of vaporisation of water:
- water:
Misconceptions
Common misconceptions students have regarding the concept when applying to various situations and solving problems.
- Students may use
during a phase change, even though temperature remains constant during melting or boiling. - Students may forget to convert mass from grams to kilograms.
- Students may think coldness transfers from the ice to the drink, rather than thermal energy transferring from the warmer drink to the colder ice.
- Students may assume the final temperature is always the average of the starting temperatures.
- Students may ignore latent heat when ice melts or water boils.
- Students may use Celsius temperature values directly instead of temperature change for
. - Students may forget that thermal equilibrium requires the final temperatures to be equal.
Further Reading
- Specific heat capacity
- Specific latent heat
- Thermal equilibrium
- Conservation of energy
- Calorimetry
- First law of thermodynamics
Explicit Instruction
Teacher Explanation
When solving thermal energy problems, first identify the process.
Case 1: Temperature Changes, but State Does Not Change
Use:
where:
is energy transferred in joules, is mass in kilograms, is specific heat capacity is change in temperature
Case 2: State Changes, but Temperature Does Not Change
Use:
where:
is the specific latent heat - for melting or freezing, use
- for boiling or condensing, use
Case 3: Two Substances Reach Thermal Equilibrium
Use conservation of energy:
For example, hot water cooling down and cold water warming up:
where:
is the initial hot temperature is the initial cold temperature is the final equilibrium temperature
Problem-Solving Method
- Draw an energy flow diagram.
- Identify what is warming, cooling, melting, freezing, boiling, or condensing.
- Write each energy term separately.
- Use
for temperature changes. - Use
for state changes. - Apply conservation of energy.
- Solve algebraically.
- Check the final answer is physically reasonable.
Worked Examples
Worked Example 1
A
Given:
Use:
Solution:
$\begin{align}
Q&=mc\Delta T \
Q&=(0.250)(4.18 \times 10^3)(55.0) \
Q&=57475 \text{ J} \
Q&=5.75 \times 10^4 \text{ J}
\end{align}$
Answer:
The energy transferred to the water is
Worked Example 2
Calculate the energy required to melt
Given:
Use:
Solution:
$\begin{align}
Q&=mL \
Q&=(0.0350)(3.34 \times 10^5) \
Q&=11690 \text{ J} \
Q&=1.17 \times 10^4 \text{ J}
\end{align}$
Answer:
The energy required to melt the ice is
Worked Example 3
A
Since both substances are water, use:
Hot water cools:
Cold water warms:
Solution:
$\begin{align}
m_hc_w(T_h-T_f)&=m_cc_w(T_f-T_c) \
(0.200)c_w(80.0-T_f)&=(0.100)c_w(T_f-20.0)
\end{align}$
Since
$\begin{align}
0.200(80.0-T_f)&=0.100(T_f-20.0) \
16.0-0.200T_f&=0.100T_f-2.00 \
18.0&=0.300T_f \
T_f&=60.0^\circ \text{C}
\end{align}$
Answer:
The final equilibrium temperature is
Worked Example 4
A
Given:
First, calculate the energy needed to melt the ice:
$\begin{align}
Q_{\text{melt}}&=m_iL_f \
Q_{\text{melt}}&=(0.0200)(3.34 \times 10^5) \
Q_{\text{melt}}&=6680 \text{ J}
\end{align}$
Next, calculate the maximum energy the warm water can release if it cools to
$\begin{align}
Q_{\text{water to }0^\circ \text{C}}&=m_wc_w\Delta T \
Q_{\text{water to }0^\circ \text{C}}&=(0.200)(4.18 \times 10^3)(25.0) \
Q_{\text{water to }0^\circ \text{C}}&=20900 \text{ J}
\end{align}$
Since
The remaining energy warms the melted ice from
Energy lost by warm water:
Energy gained by ice:
Set energy lost equal to energy gained:
$\begin{align}
m_wc_w(25.0-T_f)&=m_iL_f+m_ic_wT_f \
(0.200)(4.18 \times 10^3)(25.0-T_f)&=(0.0200)(3.34 \times 10^5)+(0.0200)(4.18 \times 10^3)T_f \
836(25.0-T_f)&=6680+83.6T_f \
20900-836T_f&=6680+83.6T_f \
14220&=919.6T_f \
T_f&=15.5^\circ \text{C}
\end{align}$
Answer:
All the ice melts and the final equilibrium temperature is
Check for Understanding
Check 1
Which equation should be used when a substance changes temperature but does not change state?
Answer:
Check 2
Which equation should be used when ice melts at
Answer:
Check 3
Why is temperature constant during melting?
Answer:
The energy added increases the internal potential energy of the particles by weakening or breaking bonds between particles. It does not increase the average kinetic energy of the particles, so the temperature remains constant.
Check 4
In a thermal equilibrium problem, why can we write
Answer:
If the system is isolated, energy is conserved. The thermal energy lost by the hotter object is equal to the thermal energy gained by the colder object.
Investigation (Alternative to Explicit)
Hypothesis
If hot water and cold water are mixed in an insulated cup, then the final equilibrium temperature can be predicted using conservation of energy because the energy lost by the hot water should equal the energy gained by the cold water.
Data Collection
Equipment:
- insulated cup or calorimeter
- digital thermometer
- balance
- measuring cylinder
- hot water
- cold water
- safety glasses
Method:
- Measure the mass of the empty insulated cup.
- Add a known mass of cold water to the cup.
- Measure the initial temperature of the cold water.
- Measure a known mass of hot water.
- Measure the initial temperature of the hot water.
- Add the hot water to the cold water.
- Stir gently and record the final stable temperature.
- Repeat with different mass ratios.
Analysis
Use:
Rearrange or substitute to calculate the theoretical final temperature.
Compare:
- theoretical final temperature
- experimental final temperature
- percentage error
Percentage error:
Evaluation
Discuss:
- energy lost to the surroundings
- energy absorbed by the cup
- uncertainty in thermometer readings
- delay in reading the final temperature
- incomplete mixing
- evaporation from hot water
Improvements:
- use a lid
- use a better insulated calorimeter
- stir consistently
- measure masses more accurately
- repeat trials and calculate a mean
Problems
The following problems are designed to build from single-step calculations to multi-step thermal equilibrium problems.
Problem 1
Calculate the energy required to heat
Problem 2
Calculate the energy released when
Problem 3
Calculate the energy required to melt
Use
Problem 4
Calculate the energy required to boil
Use
Problem 5
A
Problem 6
A
Problem 7
A
Hint:
The ice must:
- warm from
to - melt at
- warm as liquid water to
Problem 8
A student mixes hot and cold water in a cup and calculates a theoretical final temperature of
Followup
Self-check
Students should be able to answer the following:
- Can I explain the difference between specific heat capacity and specific latent heat?
- Can I identify when temperature changes and when state changes?
- Can I construct an energy balance equation?
- Can I solve thermal equilibrium problems using conservation of energy?
- Can I explain why real experimental results may differ from theoretical predictions?
Next Topic
The next topic is the relationship between thermal energy and mechanical work, including the first law of thermodynamics: