D2.10 · Solve heat, temperature-change, and phase-change problems

Ontario Grade 11 Physics

Energy and Society

Using energy, specific heat capacity, and latent heat

What you will learn

• Distinguish temperature change from a change of state.
• Choose and use the correct heat equation for a temperature change or a phase change.
• Track heat entering or leaving a chosen system using a clear sign convention.
• Solve energy-transfer problems and check units, significant figures, and physical reasonableness.

1. Prerequisite bridge: temperature and energy

• Choose and name the system before calculating.
• Temperature is a scalar; heat transfer has a direction into or out of the system.
• Convert mass to kilograms and report energy in joules.

2. Temperature changes: specific heat capacity

• Use specific heat for a temperature change with no phase change.
• Use $\Delta T=T_f-T_i$ so the sign reflects warming or cooling.
• Specific heat capacity depends on the substance.

3. Phase changes and combined energy problems

• Use latent heat for melting, freezing, boiling, or condensing at the phase-change temperature.
• A phase change does not itself produce a temperature change.
• For an insulated exchange, account for energy lost and gained by the substances in the system.

4. A reliable solving routine

• Set the system and sign convention before using an equation.
• Separate stages with different physical changes.
• Check units, signs, rounding, and physical reasonableness.

Worked example

1. Heating a metal sample

1. Define the system
   The system is the aluminum sample. Heat entering it is positive. The sample warms, so energy transfers into it. The unknown is the heat transferred, $Q$.
2. Find the temperature change
   Subtract the initial temperature from the final temperature. A positive result agrees with the stated warming.
   $$\Delta T=85.0\ ^\circ\mathrm{C}-20.0\ ^\circ\mathrm{C}=65.0\ ^\circ\mathrm{C}$$
3. Use specific heat
   There is a temperature change and no phase change, so use the specific-heat relationship. Substitute the mass, specific heat capacity, and temperature change with their units.
   $$Q=(0.250\ \mathrm{kg})(900\ \mathrm{J/(kg\cdot{}^\circ C)})(65.0\ ^\circ\mathrm{C})=14625\ \mathrm{J}$$
4. Round and check
   The mass has three significant figures, while the supplied specific heat has two. Report two significant figures. Kilograms and degrees Celsius cancel, leaving joules. The positive result agrees with energy entering a warming sample.
   $$Q=1.5\times10^4\ \mathrm{J}$$

Worked example

2. Melting ice

1. Define the system
   The system is the ice. Energy entering it is positive. The ice melts, so its state changes but its temperature does not change during the melting stage. The unknown is the heat absorbed.
2. Choose the phase-change relationship
   Because the problem describes melting at the melting temperature, use latent heat of fusion rather than the specific-heat equation.
   $$Q=mL_f$$
3. Substitute and calculate
   Use the ice mass in kilograms and the latent heat of fusion in joules per kilogram. The kilogram units cancel, leaving joules.
   $$Q=(0.0800\ \mathrm{kg})(3.34\times10^5\ \mathrm{J/kg})=26720\ \mathrm{J}$$
4. Round and check
   Both given values have three significant figures. Melting requires energy to enter the ice, so the answer is positive. The result has units of joules.
   $$Q=2.67\times10^4\ \mathrm{J}$$

Worked example

3. Warm water melting ice, then cooling

1. Define the system and energy directions
   The system is the warm water and the ice together. Energy does not leave this system in the insulated model. The warm water loses energy as it cools; the ice gains energy to melt and then the melted ice warms. The unknown is the final temperature, $T_f$.
2. Calculate energy available from the warm water
   First find the energy the original water would lose in cooling from 60.0 °C to 0.0 °C. This gives the energy available for melting and warming the ice. The magnitude is positive here because it is being compared as an amount of energy released.
   $$Q_{\mathrm{released}}=(0.200\ \mathrm{kg})(4180\ \mathrm{J/(kg\cdot{}^\circ C)})(60.0\ ^\circ\mathrm{C})=50160\ \mathrm{J}$$
3. Find energy used to melt the ice
   The ice starts at its melting temperature, so use latent heat for this stage. Subtract this energy from the amount released by the warm water to find the energy left to warm all the liquid water.
   $$Q_{\mathrm{melt}}=(0.0500\ \mathrm{kg})(3.34\times10^5\ \mathrm{J/kg})=16700\ \mathrm{J}$$
4. Find the final temperature
   After melting, the total liquid-water mass is 0.250 kg. The remaining energy warms this water from 0.0 °C to the final temperature. Equating the remaining energy to the specific-heat energy gives the result.
   $$T_f=\frac{50160\ \mathrm{J}-16700\ \mathrm{J}}{(0.250\ \mathrm{kg})(4180\ \mathrm{J/(kg\cdot{}^\circ C)})}=32.0\ ^\circ\mathrm{C}$$

Common mistakes and how to avoid them

Lesson summary

• Temperature change is final temperature minus initial temperature.
• For a temperature change, use $Q=mc\Delta T$.
• For a phase change, use $Q=mL$ and select the latent heat for that change.
• Define the system and sign convention before calculating. Heat entering is positive; heat leaving is negative.
• Check units, direction, significant figures, and whether the result makes physical sense.

Check your understanding

Key terms

System

The object or group of objects chosen for a physics calculation.

Heat

Energy transferred into or out of a system because of a temperature difference.

Specific heat capacity

The energy needed to raise the temperature of one kilogram of a substance by one degree Celsius.

Phase change

A change between solid, liquid, and gas.

Latent heat

Energy transferred per kilogram during a phase change, without a temperature change during that change.

Insulated

Modelled so that energy does not transfer between the system and its surroundings.

Continue through SPH3U

View the complete SPH3U Ontario Grade 11 Physics curriculum and lessons

• D2.9 · Determine specific heat capacity through inquiry
• D2.11 · Draw and analyse heating and cooling curves
• D1.1 · Analyse a technology that transfers or transforms thermal energy
• D1.2 · Assess societal and environmental impacts of energy technologies
• D2.1 · Use work, power, mechanical, thermal, and nuclear energy terminology
• D2.2 · Solve work, force, and displacement problems

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Physics (SPH3U), expectation D2.10. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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