D2.11 Draw and analyse heating and cooling curves

Ontario Grade 11 Physics

Energy and Society

Reading temperature changes and phase changes from a graph

A temperature–time graph can show how a material changes as it is heated or cooled. The graph may rise, fall, or stay flat. Those shapes carry information: a sloped section shows a temperature change, while a flat section shows a phase change occurring without a temperature change during that interval. In this lesson, the system is the sample of material being heated or cooled. Time increases from left to right, so the positive direction on the horizontal axis is forward in time. Temperature is a scalar: it has a value but no direction. A positive or negative graph slope describes how temperature changes as time advances; it is not a direction in space.

What you will learn

Label the axes and describe the physical system and sign convention for a heating or cooling curve.
Draw and interpret sloped sections and flat sections of a temperature–time graph.
Use graph values to compare temperature changes and calculate an average temperature-change rate.
Explain what a flat section indicates during a phase change.

1. Prerequisite bridge: reading the graph

A graph has a horizontal axis and a vertical axis. For a heating or cooling curve, put time on the horizontal axis and temperature on the vertical axis. Label each axis with its quantity and unit. Time is measured in seconds, with symbol

t

, and temperature is commonly measured in degrees Celsius, with symbol

T

. Use kelvins when a calculation requires an SI temperature difference; a change of

1\ \mathrm{K}

is the same size as a change of

1\ ^\circ\mathrm{C}

.

A point on the graph gives the sample’s temperature at a particular time. A sloped line means the temperature is changing. A horizontal line, also called a plateau, means the temperature stays constant while time passes. The slope of a graph describes how much the vertical quantity changes for a change in the horizontal quantity. Here, it is the temperature-change rate.

The sample is the physical system. The graph describes that same sample throughout the process. The sign convention is simple: time moves forward to the right. A rising graph has a positive temperature-change rate; a falling graph has a negative temperature-change rate. Temperature itself is not a vector, and neither heating nor cooling means motion in a particular physical direction.

\text{temperature-change rate}=\frac{\Delta T}{\Delta t}

Horizontal axis: time, in seconds.
Vertical axis: temperature, in degrees Celsius or kelvins.
Rising, falling, and flat sections show increasing, decreasing, and constant temperature.

2. What the curve shows

A heating curve shows a sample as it receives energy and its temperature and phase change. A cooling curve shows a sample as it loses energy and its temperature and phase change. A phase is a physical form of a material, such as solid, liquid, or gas. For a simple, idealized curve, assume the sample is a pure substance and the conditions remain steady.

On a sloped part of a heating curve, the sample remains in one phase while its temperature rises. The same kind of sloped section on a cooling curve shows a temperature decrease within one phase. Label the phase on each sloped section when it is known. The curve’s shape alone does not tell you the material’s identity.

A flat section during heating indicates that the sample is changing phase while its temperature remains constant during that interval. For example, a solid may melt into a liquid. During cooling, a flat section can show a liquid freezing into a solid. A later plateau can show a liquid changing to a gas during heating, or a gas changing to a liquid during cooling. The phase names depend on the process and the known material; do not assign a specific phase change if the information is not given.

To draw a curve, plot the supplied time and temperature values, label both axes and units, and connect the points in order. Use a smooth or straight section only as a guide to the stated trend; do not invent extra measurements. Mark a plateau where the temperature remains the same. Add phase labels only when they are supported by the information provided.

\Delta T=T_{\mathrm{final}}-T_{\mathrm{initial}}

A sloped section represents temperature change within a phase.
A plateau represents a phase change at constant temperature in the idealized curve.
A cooling curve generally falls or stays flat as time moves forward.

3. How to analyse and compare curves

Read the axes before interpreting the line. Find the time and temperature at the start and end of each section. A rising section has a positive temperature change; a falling section has a negative temperature change; a plateau has zero temperature change. State whether the sample is heating or cooling and identify any phase change only if the curve or problem gives enough information.

For a sloped section, calculate the average temperature-change rate by dividing the temperature change by the time interval. The SI unit for this rate is kelvins per second, written

\mathrm{K/s}

. A positive result means the temperature rose as time advanced. A negative result means it fell. This average describes the whole selected interval; it does not claim that every instant had exactly the same rate.

When comparing sections, use the same units and state the intervals. A larger magnitude of temperature-change rate means a larger temperature change per second. A flat section has a rate of zero, even though a phase change is taking place. A graph’s slope alone does not provide enough information to calculate the amount of energy transferred.

r=\frac{T_2-T_1}{t_2-t_1}

Identify the interval before calculating a rate.
Keep the sign: rising is positive and falling is negative.
Check that the result has units of temperature divided by time.

4. Drawing and checking your interpretation

A useful sketch shows the overall pattern clearly. For heating, draw time increasing to the right and temperature generally rising, with a horizontal section wherever a phase change is described. For cooling, draw time increasing to the right and temperature generally falling, with a horizontal section at a described phase change. Use the information in the question to decide how many sections to draw and where to place them.

Check a completed graph in three ways. First, confirm that the axis labels and units match the quantities. Second, check that each section agrees with the stated process: heating should not be shown as a temperature decrease, and cooling should not be shown as an increase. Third, check each plateau: its endpoints should have the same temperature, even though their times differ.

A graph is a model of the described process. Unless values are explicitly identified as measured data, treat them as given or illustrative values, not as results from a real investigation. A carefully labelled sketch can still communicate the sequence of temperature changes and phase changes without claiming more than the information supports.

Time advances to the right on both heating and cooling graphs.
A plateau has equal start and end temperatures.
Do not treat a sketch or supplied example as experimental evidence.

Worked example

Sketching a heating curve

A sample begins as a solid at

20\ ^\circ\mathrm{C}

. It warms to

80\ ^\circ\mathrm{C}

, melts while remaining at

80\ ^\circ\mathrm{C}

, and then warms as a liquid to

110\ ^\circ\mathrm{C}

. Draw and label the expected curve in the order described.

Set up the graph
The system is the sample. Time increases to the right, and temperature is the vertical quantity. Use seconds for time and degrees Celsius for the supplied temperatures.
Plot the sequence
Draw a rising section for the solid warming from $20\ ^\circ\mathrm{C}$ to $80\ ^\circ\mathrm{C}$ . Then draw a horizontal section at $80\ ^\circ\mathrm{C}$ for melting. Finish with a rising section for the liquid warming to $110\ ^\circ\mathrm{C}$ . The question gives no times, so do not assign numerical time values.
$20\ ^\circ\mathrm{C}\nearrow 80\ ^\circ\mathrm{C}\;\longrightarrow\;80\ ^\circ\mathrm{C}\nearrow 110\ ^\circ\mathrm{C}$

Answer: The graph has a rising solid section, a plateau at

80\ ^\circ\mathrm{C}

labelled melting, and a rising liquid section. Time runs to the right.

Check: The sample warms before and after melting, so both outer sections rise. The plateau keeps temperature constant during the stated phase change. No unsupported time measurements are added.

Worked example

Finding a cooling rate

A sample’s temperature changes from

60.0\ ^\circ\mathrm{C}

at

10.0\ \mathrm{s}

to

36.0\ ^\circ\mathrm{C}

at

70.0\ \mathrm{s}

. Find its average temperature-change rate for this interval.

Define the interval and sign
The system is the sample, and time increases to the right. The initial point is $10.0\ \mathrm{s}$ and $60.0\ ^\circ\mathrm{C}$ ; the final point is $70.0\ \mathrm{s}$ and $36.0\ ^\circ\mathrm{C}$ . A decrease in temperature should give a negative rate.
Use the graph relationship
Average temperature-change rate is final temperature minus initial temperature, divided by final time minus initial time. Convert the temperature difference to kelvins; its numerical size is $24.0\ \mathrm{K}$ .
$r=\frac{(36.0-60.0)\ \mathrm{K}}{(70.0-10.0)\ \mathrm{s}}$
Calculate and interpret
The temperature change is $-24.0\ \mathrm{K}$ and the elapsed time is $60.0\ \mathrm{s}$ . Dividing gives a negative rate. Three significant figures match the given values.
$r=-0.400\ \mathrm{K/s}$

Answer: The average temperature-change rate is

-0.400\ \mathrm{K/s}

, meaning the sample cools by an average of

0.400\ \mathrm{K}

each second over the interval.

Check: The units are temperature divided by time. The negative sign agrees with the falling temperature, and a change of

24.0\ \mathrm{K}

over

60.0\ \mathrm{s}

is consistent with the calculated rate.

Worked example

Interpreting a plateau while cooling

A cooling curve falls from

95\ ^\circ\mathrm{C}

to

40\ ^\circ\mathrm{C}

, stays at

40\ ^\circ\mathrm{C}

for an interval, and then falls from

40\ ^\circ\mathrm{C}

to

15\ ^\circ\mathrm{C}

. The material is known to freeze at

40\ ^\circ\mathrm{C}

. Describe the stages and the temperature-change rate during the plateau.

Read the falling sections
Time advances to the right. The first and last sections slope downward, so the sample’s temperature decreases during those intervals. The prompt does not identify the phases in those sections, so do not add phase labels.
Interpret the flat section
The sample remains at $40\ ^\circ\mathrm{C}$ while time passes. Since the material is stated to freeze at that temperature and is cooling, the plateau represents freezing.
$\Delta T=40\ ^\circ\mathrm{C}-40\ ^\circ\mathrm{C}=0\ \mathrm{K}$
State the rate
The temperature change is zero over a nonzero time interval. Therefore, the temperature-change rate is zero during the plateau, even though the phase is changing.
$r=\frac{0\ \mathrm{K}}{\Delta t}=0\ \mathrm{K/s}$

Answer: The sample cools to its freezing temperature, freezes at constant temperature, and then cools further. Its temperature-change rate during the plateau is

0\ \mathrm{K/s}

.

Check: A horizontal segment has no vertical change, so its rate is zero. The freezing label is justified by the stated material and process.

Common mistakes and how to avoid them

Putting temperature on the horizontal axis and time on the vertical axis without following the graph convention used in the question.

Correction: Check the instructions. For the curves in this lesson, put time on the horizontal axis and temperature on the vertical axis, with units.

Drawing a plateau as a sloped line because the sample is still changing during a phase change.

Correction: The phase changes, but the temperature stays constant during the idealized plateau. Draw a horizontal section.

Reporting a cooling rate as positive after calculating final temperature minus initial temperature.

Correction: Keep the sign from the subtraction. A falling temperature over forward time gives a negative rate.

Naming a phase change from a curve when the substance or process is not identified.

Correction: Describe the plateau as a phase change unless the given information supports a specific label such as melting or freezing.

Assuming a steeper curve proves that more energy is being transferred.

Correction: A steeper section shows a greater temperature-change rate. The curve alone does not give the amount of energy transferred.

Lesson summary

A heating or cooling curve plots temperature against time for one sample.
Time increases to the right. A rising or falling section shows changing temperature within a phase.
A horizontal plateau shows a phase change at constant temperature in the idealized curve.
Average temperature-change rate is temperature change divided by elapsed time. Its sign indicates whether temperature rises or falls.
Label axes, units, sections, and supported phase changes. Check the sign and units of any calculated rate.

Check your understanding

Question 1

On a temperature–time graph, what does a horizontal section indicate in an idealized heating curve?

The sample’s temperature is constant during a phase change.
The sample’s temperature is rising at a constant rate.
Time has stopped passing.
The sample has reached a temperature of zero.

Show answer and explanation

The sample’s temperature is constant during a phase change.

A horizontal section has no temperature change as time advances. In the idealized curve, it represents a phase change at constant temperature.

Question 2

A sample cools from

30.0\ ^\circ\mathrm{C}

to

18.0\ ^\circ\mathrm{C}

in

40.0\ \mathrm{s}

. What is its average temperature-change rate?

$+0.300\ \mathrm{K/s}$
$-0.300\ \mathrm{K/s}$
$-12.0\ \mathrm{K/s}$
$+12.0\ \mathrm{K/s}$

Show answer and explanation

-0.300\ \mathrm{K/s}

The temperature change is

18.0-30.0=-12.0\ \mathrm{K}

. Dividing by

40.0\ \mathrm{s}

gives

-0.300\ \mathrm{K/s}

. The negative sign shows cooling.

Question 3

Which axis arrangement is used for the curves in this lesson?

Time on the vertical axis and temperature on the horizontal axis.
Temperature on both axes.
Time on the horizontal axis and temperature on the vertical axis.
Phase on the horizontal axis and energy on the vertical axis.

Show answer and explanation

Time on the horizontal axis and temperature on the vertical axis.

Time is shown horizontally and temperature vertically. Label both axes with their units.

Key terms

System: The object or sample being studied. Here, it is the material represented by the curve.
Scalar: A quantity with a value but no direction. Temperature is a scalar.
Phase: A physical form of a material, such as solid, liquid, or gas.
Plateau: A horizontal part of a graph. On an idealized heating or cooling curve, it shows constant temperature during a phase change.
Temperature-change rate: The change in temperature divided by the time interval. It is measured in kelvins per second when using SI units.

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About this lesson and its review

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

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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D2.11 · Draw and analyse heating and cooling curves

Ontario Grade 11 Physics

Reading temperature changes and phase changes from a graph

What you will learn

1. Prerequisite bridge: reading the graph

2. What the curve shows

3. How to analyse and compare curves

4. Drawing and checking your interpretation

Sketching a heating curve

Finding a cooling rate

Interpreting a plateau while cooling

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SPH3U

About this lesson and its review