Thermal Equilibrium Examples in Physics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Thermal Equilibrium.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Thermal equilibrium is the state in which objects in thermal contact have reached the same temperature, so there is no net transfer of thermal energy between them.

When two objects stop getting warmer or cooler because they have matched temperatures, they are in thermal equilibrium.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Thermal Equilibrium starts by identifying what is warmer, what is cooler, and what energy or state variable changes.

Common stuck point: Students often know a formula related to thermal equilibrium but skip the recognition step: Am I tracking thermal energy transfer, particle motion, temperature change, or pressure-volume-temperature relationships? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I tracking thermal energy transfer, particle motion, temperature change, or pressure-volume-temperature relationships?

Worked Examples

Example 1

medium

0.2

kg of aluminium (

c = 900

) at

200^\circ

C is dropped into

0.5

kg of water (

c = 4200

) at

20^\circ

C. Find the equilibrium temperature.

Answer

T_f \approx 34.1^\circ\text{C}

First step

Heat lost by Al = heat gained by water.

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Example 2

medium

A hot piece of metal loses

1200

J as it reaches equilibrium with

200

g of water (ignore losses). Find the temperature rise of the water. (

c_w = 4200

J/kg·

^\circ

C.)

Example 3

medium

1

kg of water at

20^\circ

C is in a

0.3

kg aluminium pan (

c = 900

) at

20^\circ

C. A burner adds

20000

J. Find the final temperature.

Example 4

hard

0.3

kg of metal (

c = 500

) at

250^\circ

C is dropped into

0.5

kg water (

c = 4200

) at

15^\circ

C in a

0.15

kg aluminium calorimeter (

c = 900

) also at

15^\circ

C. Find the equilibrium temperature.

Example 5

hard

Two cups of water are mixed:

0.3

kg at

T_1

and

0.2

kg at

T_2

reach

40^\circ

C. The same waters, used separately to cool a

0.1

kg piece of metal, give different final temperatures. Find

T_1

T_2 = 25^\circ

Example 6

hard

0.4

kg of ice at

-10^\circ

C is added to

1

kg of water at

40^\circ

C. (

c_{\text{ice}} = 2100

L_f = 334000

c_w = 4200

.) Find the equilibrium temperature.

Example 7

challenge

0.1

kg of steam at

100^\circ

C is bubbled through

1

kg of water at

20^\circ

C until all the steam condenses and equilibrium is reached. (

L_v = 2.26\times 10^6

c_w = 4200

.) Find the final temperature.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Two objects are in thermal contact. Thermal equilibrium is reached when what is true of their temperatures?

Example 2

easy

A cup of

80^\circ\text{C}

coffee sits in a

20^\circ\text{C}

room. What final temperature does it approach?

Example 3

easy

At thermal equilibrium between two objects, is there net heat flow between them?

Example 4

easy

Object A is at

50^\circ\text{C}

, object B at

30^\circ\text{C}

. Which way does heat flow when they touch?

Example 5

easy

If three objects are each in thermal equilibrium with a fourth, what can be said about their temperatures?

Example 6

easy

Equal masses of two metals start at different temperatures and reach equilibrium. Is their final temperature necessarily the midpoint?

Example 7

easy

Hot water at

60^\circ\text{C}

is mixed with equal-mass water at

20^\circ\text{C}

. Find the equilibrium temperature.

Example 8

easy

A thermometer reads a system's temperature by reaching equilibrium with it. Why does this work?

Example 9

medium

2 \text{ kg}

of water at

80^\circ\text{C}

mixes with

3 \text{ kg}

30^\circ\text{C}

. Find the equilibrium temperature.

Example 10

medium

0.5 \text{ kg}

metal (

c=400

) at

200^\circ\text{C}

is dropped into

1 \text{ kg}

water (

c=4200

) at

20^\circ\text{C}

. Find the equilibrium temperature.

Example 11

medium

Why does a large lake change temperature more slowly than a small puddle under the same sunlight?

Example 12

medium

1 \text{ kg}

of ice-cold water at

0^\circ\text{C}

mixes with

4 \text{ kg}

25^\circ\text{C}

(same substance). Find the equilibrium temperature.

Example 13

medium

Two blocks reach equilibrium. Block A lost

500 \text{ J}

. Ignoring losses to surroundings, how much heat did block B gain?

Example 14

medium

A hot stone is placed in an insulated bucket of water. Will the water and stone reach the same temperature?

Example 15

medium

3 \text{ kg}

of water at

T

mixes with

1 \text{ kg}

60^\circ\text{C}

to reach

30^\circ\text{C}

. Find

T

Example 16

medium

1 \text{ kg}

of water at

10^\circ\text{C}

is mixed with

1 \text{ kg}

50^\circ\text{C}

and

2 \text{ kg}

30^\circ\text{C}

(all same substance). Find the equilibrium temperature.

Example 17

medium

0.3 \text{ kg}

aluminium spoon (

c=900

) at

20^\circ\text{C}

is placed in

0.5 \text{ kg}

of soup (

c=4000

) at

80^\circ\text{C}

. Find the equilibrium temperature.

Example 18

challenge

0.4 \text{ kg}

of metal (

c=900

) at

150^\circ\text{C}

is dropped into

0.6 \text{ kg}

water (

c=4200

) at

15^\circ\text{C}

in a

0.2 \text{ kg}

aluminium cup (

c=900

) also at

15^\circ\text{C}

. Find the equilibrium temperature.

Example 19

challenge

1 \text{ kg}

block at

100^\circ\text{C}

is placed in

1 \text{ kg}

water at

20^\circ\text{C}

and they reach

40^\circ\text{C}

. Find the block's specific heat. (

c_{water}=4200

)

Example 20

challenge

Equal masses of substance X (

c=500

) at

90^\circ\text{C}

and substance Y (

c=2500

) at

10^\circ\text{C}

are mixed. Find the equilibrium temperature.

Example 21

easy

1

kg of water at

80^\circ

C is mixed with

1

kg of water at

20^\circ

C. Find the equilibrium temperature.

Example 22

easy

1

kg of water at

40^\circ

C is mixed with

3

kg of water at

0^\circ

C. Find the equilibrium temperature.

Example 23

easy

Three objects, each in thermal equilibrium with a fourth, are mixed together. Their temperatures are _____.

Example 24

easy

200

g of water at

80^\circ

C is poured into

200

g of water at

20^\circ

C. Find the equilibrium temperature.

Example 25

medium

3

kg of water at

T

is mixed with

1

kg at

80^\circ

C to reach

50^\circ

C. Find

T

Example 26

medium

0.3

kg copper block (

c = 390

) at

100^\circ

C is dropped into

0.4

kg of water (

c = 4200

) at

25^\circ

C. Find the equilibrium temperature.

Example 27

medium

Equal masses of two liquids (

c_A = 2000

c_B = 4000

) at

80^\circ

C and

20^\circ

C respectively are mixed. Find the equilibrium temperature.

Example 28

medium

2

kg of water at

50^\circ

C is added to

0.5

kg of water at

90^\circ

C in an insulated container. Find the equilibrium temperature.

Example 29

medium

0.1

kg of iron (

c = 450

) at

300^\circ

C is dropped into

0.5

kg of water at

20^\circ

C (

c = 4200

). Find the equilibrium temperature.

Example 30

medium

A heater delivers

4200

J to

0.5

kg of water. Find the temperature rise. (

c_w = 4200

J/kg·

^\circ

C.)

Example 31

medium

200

g of water at

90^\circ

C and

300

g at

30^\circ

C are mixed in an insulated cup. Find the equilibrium temperature.

Example 32

medium

0.5

kg block at

80^\circ

C is placed in

0.5

kg water (

c = 4200

) at

20^\circ

C and they settle at

25^\circ

C. Find the block's specific heat.

Example 33

medium

Heat flows between two objects until equilibrium. Object A loses

800

J. Object B's temperature rises by

4^\circ

C with

c = 1000

J/kg·

^\circ

C. Find object B's mass.

Example 34

hard

1

kg of ice at

0^\circ

C is added to

2

kg of water at

80^\circ

C. Assume the ice melts completely. (

L_f = 334000

J/kg,

c_w = 4200

J/kg·

^\circ

C.) Find the equilibrium temperature.

Example 35

hard

0.5

kg of steam at

100^\circ

C is condensed into

2

kg of water at

20^\circ

C. (

L_v = 2.26\times 10^6

J/kg,

c_w = 4200

J/kg·

^\circ

C.) Find the equilibrium temperature.

Example 36

hard

0.2

kg block of aluminium (

c = 900

) at

T

is placed in

0.6

kg water at

25^\circ

C (

c = 4200

). Equilibrium is

30^\circ

C. Find

T

Example 37

hard

Equal masses of two metals X (

c_X = 600

) at

100^\circ

C and Y (

c_Y = 200

) at

20^\circ

C are placed in contact (no other losses). Find the equilibrium temperature.

Example 38

challenge

An insulated tank holds

5

kg of water at

60^\circ

C. Hot water at

80^\circ

C is added until the tank temperature is

70^\circ

C. Find the mass of hot water added.

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Related Concepts

Heat Transfer Temperature Specific Heat Capacity

Background Knowledge

These ideas may be useful before you work through the harder examples.

heat transfertemperature