Lesson 3 of 5Energy and Society

Unit 3 · Lesson 3 of 5

Thermal Energy, Heat Capacity and Phase Change

Why temperature and thermal energy are different, how much energy a substance needs to warm up, and why it takes so much more to melt or boil it.

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Key things to remember

1Temperature is average particle kinetic energy; thermal energy is the total; heat is thermal energy in transit.
2Heat moves by conduction, convection and radiation — only radiation works through a vacuum.
3 $Q = m c Δ T$ for a temperature change, with $Δ T$ as final minus initial.
4Water's specific heat is $4180\ \text{J/{(kg}\cdot^\circ\text{C)}}$ — unusually high, which is why it is a good coolant.
5 $Q_{re l e a se d} + Q_{ab sor b e d} = 0$ when two objects reach a common final temperature.
6 $Q = m L_{f}$ to melt or freeze, $Q = m L_{v}$ to boil or condense. Neither contains $Δ T$ .
7For water, $L_{f} = 3.34 \times 1 0^{5} J/kg$ and $L_{v} = 2.26 \times 1 0^{6} J/kg$ .
8During a phase change, temperature does not rise even though energy is being added.

Three words that everyday speech treats as one thing, and physics does not. Temperature is the average kinetic energy of the particles. Thermal energy is the total kinetic energy of all of them. Heat is thermal energy in transit from a hotter object to a cooler one.

A bathtub and a spark

A spark from a grinder is hotter than a bath at

40^{\circ} C

— far higher temperature. But the bath contains vastly more thermal energy, because it contains vastly more particles. Temperature is an average per particle; thermal energy is a total. This is why the spark is harmless and the bath could scald you.

Fig 1Three ways heat moves. Only radiation needs no medium at all.

Mechanism	How it works	Needs a medium?	Example
Conduction	Particle-to-particle collisions transfer energy through a material	Yes — solids especially	A metal spoon heating in soup
Convection	Warmer, less dense fluid rises and carries energy with it	Yes — fluids only	A radiator warming a room
Radiation	Infrared electromagnetic waves carry energy	No — works in vacuum	Sunlight reaching Earth

Heating a substance: specific heat capacity

Q = m c Δ T

Q is the heat in joules, m the mass, c the specific heat capacity, and ΔT the temperature change.

Specific heat capacity $c$ is the energy needed to raise one kilogram of a substance by one degree Celsius. Water's value is unusually large at $4180\ \text{J/{(kg}\cdot^\circ\text{C)}}$ — which is why the ocean moderates climate, why water is used as a coolant, and why a pot takes so long to come to the boil while the pan handle heats instantly.

Substance	$c$ in $\text{J/{(kg}\cdot^\circ\text{C)}}$	Consequence
Water (liquid)	4180	Excellent coolant; moderates coastal climate
Ice	2100	Takes half as much energy per degree as liquid water
Steam	2010	Similar to ice, and very different from liquid
Aluminium	900	Heats and cools quickly — good for cookware
Iron / steel	450	A pan handle gets hot fast
Copper	385	Superb conductor, low capacity — ideal for heat sinks
Lead	130	Warms very easily

Celsius and kelvin give the same ΔT

A change of one degree Celsius is exactly a change of one kelvin, so

Δ T

is numerically identical in both. You only need to convert to kelvin for absolute temperatures, which this equation never uses. Do keep

Δ T

as final minus initial, so cooling gives a negative

Q

Mixing: where the energy goes

Q_{re l e a se d} + Q_{ab sor b e d} = 0

Energy lost by the hot object exactly equals energy gained by the cold one, so the signed total is zero.

Drop a hot metal block into cool water and energy flows until both reach the same final temperature. The block's $Q$ is negative (it cools, so $Δ T$ is negative) and the water's is positive, and they cancel. This course assumes a perfectly insulated container, so nothing escapes to the surroundings.

Changing state: latent heat

Fig 2The flat sections are phase changes — energy going in with no temperature rise at all.

Heat ice steadily and its temperature climbs to $0^{\circ} C$ , then stops while the ice melts, even though energy is still pouring in. The energy is going into breaking the bonds holding the solid together rather than into speeding the particles up. Only once all the ice is gone does the temperature start rising again.

Q = m L_{f} Q = m L_{v}

Latent heat of fusion for melting or freezing; latent heat of vaporization for boiling or condensing. No ΔT appears — there isn't one.

Quantity for water	Value	Meaning
$L_{f}$ (fusion)	$3.34 \times 1 0^{5} J/kg$	To melt 1 kg of ice at $0^{\circ} C$
$L_{v}$ (vaporization)	$2.26 \times 1 0^{6} J/kg$	To boil 1 kg of water at $100^{\circ} C$

Why steam burns are so much worse

Boiling takes nearly seven times the energy of melting, and steam releases all

2.26 \times 1 0^{6} J/kg

the instant it condenses on your skin — on top of the energy it then gives up cooling from

100^{\circ} C

. Water at

100^{\circ} C

only delivers the second part.

Multi-stage problems

Taking ice at

- 10^{\circ} C

to steam at

110^{\circ} C

is five separate calculations: warm the ice (

m c Δ T

), melt it (

m L_{f}

), warm the water (

m c Δ T

), boil it (

m L_{v}

), warm the steam (

m c Δ T

). Each uses a different

c

, and the two phase changes have no

Δ T

at all. Add the five results.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

How much energy is needed to heat

2.5 kg

of water from

20^{\circ} C

85^{\circ} C

? Take

c = 4180\ \text{J/{(kg}\cdot^\circ\text{C)}}

Example 2

How much energy is needed to melt

0.40 kg

of ice that is already at

0^{\circ} C

? Take

L_{f} = 3.34 \times 1 0^{5} J/kg

Example 3

0.30 kg

iron block at

180^{\circ} C

is dropped into

1.2 kg

of water at

18^{\circ} C

in an insulated container. Find the final temperature. Take

c_{i ro n} = 450

and

c_{water} = 4180\ \text{J/{(kg}\cdot^\circ\text{C)}}

Example 4

How much energy converts

0.20 kg

of ice at

0^{\circ} C

into water at

50^{\circ} C

Where marks get lost

Using $Δ T$ in $Q = m L$ . There is no temperature change during a phase change, which is the whole point.
Treating a multi-stage problem as one calculation. Each warming stage and each phase change is separate.
Using the specific heat of liquid water for ice or steam. All three values differ.
Writing $Δ T$ as initial minus final, which flips every sign in a mixing problem.
Confusing temperature with thermal energy — a spark is hotter than a bath but carries far less energy.
Forgetting that mass must be in kilograms when $c$ is quoted per kilogram.
Assuming heat always flows to the object with less thermal energy. It flows from higher temperature to lower, regardless of totals.

Flashcards 11

Temperature

Thermal energy

Heat

Conduction, convection, radiation

+ 7 more

Practice

How much heat, in joules, is needed to raise

0.50 kg

of water by

30^{\circ} C

? Take

c = 4180\ \text{J/{(kg}\cdot^\circ\text{C)}}

Which statement correctly distinguishes temperature from thermal energy?

How much energy, in joules, is needed to melt

1.5 kg

of ice at

0^{\circ} C

? Take

L_{f} = 3.34 \times 1 0^{5} J/kg

A heating curve shows a flat horizontal section. What is happening there?

Which method of heat transfer can operate through a vacuum?

2.0 kg

aluminium block cools from

95^{\circ} C

25^{\circ} C

. How much energy does it release, in joules? Take

c = 900\ \text{J/{(kg}\cdot^\circ\text{C)}}

Why does water make a good coolant in car engines?

How much energy, in joules, is needed to boil

0.25 kg

of water already at

100^{\circ} C

? Take

L_{v} = 2.26 \times 1 0^{6} J/kg

Match each process to the equation that describes it.

A. $Q = m c Δ T$
B. $Q = m L_{f}$
C. $Q = m L_{v}$
D. $Q_{re l e a se d} + Q_{ab sor b e d} = 0$

Warming water from 20 to 60 °C

Melting ice at 0 °C

Boiling water at 100 °C

Hot metal cooling in water

Order the stages of taking ice at

- 10^{\circ} C

to steam at

110^{\circ} C

1Warm the steam from $100^{\circ} C$ to $110^{\circ} C$
2Boil the water at $100^{\circ} C$
3Warm the water from $0^{\circ} C$ to $100^{\circ} C$
4Warm the ice from $- 10^{\circ} C$ to $0^{\circ} C$
5Melt the ice at $0^{\circ} C$

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