Quantity of Heat and Heat Transfer

Lessons

Senior 3 · Physics

Quantity of Heat and Heat Transfer

5 lessons

Lesson 3 of 5

Latent Heat and the Heating Curve of Water

25 min

More heat, same temperature

Put a thermometer in a pot of water on a stove in Kigali. The reading climbs steadily until the water boils, then stops at about 95 °C. Turn the stove up and the water bubbles more fiercely, but the thermometer does not move. Heat is still pouring in, so where is it going?

The answer is that the heat is changing the state of the water, from liquid to gas, instead of raising its temperature. Heat that changes the state of a substance without changing its temperature is called latent heat, from the Latin for "hidden".

The heating curve of water

Imagine a block of ice at −20 °C heated at a steady rate, with a thermometer in it, until it has all turned to steam. A graph of temperature against time looks like this:

The graph has five parts:

Ice warms from −20 °C to 0 °C. The line is steep because ice has a low specific heat capacity (2100).
Melting at 0 °C: the line is flat. Ice and water are both present, and the heat supplied melts the ice.
Water warms from 0 °C to the boiling point. The line is less steep than for ice because water's specific heat capacity (4200) is twice as large.
Boiling: the line is flat again, and for much longer, because it takes far more heat to boil water away than to melt ice.
Steam warms if it is kept in a closed container and heated further.

The same curve read backwards is a cooling curve: steam condenses and water freezes at the same fixed temperatures, giving out the same latent heat.

Where does the latent heat go?

Temperature measures the average kinetic energy of the particles. During melting, the heat supplied does not make the molecules move faster; it pulls them apart against the forces holding them in the solid's fixed pattern. The energy is stored as potential energy of the particles. Boiling needs much more energy than melting, because in a gas the molecules must be separated completely, far beyond the short distances between molecules in a liquid.

Definition

Latent heat: heat absorbed or given out by a substance when it changes state at constant temperature.

Specific latent heat of fusion ( $l_{f}$ ): the heat needed to change 1 kg of a solid at its melting point into liquid at the same temperature. For ice, $l_f = 3.34 \times 10^{5}\ \mathrm{J\,kg^{-1}}$ .

Specific latent heat of vaporisation ( $l_{v}$ ): the heat needed to change 1 kg of a liquid at its boiling point into vapour at the same temperature. For water, $l_v = 2.26 \times 10^{6}\ \mathrm{J\,kg^{-1}}$ (at 100 °C).

For a mass $m$ changing state: $Q = m\,l$ . SI unit of $l$ : the joule per kilogram ( $\mathrm{J\,kg^{-1}}$ ).

Worked example 1: melting ice

How much heat is needed to melt 0.50 kg of ice at 0 °C into water at 0 °C?

Q = m\,l_f = 0.50 \times 3.34 \times 10^{5} = 1.67 \times 10^{5}\ \text{J} = 167\ \text{kJ}

Worked example 2: from ice to steam

Find the total heat needed to turn 0.20 kg of ice at −10 °C into steam at 100 °C. Take $c_{ice} = 2100$ , $c_{water} = 4200\ \mathrm{J\,kg^{-1}\,K^{-1}}$ .

Stage	Calculation	Heat (J)
Ice warms −10 °C → 0 °C	$0.20 \times 2100 \times 10$	4 200
Ice melts at 0 °C	$0.20 \times 3.34 \times 10^{5}$	66 800
Water warms 0 °C → 100 °C	$0.20 \times 4200 \times 100$	84 000
Water boils at 100 °C	$0.20 \times 2.26 \times 10^{6}$	452 000
Total		607 000

So $Q \approx 6.07 \times 10^{5}\ \text{J}$ . Boiling alone takes about three quarters of it.

Worked example 3: measuring $l_{v}$

A 500 W immersion heater keeps water boiling in a beaker that stands on a top-pan balance. In 4.0 min the balance reading falls by 0.050 kg. Find the specific latent heat of vaporisation.

E = P\,t = 500 \times 240 = 120\,000\ \text{J}

l_v = \frac{E}{m} = \frac{120\,000}{0.050} = 2.4 \times 10^{6}\ \mathrm{J\,kg^{-1}}

This is higher than the accepted $2.26 \times 10^{6}$ , because part of the heater's energy is lost to the air and the beaker instead of boiling water away.

Worked example 4: ice in a drink

An ice cube of mass 0.050 kg at 0 °C is dropped into 0.30 kg of juice at 25 °C (take $c = 4200$ ). Find the final temperature, ignoring the cup and the air.

The ice first melts, then the melt-water warms to $T$ ; the juice cools to $T$ :

0.050 \times 3.34 \times 10^{5} + 0.050 \times 4200 \times T = 0.30 \times 4200 \times (25 - T)

16\,700 + 210\,T = 31\,500 - 1260\,T \quad\Rightarrow\quad 1470\,T = 14\,800 \quad\Rightarrow\quad T \approx 10\ ^\circ\text{C}

Warning

Steam scalds worse than boiling water. When 10 g of steam condenses on skin and cools to 37 °C, it gives out $0.010 \times 2.26 \times 10^{6} + 0.010 \times 4200 \times 63 \approx 25\,000\ \text{J}$ ; 10 g of water at 100 °C gives out only about 2600 J. Point a kettle's spout away from you and lift saucepan lids away from your face.

Boiling point and altitude

Water boils when the pressure of its vapour equals the pressure of the air above it. Kigali lies at about 1,570 m above sea level (REMA gives the city's average elevation as 1,567 m), where the air pressure is only about 84 kPa instead of 101 kPa. So water boils there at about 95 °C (a value calculated from the standard atmosphere, not measured). Food in boiling water cooks more slowly, which is one reason pressure cookers are useful: they trap the steam, raise the pressure, and so raise the boiling point above 110 °C. Unit 5 looks at atmospheric pressure in detail.

Water boiling in a pan. The bubbles form throughout the liquid, not just at the surface: they are pockets of water vapour. While boiling continues, the water's temperature stays at its boiling point. Photo: Tomwsulcer, CC0, via Wikimedia Commons

Evaporation: cooling without boiling

Water also turns to vapour below its boiling point. The fastest molecules at the surface escape, so the average kinetic energy of those left behind falls and the liquid cools. The liquid then takes heat from its surroundings. This is why sweat cools the body and why wet clothes feel cold in the wind.

Evaporation	Boiling
Happens at any temperature	Happens only at the boiling point
Only from the surface	Throughout the liquid, with bubbles
Faster with a larger surface, wind, higher temperature and drier air	Faster only if heat is supplied faster
Cools the liquid that is left	Temperature stays constant while heat is supplied

Tip

Flat line = changing state. On a heating or cooling curve, every sloping part uses $Q = mc\,\Delta T$ and every flat part uses $Q = m\,l$ . A longer flat part means a larger latent heat.

Heating curve of water, time lapse with graph (MrGrodskiChemistry). A temperature probe draws the curve live as ice is heated to steam; watch for the two flat parts, and notice how much longer the boiling one is.

✦

Practice

Define the specific latent heat of vaporisation and give its SI unit.
How much heat is needed to melt 2.0 kg of ice at 0 °C?
How much heat is needed to boil away 0.40 kg of water already at its boiling point?
A 1500 W kettle is left boiling for 5.0 min. What mass of water boils away, if all the energy goes into vaporisation?
How much heat is needed to change 0.10 kg of ice at −5 °C into water at 30 °C?
On the heating curve of water, explain why the temperature stays constant during boiling although heat is still being supplied.
5.0 g of steam at 100 °C condenses on a hand and cools to 35 °C. Compare the heat given to the skin with that from 5.0 g of water at 100 °C cooling to 35 °C.
40 g of ice at 0 °C is added to 0.25 kg of water at 30 °C. Find the final temperature, ignoring heat losses.

Show answersHide answers

The heat needed to change 1 kg of a liquid at its boiling point into vapour at the same temperature; $\mathrm{J\,kg^{-1}}$ .
$Q = 2.0 \times 3.34 \times 10^{5} = 6.68 \times 10^{5}\ \text{J}$ (668 kJ).
$Q = 0.40 \times 2.26 \times 10^{6} = 9.04 \times 10^{5}\ \text{J}$ (904 kJ).
$m = \frac{1500 \times 300}{2.26 \times 10^{6}} \approx 0.20\ \text{kg}$ .
$0.10 \times 2100 \times 5 + 0.10 \times 3.34 \times 10^{5} + 0.10 \times 4200 \times 30 = 1050 + 33\,400 + 12\,600 = 47\,050\ \text{J} \approx 47\ \text{kJ}$ .
The heat is used to separate the water molecules completely (it increases their potential energy), not to make them move faster, so their average kinetic energy, and hence the temperature, does not rise.
Steam: $0.0050 \times 2.26 \times 10^{6} + 0.0050 \times 4200 \times 65 = 11\,300 + 1365 \approx 12\,700\ \text{J}$ . Water: $1365\ \text{J}$ . The steam gives about 9 times as much heat.
$0.040 \times 3.34 \times 10^{5} + 0.040 \times 4200\,T = 0.25 \times 4200 \times (30 - T) \Rightarrow 13\,360 + 168\,T = 31\,500 - 1050\,T \Rightarrow T = \frac{18\,140}{1218} \approx 14.9\ ^\circ\text{C}$ .

Key points

During a change of state the temperature stays constant; the latent heat changes the particles' potential energy.
$Q = m\,l$ : for water $l_f = 3.34 \times 10^{5}$ and $l_v = 2.26 \times 10^{6}\ \mathrm{J\,kg^{-1}}$ .
Heating curve: sloping parts use $mc\,\Delta T$ ; flat parts use $m l$ .
Lower air pressure lowers the boiling point: about 95 °C in Kigali. Evaporation cools the liquid left behind.

Sources and credits

14.3: Phase Change and Latent Heat, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Consulted for the physics; no text was copied.
Kigali's elevation: Kigali: State of Environment and Outlook Report 2013, Rwanda Environment Management Authority (REMA), 2013, p. 7. The pressure and boiling point at that height were calculated by Ganzaa from the International Standard Atmosphere and the Antoine equation for water.
Photos: Water boiling in a pot on a stove.jpg by Tomwsulcer, and Ikibindi cya kinyarwanda gikozwe mu ibumba hamwe nimiheha.jpg by Elysee 2006, both Wikimedia Commons, dedicated to the public domain under CC0 1.0.
Video: Heating Curve of Water, Time lapse with Graph, MrGrodskiChemistry, embedded from YouTube.

Written by Ganzaa: the lesson text, the heating-curve diagram, the tables, worked examples and practice exercises.

1More heat, same temperature

2The heating curve of water

3Where does the latent heat go?

4Boiling point and altitude

5Evaporation: cooling without boiling

More heat, same temperature

The heating curve of water

Where does the latent heat go?

Boiling point and altitude

Evaporation: cooling without boiling