Exam practice

Each blow of the hammer does work on the bar. That energy makes the iron’s molecules vibrate faster, which increases their average kinetic energy. A higher average kinetic energy means a higher temperature, so the bar gets hot. Mechanical energy has been changed into thermal energy.

Heat flows from the tea to the surroundings only while the tea is hotter than the room. As it cools, the difference gets smaller, and when the tea reaches room temperature, the two are in thermal equilibrium. With no temperature difference, no more heat flows, so the tea stops cooling.

When water cools below 4 °C, it expands (the anomalous expansion of water), and when it freezes into ice, its volume increases by about 9% more. The pipe is already full, so the expanding water and ice push on its walls with a huge pressure, and the pipe cracks.

Water has a very high specific heat capacity (4,200 J/kg °C). It can take in a large amount of heat from the hot engine while its own temperature rises only a little. It then gives this heat off to the air through the radiator, and comes back to cool the engine again. Water is also cheap and easy to get.

Because of its high specific heat capacity, hot water gives out a large amount of heat for each degree it cools. So the water in the bag stays warm for a long time, giving a steady, gentle warmth to the body. (The rubber bag is a poor conductor, so the heat flows out slowly and doesn’t burn the skin.)

The sea has a high specific heat capacity, so it warms slowly by day and cools slowly by night, and stays at a nearly steady temperature. By day, a cool sea breeze blows onto the land; at night, a land breeze blows the other way and the warm sea keeps the air from getting very cold. So the air temperature on the coast doesn’t rise or fall very much.

A desert is mostly dry sand and rock, with very little water in the ground or the air. Sand has a low specific heat capacity, so it heats up quickly in the strong sun by day and loses its heat quickly at night. With no water nearby, and usually no clouds to keep the heat in, nothing slows these changes, so days are very hot and nights very cold.

The lukewarm water has the higher temperature, which means the average kinetic energy of its molecules is higher. But thermal energy is the total kinetic energy of all the molecules. The tank holds far more water, so it has many more molecules. Even though each moves a little more slowly, together they have much more energy. So the large tank of cold water has more thermal energy than the bucket, even though it is at a lower temperature. A higher temperature does not mean more thermal energy.

The molecules of the hot tea (and the cup) move faster, on average, than the molecules of our skin. When they collide, the faster molecules give some of their energy to the slower ones, so heat flows into our hand, and we feel hot. The molecules of the ice move more slowly than those of our skin, so energy flows from our skin into the ice; our hand loses heat, and we feel cold.

Pour hot water over the lid only (or hold it under a hot tap) for a minute. The heat increases the kinetic energy of the molecules of the lid, so they vibrate faster and move farther apart: the lid expands. Metal expands more than glass, and glass is a poor conductor so the bottle’s neck stays cooler, so the lid becomes slightly looser than the neck and can be turned easily.

• i.The anomalous expansion of water.
• ii.From 0 °C to 4 °C the volume decreases (from about 1000.13 cm³ to 1000.00 cm³). From 4 °C to 10 °C it increases again (to about 1000.27 cm³). The volume is smallest at 4 °C.
• iii.Other liquids expand steadily whenever they are heated, at all temperatures. Water contracts when heated from 0 °C to 4 °C, and only expands above 4 °C. (It also expands when it freezes, while most liquids contract when they become solid.)
• iv.It explains why lakes and ponds freeze from the top, so fish survive under the ice; why water pipes burst in winter in cold places; why bottles of water crack in the freezer; and why freezing water cracks rocks.
• v.Density is mass ÷ volume, so where the volume is smallest, the density is greatest. The graph rises from 0 °C to a peak at 4 °C (about 1000 kg/m³) and falls after that, like the right-hand graph of Figure 9.4.

On a very cold winter morning, the water in the jug has cooled close to 0 °C. Because of the anomalous expansion of water, water is densest at 4 °C. The water at about 4 °C, being the densest, sinks to the bottom, while water colder than 4 °C is less dense and stays on top. So the bottom of the jug holds the “warmest” water (near 4 °C), and the top the coldest (near 0 °C). The steel jug, a good conductor, lets the water lose heat to the cold air quickly, which makes these layers form.

Ghee behaves like most substances: as it cools and becomes solid (in a cold Nepali winter, ghee in its jar sets solid), its volume decreases and its density increases. Solid ghee is denser than melted ghee, so it sinks in it, and the surface of set ghee often dips in the middle where it shrank last.
Honey is a thick solution of sugars in a little water. In a home freezer it doesn’t freeze into a solid at all; it just becomes very thick and stiff, while its volume decreases slightly and its density increases slightly. (Over time, cool honey can form sugar crystals, called granulating.)
Both are unlike water, which expands and becomes less dense when it freezes, which is why ice floats on water but solid ghee sinks in melted ghee.

The ice in the aluminium box melts faster. To melt, ice must take in heat from its surroundings. Aluminium is a good conductor of heat, so heat from the warm air flows quickly through the box walls into the ice. Wood is a poor conductor (an insulator), so heat reaches the ice in the wooden box only slowly, and it melts more slowly.

The specific heat capacity of a substance is the amount of heat needed to raise the temperature of 1 kg of it by 1 °C (or 1 K). Its SI unit is the joule per kilogram per degree Celsius, J/kg °C (or J/kg K).

$Q = ms(T_2 - T_1)$: the heat taken in or given out by a substance equals its mass × its specific heat capacity × its change in temperature. Here $Q$ is the heat (J), $m$ the mass (kg), $s$ the specific heat capacity (J/kg °C), and $T_1$ and $T_2$ the starting and final temperatures (°C).

Any two of: water is used as a coolant in car radiators and power stations; hot water bags keep us warm for a long time; a wet cloth cools a feverish forehead; the sea keeps coastal climates mild (sea and land breezes); water in our bodies keeps our temperature steady; farmers flood fields on frosty nights so that the water, giving out heat slowly, protects the crops.

Water boils at 100 °C only at standard sea-level pressure. Where the air pressure is lower, it boils at a lower temperature. At high altitude the air is thinner and its pressure lower: in Kathmandu (about 1,400 m) water boils at about 95 °C, at Namche Bazaar about 89 °C, and on Everest about 70 °C. Water also boils below 100 °C in a vacuum or in a syringe whose piston is pulled out while its nozzle is closed.

Liquid-in-glass thermometer: a liquid (mercury or alcohol) in a bulb expands when heated and rises up a fine capillary tube; the height of the thread against the scale gives the temperature.

Digital thermometer: a thermistor, whose electrical resistance changes with temperature, changes the current in a circuit; a chip shows this as a temperature on a display.

Radiation (infrared) thermometer: a lens focuses the infrared radiation given out by an object onto a sensor, which makes an electrical signal; the temperature is shown without touching the object.

Calibration is marking the scale on a thermometer. Two fixed points are marked first: the lower fixed point, by putting the bulb in pure melting ice and marking where the thread stops (0 °C); and the upper fixed point, by holding the bulb in the steam above water boiling at one atmosphere and marking where the thread stops (100 °C). The distance between the two marks is then divided into 100 equal parts, each 1 °C (or into 10 parts of 10 °C each, as in Activity 9.4).

$m$ = 500 g = 0.5 kg, $s$ = 4,200 J/kg °C, $T_1$ = 15 °C, $T_2$ = 85 °C
$$Q = ms(T_2 - T_1) = 0.5 \times 4200 \times (85 - 15) = 0.5 \times 4200 \times 70 = 147{,}000\ \text{J}$$
Q = 147,000 J = 147 kJ.

$m$ = 5 kg, $s$ = 460 J/kg °C, fall in temperature = 430 − 30 = 400 °C
$$Q = ms(T_1 - T_2) = 5 \times 460 \times 400 = 920{,}000\ \text{J}$$
Q = 920,000 J = 920 kJ.

First find the specific heat capacity of paraffin from the first sentence.

Specific heat capacity of paraffin:
$$s = \frac{Q}{m(T_2 - T_1)} = \frac{42000}{2 \times 10} = 2100\ \text{J/kg °C}$$
Heat for 4 kg, from 20 °C to 40 °C:
$$Q = ms(T_2 - T_1) = 4 \times 2100 \times (40 - 20) = 168{,}000\ \text{J}$$
Q = 168,000 J = 1.68 × 10⁵ J. (A quicker way: twice the mass and twice the rise need 2 × 2 = 4 times the heat, 4 × 42,000 J.)
Some books print this question with 4,200 J, which gives paraffin 210 J/kg °C, a tenth of its real value (about 2,100 J/kg °C, half of water’s); the method is the same, and the answer is then 16,800 J.

$m$ = 0.5 kg, $Q$ = 7,938 J, rise = 226 − 100 = 126 °C
$$s = \frac{Q}{m(T_2 - T_1)} = \frac{7938}{0.5 \times 126} = \frac{7938}{63} = 126\ \text{J/kg °C}$$
s = 126 J/kg °C, the same as mercury in the table.

Let the final temperature be $T$. Both are water, so $s$ cancels.
Heat lost by the hot water = heat gained by the cold water:
$$4 \times s \times (80 - T) = 16 \times s \times (T - 25)$$
$320 - 4T = 16T - 400$
$720 = 20T$, so $T = 36\ °\text{C}$
The mixture is at 36 °C. Check: it lies between 25 °C and 80 °C, closer to 25 °C because there is more cold water.