Thermal Properties of Matter: Class 11 Physics · Chapter 10

10.1 Introduction

A mechanic fits a steel ring gear onto a flywheel by warming the ring until it slides over, then letting it cool into a tight grip. On Chennai's Marina beach the afternoon breeze blows in from the sea, but late at night it drifts out. A pressure cooker matters far more in Leh than in Kochi, and a puff of steam can hurt more than a splash of boiling water.

Each puzzle depends on how matter responds when energy flows in or out because of a temperature difference. This chapter shows how temperature is measured, how matter expands, how much energy warming and changes of state take, how heat travels, and how fast a cup of chai cools.

10.2 Temperature and heat

Dip a finger into bath water and you know at once whether it is hot or cold, but you could not tell 38 °C from 41 °C, and you could hardly test molten iron or liquid nitrogen with your hand. Our sense of hot and cold is also easily fooled, as the activity below shows. Physics replaces it with temperature (ताप): a number, read from an instrument on an agreed scale, that says how hot a body is.

Put a pan of just-boiled milk and a frozen gel pack from a lunch bag on the same kitchen shelf on a May afternoon. The milk cools, the pack thaws and warms, and after an hour or two both are at room temperature (Fig. 10.1). Energy has flowed out of the milk into the room, and out of the room into the pack. Energy that passes between a system and its surroundings because their temperatures differ is called heat (ऊष्मा). Once the temperatures match, the net flow stops, and we say the bodies are in thermal equilibrium.

Heat is energy in transit, so its SI unit is the joule (J). A body does not contain heat, any more than a bank account contains deposits: it contains internal energy, and heat is one way of changing it. Energy that crosses a boundary for any other reason, such as the work done by a stirrer churning water, is not heat (Chapter 11 develops this). The SI unit of temperature is the kelvin (K); in daily life we also use the degree Celsius (°C). The rest of the chapter follows what heat does to matter: it warms it, swells it, and melts or boils it.

10.3 Measurement of temperature

A digital fever thermometer reads the electrical resistance of a tiny bead that changes with temperature; a laboratory thermometer reads the length of a thread of coloured alcohol; an infrared 'gun' reads the radiation a surface gives off. Each relies on a thermometric property, a measurable quantity that changes steadily and reproducibly with temperature. For a liquid-in-glass thermometer the liquid, usually mercury or alcohol, must expand almost linearly, so that equal steps along the thread mean equal steps of temperature.

To turn a thread length into a number you need a scale, and a scale needs two fixed points: physical events that happen at one fixed temperature every time and are easy to reproduce. The traditional pair is the ice point, where pure ice and water coexist, and the steam point, where pure water boils, both at standard atmospheric pressure. The Celsius scale labels them 0 °C and 100 °C and divides the gap into 100 degrees. The Fahrenheit scale labels them 32 °F and 212 °F, so the same gap holds 180 degrees (Fig. 10.2).

Both scales are linear, so a temperature lies the same fraction of the way from the ice point to the steam point on each:

Rearranged, $t_F=\frac{9}{5}{t}_C+32.$ A graph of $t_F$ against $t_C$ is a straight line with slope $9\mathrm{/}5$ that cuts the $t_F$ axis at 32. Remember that changes convert with the slope alone: a rise of 10 Celsius degrees is a rise of 18 Fahrenheit degrees.

10.4 Ideal-gas equation and absolute temperature

Calibrate a mercury thermometer and an alcohol thermometer at 0 °C and 100 °C, then dip both into water that the mercury one calls 50 °C. The alcohol one reads slightly differently, because the two liquids do not expand in exactly proportional ways, and neither has a better claim to be right. Gases settle the question. At low density, hydrogen, helium and nitrogen all respond to temperature in the same way, so thermometers built on them agree with one another, and a gas becomes the natural standard.

For a fixed amount of gas, three quantities describe its state: pressure $P$, volume $V$ and temperature. In 1662 Robert Boyle reported that at a steady temperature the product $PV$ stays constant: Boyle's law. In the 1780s Jacques Charles found that at steady pressure the volume grows in step with temperature, so that $V\mathrm{/}T$ is constant, provided $T$ is counted from the right zero (we find it below): Charles' law. Together they make $PV\mathrm{/}T$ constant for a given sample. Doubling the amount of gas doubles $PV\mathrm{/}T$, so it is proportional to the number of moles $\mu$:

Real gases follow Eq. (10.2) closely when their density is low; a gas that would follow it exactly is called an ideal gas. Hold the volume fixed and the equation gives $P\propto T$: the pressure of the trapped gas becomes the thermometric property. This is the principle of the constant-volume gas thermometer.

Plot pressure against Celsius temperature for such a thermometer and you get a straight line (Fig. 10.3). A different gas or a different amount gives a different slope. Real gases liquefy long before their pressure falls anywhere near zero, but if you extend each line downwards they all reach zero pressure at the same temperature, −273.15 °C. An ideal gas would exert no pressure there, and no lower temperature makes sense. This limit is absolute zero.

Lord Kelvin (William Thomson) proposed a scale whose zero is absolute zero and whose degree is the same size as the Celsius degree. Temperatures on it are written in kelvin, with no degree sign:

A temperature difference has the same value in kelvin and in Celsius degrees, so a unit like $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$ works with Celsius changes. But in $PV=\mu RT$, and in every law that multiplies or divides by $T$, you must use kelvin.

10.5 Thermal expansion

Electricity cables strung between poles sag visibly more on a May afternoon than on a December morning. Concrete roads are laid as separate slabs with narrow gaps filled with a soft sealant. A thick glass tumbler can crack if boiling water is poured into it straight from the fridge, because its inner surface swells before the outer surface has warmed. Most substances expand when heated and contract when cooled; this change of size with temperature is thermal expansion.

We call the change in a length linear expansion, the change in a surface area expansion and the change in a volume volume expansion (Fig. 10.4). For a rod of length $l$ warmed by a small $\mathrm{\Delta }T$, experiments show that the fractional change in length is proportional to $\mathrm{\Delta }T$:

The constant ${\alpha }_l$, the coefficient of linear expansion or linear expansivity, depends on the material and has the unit ${\mathrm{K}}^{-1}$. Table 10.1 lists typical values. Copper expands about five times as much as Pyrex glass for the same warming, which is why Pyrex dishes survive going from a fridge into an oven. Invar, an iron–nickel alloy, hardly changes at all, so it is used in measuring tapes and precision instruments.

For volume we define the coefficient of volume expansion in the same way:

Like ${\alpha }_l$, ${\alpha }_V$ is a property of the material, but it is not strictly constant. For a solid such as copper it is tiny at very low temperatures, grows as the solid warms and levels off only at high temperatures. Over everyday ranges we treat both coefficients as constants. Table 10.2 shows that liquids expand far more than solids. Ethanol expands about six times as much as mercury, which is why an alcohol thermometer's thread moves so far for each degree.

Water: the odd one out

Water is the famous exception. Its density peaks near 4 °C (more precisely 3.98 °C, at 999.97 kg m⁻³), so between 0 °C and 4 °C water shrinks as it warms and swells as it cools. Freezing adds a further jump of about 9 % in volume, which is why ice floats and why a full glass bottle left in a freezer cracks. Both quirks come from hydrogen bonds, which lock the molecules of ice into an open, roomy framework; fragments of that framework already appear in water just above freezing.

Picture a lake in Ladakh as winter sets in (Fig. 10.5). Density decides which water lies on top. While the surface is warmer than 4 °C, chilling makes it heavier, so it sinks and the lake turns over until all of it is near 4 °C. Water chilled further is now lighter than the 4 °C water beneath, so the coldest layer floats, reaches 0 °C and freezes into a lid. Ice conducts heat poorly, so the lid slows further cooling, and the bed of the lake stays near 4 °C all winter, with its fish alive. A liquid that grew steadily denser on cooling would freeze solid from the bed upwards.

Gases

Gases expand much more than solids or liquids, and their expansivity depends strongly on temperature. For an ideal gas it follows directly from the gas equation.

Linking volume and linear expansion

Thermal stress

Now stop the expansion. Fix a steel rod between two immovable walls and warm it (Fig. 10.7). The walls must push it back by exactly the length it would have gained, so the rod carries a compressive strain $\mathrm{\Delta }l\mathrm{/}l={\alpha }_l\mathrm{\Delta }T\mathrm{.}$ By Hooke's law (Chapter 8) the thermal stress in the rod is

The stress depends only on the material and the temperature change, not on the rod's length or thickness. The force, $F=AY{\alpha }_l\mathrm{\Delta }T,$ grows with the cross-section and can be enormous. That is why railway lines were once laid with gaps between rails, why modern welded rails are laid at a carefully chosen temperature and anchored to heavy concrete sleepers, and why bridges rest on rollers and expansion joints.

10.6 Specific heat capacity

Why does an empty steel tawa get too hot to touch within a minute on the gas, while a pan of water on the next burner takes several minutes just to get warm? The heat needed to warm something depends on more than the flame. A heater that supplies energy at a steady rate lets you compare amounts of heat simply by timing them.

Three things matter, then: how much material there is ($m$), how far it is warmed ($\mathrm{\Delta }T$) and what it is made of. The heat capacity $S$ of a body is the heat it needs per kelvin of warming:

Heat capacity belongs to an object: a full bucket has more than a cup. To get a property of the material, divide by the mass. The specific heat capacity $s$ is the heat needed per kilogram per kelvin:

Equivalently $\mathrm{\Delta }Q=ms\mathrm{\Delta }T,$ which holds while the substance stays in one state. The SI unit of $s$ is $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}\mathrm{.}$ It depends on the substance and, slightly, on its temperature.

When the amount of substance is counted in moles $\mu$, the heat needed per mole per kelvin is the molar specific heat capacity $C$, in $\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$:

For a gas the answer depends on how you heat it. Warmed in a rigid container, a gas uses all the heat to raise its internal energy; this gives $C_V$, the molar specific heat at constant volume. Warmed at constant pressure, it also expands and pushes back its surroundings, so it needs more heat for each kelvin; this gives $C_P$. In Table 10.5, $C_P-C_V$ comes out close to $8.3\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1},$ the gas constant $R$, for every gas. Chapter 11 explains why.

Compare a day in coastal Mumbai with one in Jaisalmer, in the Thar desert. In Mumbai the night is barely cooler than the afternoon; in Jaisalmer the temperature often swings by more than 15 °C. The difference is the sea. Water's specific heat is about five times that of sand or rock, so the Arabian Sea soaks up the day's sunshine with only a small rise in temperature and gives the heat back slowly through the night, while dry sand heats and cools fast. The same property makes water the coolant in car engines and the filling of a rubber hot-water bottle.

10.7 Calorimetry

Wrap a system so well that no heat can get in or out, and we call it isolated. Energy cannot leave it, so when heat flows between its parts, every joule lost by the hot parts turns up in the cold ones. Calorimetry rests on this bookkeeping, also called the method of mixtures:

Calorimetry means the measurement of heat, and a calorimeter is the vessel used for it (Fig. 10.9). Each part has a job:

• A thin cup of copper or aluminium. It takes little heat itself and quickly matches the temperature of its contents.
• A stirrer of the same metal, so that the water, the cup and any sample reach one common temperature.
• An outer jacket, with glass wool or foam packed into the gap, to slow heat leaking in from or out to the room.
• A lid with a hole for the thermometer.

The cup warms along with its contents, so it must be counted. A cup of mass $m_c$ and specific heat $s_c$ absorbs as much heat per kelvin as a mass $m_c{s}_c\mathrm{/}{s}_w$ of water would; this mass is the cup's water equivalent. In practice some heat still leaks to the room, so careful experiments are quick and start a little below room temperature.

10.8 Change of state

Ice cubes in a glass of nimbu-paani turn to water; water in a kettle turns to steam; on a winter morning, the water vapour in your breath turns back into tiny droplets. Each is a change of state, a transition between solid, liquid and gas, and each happens when the substance takes heat from, or gives heat to, its surroundings. Changes of state share a feature that you can check for yourself.

Names first: solid to liquid is melting (or fusion), and liquid to solid is freezing. Your graph shows what is special about them. The flat step means that for as long as both ice and water are present, they sit together in thermal equilibrium at one fixed temperature, the melting point. Each pure substance has its own. Because it shifts a little with pressure, data tables quote the normal melting point, measured at $1\mathrm{a}\mathrm{t}\mathrm{m}=1.013\times {10}^5\mathrm{P}\mathrm{a}\mathrm{.}$

Two states in equilibrium share one temperature and one pressure; what differs between them is the density, that is, the volume per mole. Ice floating in water at 0 °C is the everyday example: same temperature, same pressure, but the ice is about 9 % less dense.

Pressure changes melting points, and for ice the effect runs the 'wrong' way. Ice is less dense than water, so squeezing it favours the liquid and lowers the melting point, by about 0.007 °C for each extra atmosphere. A neat experiment shows the result.

Ice skating is often explained by the same effect, with the blade gliding on water melted by pressure. A skater's weight on a narrow blade does lower the melting point, but only slightly, while skating works well even at −20 °C. Measurements show that most of the slippery layer comes from frictional heating and from a thin film at the surface of ice that is liquid-like even below 0 °C.

Carry on heating and a second flat step appears near 100 °C. Now liquid is turning into vapour, which is vaporisation; in boiling, bubbles of vapour form inside the liquid itself. Liquid and vapour coexist in equilibrium at a fixed temperature, the boiling point, until the liquid is used up.

Ask anyone who has cooked rajma in Leh. About 3500 m up, the air pressure is roughly two-thirds of its sea-level value and water boils near 88 °C, so dal simmers for ages. A pressure cooker does the opposite: its weighted valve holds the pressure inside at about twice atmospheric, water boils near 120 °C, and food cooks in a fraction of the time. The boiling point at 1 atm is the normal boiling point.

Some solids skip the liquid stage altogether. Camphor (कपूर) left in an open puja box slowly disappears, and solid carbon dioxide, dry ice, turns straight into gas at −78.5 °C; the white fog around it is water vapour from the air condensing in the cold. A direct change from solid to vapour is sublimation; iodine and naphthalene balls also sublime.

10.8.1 Latent heat

Look again at the flat steps of Fig. 10.13: the heater is still pouring energy in, yet the thermometer does not move. That energy goes into rearranging the molecules, loosening the solid's framework or pulling the liquid's molecules apart, and it is called latent heat, from the Latin for hidden. The heat needed for a change of state is proportional to the mass that changes:

The latent heat $L$ depends on pressure and is quoted at 1 atm. For solid to liquid it is the latent heat of fusion $L_f$; for liquid to vapour, the latent heat of vaporisation $L_v$. The same amount comes back out when the change reverses: freezing water releases $L_f$ per kilogram and condensing steam releases $L_v$. Table 10.7 gives values for a few substances.

On a graph of temperature against heat supplied, the flat steps are the changes of state and the sloping parts have slope $\mathrm{\Delta }T\mathrm{/}\mathrm{\Delta }Q=1\mathrm{/}(ms)\mathrm{.}$ The ice and steam lines are steeper than the water line because ice and steam have about half the specific heat of liquid water.

For water, $L_f=3.34\times {10}^5\mathrm{J}\mathrm{/}\mathrm{k}\mathrm{g}$ and $L_v=2.26\times {10}^6\mathrm{J}\mathrm{/}\mathrm{k}\mathrm{g}\mathrm{.}$ Turning a kilogram of boiling water into steam takes almost seven times as much heat as melting a kilogram of ice, and more than five times as much as warming that water from 0 °C to 100 °C. Turn that around: a kilogram of steam condensing on skin dumps $2.26\times {10}^6\mathrm{J}$ there before it has even begun to cool, which water already at 100 °C cannot do. A puff of steam from an idli steamer or a pressure-cooker whistle therefore burns worse than a splash of boiling water.

10.9 Heat transfer

Heat is energy on the move from hot to cold, between two bodies or between two parts of one body. It travels in three distinct ways, and a pan of milk on a gas stove shows all of them at once (Fig. 10.14): conduction through the steel ladle resting in it, convection in the milk as it circulates, and radiation that you feel on your hand held beside the flame.

10.9.1 Conduction

Leave a steel ladle in a pot of simmering dal and soon its handle is too hot to hold, although nothing has flowed up it. Energy has been handed along the steel from particle to particle: fast-jiggling particles at the hot end bump their slower neighbours, and in metals free electrons carry most of the energy, which is why metals are such good conductors. This is conduction, the transfer of heat through a material from hotter to cooler regions with no bulk motion of the material. Liquids conduct less well than solids, and gases very poorly.

To put numbers on conduction, use a standard arrangement (Fig. 10.15): a bar with a uniform cross-section $A$ and a length $L$, its sides lagged so that heat can leave only through the ends, which are pressed against two large reservoirs at $T_C$ and $T_D$ ($T_C>T_D$). Leave it long enough and the bar settles into a steady state in which nothing changes with time: the temperature falls evenly from one end to the other, and heat leaves the cold end exactly as fast as it enters the hot end. Experiments on such bars give a simple rule. The rate of flow, the heat current $H$, doubles if you double the cross-section or the temperature difference, and halves if you double the length:

The constant $K$ is the thermal conductivity (ऊष्मा चालकता) of the material. Its SI unit is $\mathrm{J}{\mathrm{s}}^{-1}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1},$ that is, $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\mathrm{.}$ It varies a little with temperature, but over ordinary ranges we treat it as constant. Table 10.8 shows its enormous range: copper conducts heat about 15 000 times better than still air.

Designers use both ends of the table. A steel tumbler of hot chai is uncomfortable to hold while a clay kulhad of the same chai is fine, because the kulhad's wall conducts heat to your fingers far more slowly. Ice-cream sellers pack their stock in thermocol boxes, and a razai keeps you warm, mainly because both trap air, one of the poorest conductors there is. At the other extreme, a laptop moves heat from its processor through copper heat pipes to aluminium fins, so that the chip never overheats.

10.9.2 Convection

Sprinkle a pinch of haldi into a pan of water on the stove and watch the specks: they rise up the middle and sink down the sides, round and round. The water itself is moving and carrying heat with it. This is convection (संवहन), and it can happen only in liquids and gases. Water at the bottom is warmed, expands and becomes less dense, so buoyancy pushes it up; denser, cooler water flows down to replace it and is warmed in turn. Such a convection current spreads heat through the pan far faster than conduction alone. Because it relies on buoyancy, this natural convection needs gravity: on the International Space Station a candle burns as a dim, round glow instead of a tall flame.

In forced convection a fan or pump makes the fluid move. A ceiling fan, a convection oven and the fan inside a laptop all push air past hot surfaces to carry heat away faster. Your body runs a forced-convection system too: the heart pumps blood, which collects heat from busy muscles and organs and carries it to the skin, keeping the whole body at nearly the same temperature.

Stand on Marina beach at four in the afternoon and the wind blows steadily in off the Bay of Bengal; stand there after midnight and a gentler breeze drifts out to sea (Fig. 10.17). The cause is a convection loop driven by uneven heating. Under the afternoon sun the sand gets much hotter than the sea surface, because the sea's large specific heat and its constant stirring keep its temperature nearly steady. The hot sand warms the air above it; that air turns buoyant and climbs, spreads seawards high up, sinks over the cooler water and returns along the surface as the sea breeze. After dark the sand loses its heat quickly while the sea stays warm, so the loop runs backwards and a land breeze blows off the shore.