NotesShala Class 11 · Physics

Chapter10

Thermal Properties of Matter

Temperature and Thermal Expansion · Calorimetry and Change of State · Heat Transfer

Reading key Definition Note Remember Derivation Strategy Formulas sit on graph paper · coloured words are the key terms

Part 1

Temperature and Thermal Expansion

We all have a common-sense idea of hot and cold. When a body gets heat, its size or its physical state may change. These changes follow definite laws. All the properties of matter that change because of heat are called its thermal properties. This chapter covers heat, how heat is measured and how heat moves from one place to another.

1.1Temperature and Heat

Temperature

Definition

Temperature is the degree of hotness or coldness of a body. It tells how hot or cold the body is. Temperature is always a relative measure.

Take two bodies at temperatures $T_1$ and $T_2$ , with $T_1 > T_2$ .

iBody at

T_1

It is the hotter body of the two.

iiBody at

T_2

It is the colder body of the two.

Note

When a body is heated, it may expand. Its physical state may also change.

Heat

Definition

Heat is the form of energy that flows from a hotter body to a colder body because of the temperature difference between them.

Heat transfer is a non-mechanical process. No mechanical work is done when heat flows.

Hotter body (

T_1

)→ heat flows →Colder body (

T_2

)

SI unit of heatjoule (J)

CGS unit of heat (widely used)calorie (cal)

Relation between the two units1 cal = 4.2 J

Definition

One Calorie

One calorie is the heat required to warm 1 g of water by 1 °C, from 14.5 °C to 15.5 °C.

1.2Caloric and Dynamic Theory of Heat

Caloric Theory of Heat

This old theory treats heat as a fluid called caloric. Caloric was thought to be invisible, weightless and odourless.

Caloric is added to a bodyTemperature rises ↑

Caloric is taken out of a bodyTemperature falls ↓

Dynamic Theory of Heat

Every substance is made of molecules. This is true for solids, liquids and gases. The molecules are always in continuous random motion.

A molecule can have three types of motion. Which types it has depends on the nature and the temperature of the substance.

iTranslatory Motion

The molecule moves along a straight line. It is common in gases.

iiVibrational Motion

The molecule moves to and fro about its mean position. It is the main motion in solids. It also occurs in liquids and gases.

iiiRotational Motion

The molecule spins about its own axis. This motion usually occurs at high temperature.

Note

Each type of motion gives kinetic energy to the molecules. The heat of a body is its total thermal energy. It is the sum of the kinetic energies of all its molecules, due to their translational, vibrational and rotational motion.

1.3Measurement of Temperature

Temperature is a fundamental physical quantity, so we need to measure it. It is measured on fixed scales.

Definition

Thermometer

A thermometer is an instrument that measures temperature. It is calibrated, so that each temperature gets a number.

Temperature is measured in these units:

Kkelvin (SI unit) °Cdegree Celsius (centigrade) °Fdegree Fahrenheit

Many physical properties of materials change enough with temperature. Such a property can be used to make a thermometer. The common ones are:

change in the volume of a liquid with temperature
change in the pressure or volume of a gas with temperature
change in the resistance of a metal with temperature
change in the thermo-emf of a thermocouple junction with temperature, etc.

Note

Fixed Reference Points

Every standard scale needs two reference points. For water, these are the ice point and the steam point. At these points pure water freezes and boils, under standard pressure.

Fahrenheit Scale

Ice point 32 °F, steam point 212 °F. The gap is split into 180 equal parts.

Celsius Scale

Ice point 0 °C, steam point 100 °C. The gap is split into 100 equal parts.

Plot the Fahrenheit temperature $T_F$ against the Celsius temperature $T_C$ . The graph is a straight line.

A change of 100 °C on the Celsius scale is the same as a change of 180 °F on the Fahrenheit scale. So $\Delta T_F = 180°$ when $\Delta T_C = 100°$ .

T_F

against

T_C

is a straight line

Equation of the Straight Line

T_F = \dfrac{9}{5}\,T_C + 32

Note

Kelvin and Celsius

One kelvin and one Celsius degree have the same size. So the two scales differ only by a fixed number:

T_K = T_C + 273.15

Different Scales to Measure Temperature

S.No.	Scale	Unit	Ice point (lower fixed point)	Steam point (upper fixed point)
1.	Celsius scale	degree Celsius (°C)	0 °C	100 °C
2.	Fahrenheit scale	degree Fahrenheit (°F)	32 °F	212 °F
3.	Réaumur scale	degree Réaumur (°R)	0 °R	80 °R
4.	Kelvin scale	kelvin (K)	273.15 K	373.15 K

Relation Among the Different Scales

The same two fixed points read on the four scales

Relation Among the Scales

\dfrac{C-0}{100} = \dfrac{F-32}{180} = \dfrac{R-0}{80} = \dfrac{K-273.15}{100}

where, $C$ , $F$ , $R$ and $K$ are the readings of the same temperature on the Celsius, Fahrenheit, Réaumur and Kelvin scales.

1.4Constant-Volume Gas and Platinum Resistance Thermometers

Constant-Volume Gas Thermometer

The volume of the gas is kept constant. Its pressure is noted at four temperatures:

$p_0$ at 0 °C $p_{100}$ at 100 °C $p_{tr}$ at the triple point of water $p$ at the unknown temperature

t

Unknown Temperature in °C

t = \left(\dfrac{p - p_0}{p_{100} - p_0} \times 100\right){}^{\circ}\text{C}

In Kelvin

T = \left(273.16\,\dfrac{p}{p_{tr}}\right)\text{K}

Platinum Resistance Thermometer

Here the resistance of a platinum wire is noted at four temperatures:

$R_0$ at 0 °C $R_{100}$ at 100 °C $R_{tr}$ at the triple point of water $R_t$ at the unknown temperature

t

Unknown Temperature in °C

t = \left(\dfrac{R_t - R_0}{R_{100} - R_0} \times 100\right){}^{\circ}\text{C}

In Kelvin

T = \dfrac{R_t}{R_{tr}}\,T_{tr} = \left(273.16\,\dfrac{R_t}{R_{tr}}\right)\text{K}

1.5Ideal Gas Equation

iLiquid-in-Glass Thermometer

Different liquids expand differently. So two such thermometers give different readings, except at the fixed points.

iiGas Thermometer

It gives the same reading, no matter which gas is used.

At low density, all gases expand in the same way. Three quantities describe the behaviour of a gas:

$p$ pressure $V$ volume $T$ temperature

Law

Boyle's Law

At constant temperature, the pressure of a given mass of gas is inversely proportional to its volume. So $p \propto 1/V$ , that is, $pV$ = constant.

Given by the English chemist Robert Boyle (1627–1691).

Law

Charles' Law

At constant pressure, the volume of a given mass of gas is directly proportional to its temperature. So $V \propto T$ , that is, $V/T$ = constant.

Given by the French scientist Jacques Charles (1747–1823).

Boyle's law+Charles' law→

\dfrac{pV}{T} =

constant (for a given amount of gas)

This relation is called the ideal gas law. For a given amount of gas, its general form is:

Ideal Gas Equation

pV = \mu RT

where, $\mu$ = number of moles of the gas, $R$ = universal gas constant $= 8.31\ \text{J mol}^{-1}\,\text{K}^{-1}$

Note

The ideal gas equation gives $pV \propto T$ . If the volume is kept constant, then $p \propto T$ . That is why a gas can measure temperature in a constant-volume gas thermometer.

1.6Absolute Zero Temperature

Definition

There is no upper limit to temperature. But there is a sharp lower limit. No body can be colder than this lowest temperature. It is called absolute zero temperature.

Maximum temperatureNo limit

Minimum temperatureA sharp limit: absolute zero

Absolute Zero from the p–T Graph

Keep a low-density gas at constant volume. Plot its pressure $p$ against its temperature in °C.
Extend the straight line backwards (extrapolate it). It meets the temperature axis at −273.15 °C.
The British scientist Lord Kelvin named this temperature absolute zero. It is written as 0 A or 0 K.
At this temperature, the pressure and volume of the gas would become zero.

Pressure against temperature for a low-density gas at constant volume

Definition

Absolute Zero of Temperature

Absolute zero is the lowest temperature, −273.15 °C. At this temperature a gas would have zero volume (or zero pressure), and all molecular motion stops.

Remember

Why Is It the Lowest?

Imagine a temperature below absolute zero. The volume of the gas would then be negative, which is impossible. So absolute zero is the lowest temperature that can exist.

Repeat the experiment with different low-density gases $A$ , $B$ and $C$ . Each gas gives a straight line with its own slope.

When extended, all the lines meet the temperature axis at the same point, −273.15 °C. So every gas gives the same absolute zero.

p–T lines of low-density gases

A

B

and

C

Absolute zero=0 K=−273.15 °C⇒0 °C = 273.15 K

Absolute Temperature

A temperature measured on the Kelvin scale is called the absolute temperature.

Absolute Scale

The Kelvin scale itself is called the absolute scale.

1.7Thermal Expansion

Example

The metal lid of a glass bottle is sometimes screwed on very tightly. To open it, we dip the lid in hot water for a while. The metal lid expands and becomes loose. Now it unscrews easily.

Tight metal lid→dipped in hot water→lid expands→opens easily

So, when a material gets heat, its temperature rises. Then its length, area and volume can change.

Definition

Thermal expansion is the change in the size (dimensions) of a body when it is heated.

Thermal expansion is of three types:

iLinear Expansion

Increase in length.

iiArea (Superficial) Expansion

Increase in area.

iiiVolume (Cubical) Expansion

Increase in volume.

1.8Linear Expansion

Definition

Linear expansion is the increase in the length of a body when its temperature rises.

A rod has length $l_1$ at temperature $T_1$ . On heating to $T_2$ , its length becomes $l_2$ . The fractional change in length is:

\dfrac{\Delta l}{l_1} = \dfrac{l_2 - l_1}{l_1} = \alpha_l\,(T_2 - T_1) = \alpha_l\,\Delta T

The increase in length is proportional to the original length. It is also proportional to the rise in temperature.

On heating, the rod becomes longer by

\Delta l

\Delta l \propto l_1

and

\Delta l \propto \Delta T

⇒

\Delta l \propto l_1\,\Delta T

Expansion in Length

\Delta l = \alpha_l\, l_1\, \Delta T

where, $\alpha_l$ = coefficient of linear expansion, $l_1$ = original length, $\Delta T$ = rise in temperature

Definition

Coefficient of Linear Expansion

The coefficient of linear expansion $(\alpha_l)$ of a solid is the change in length of a rod of unit length when its temperature rises by one degree. It is measured in °C⁻¹ or K⁻¹.

\alpha_l = \dfrac{\Delta l}{l_1\,\Delta T} = \dfrac{\text{Increase in length}}{\text{Original length} \times \text{Rise in temperature}}

\alpha_l

depends on the material of the rod. Metals expand more.Metals: high

\alpha_l

1.9Area Expansion

Definition

Area expansion is the increase in the area of a surface when its temperature rises.

A plate has area $A_1$ at temperature $T_1$ . On heating to $T_2$ , its area becomes $A_2$ . The fractional change in area is:

\dfrac{\Delta A}{A_1} = \dfrac{A_2 - A_1}{A_1} = \alpha_A\,(T_2 - T_1) = \alpha_A\,\Delta T

New area

A_2 = A_1 + \Delta A

\Delta A \propto A_1

and

\Delta A \propto \Delta T

⇒

\Delta A \propto A_1\,\Delta T

Expansion in Area

\Delta A = \alpha_A\, A_1\, \Delta T

where, $\alpha_A$ = coefficient of area expansion. It depends on the nature of the material of the plate.

Definition

Coefficient of Area Expansion

The coefficient of area expansion $(\alpha_A)$ of a metal sheet is the change in area of a sheet of unit area when its temperature rises by one degree. It is measured in °C⁻¹ or K⁻¹.

\alpha_A = \dfrac{\Delta A}{A_1\,\Delta T} = \dfrac{\text{Increase in surface area}}{\text{Original surface area} \times \text{Rise in temperature}}

1.10Volume Expansion

Definition

Volume expansion is the increase in the volume of a body when its temperature rises.

A body has volume $V_1$ at temperature $T_1$ . On heating to $T_2$ , its volume becomes $V_2$ . The fractional change in volume is:

\dfrac{\Delta V}{V_1} = \dfrac{V_2 - V_1}{V_1} = \alpha_V\,(T_2 - T_1) = \alpha_V\,\Delta T

New volume

V_2 = V_1 + \Delta V

\Delta V \propto V_1

and

\Delta V \propto \Delta T

⇒

\Delta V \propto V_1\,\Delta T

Expansion in Volume

\Delta V = \alpha_V\, V_1\, \Delta T

where, $\alpha_V$ = coefficient of volume expansion, so that $\alpha_V = \dfrac{\Delta V}{V_1\,\Delta T}$

Definition

Coefficient of Volume Expansion

The coefficient of volume (cubical) expansion $(\alpha_V)$ of a substance is the change in volume of a body of unit volume when its temperature rises by one degree. It is measured in °C⁻¹ or K⁻¹.

$\alpha_V$ is a property of the material. But it is not strictly a constant.
In general, $\alpha_V$ depends on temperature. The graph shows this for copper.
Its value first increases with temperature.
It becomes constant at high temperature, above about 500 K.

\alpha_V

of copper against temperature

Note

All coefficients of expansion have the dimension $[\mathrm{K^{-1}}]$ . Their SI unit is per kelvin $(\mathrm{K^{-1}})$ .

1.11Relation Among the Coefficients of Expansion

Write the original size simply as $l$ , $A$ or $V$ (that is, $l_1 = l$ , $A_1 = A$ and $V_1 = V$ ). Then the three coefficients are:

Linear

\alpha_l = \dfrac{\Delta l}{l_1\,\Delta T} = \dfrac{\Delta l}{l\,\Delta T}

Area

\alpha_A = \dfrac{\Delta A}{A_1\,\Delta T} = \dfrac{\Delta A}{A\,\Delta T}

Volume

\alpha_V = \dfrac{\Delta V}{V_1\,\Delta T} = \dfrac{\Delta V}{V\,\Delta T}

Ratio of the Coefficients

\alpha_l : \alpha_A : \alpha_V = 1 : 2 : 3

So, $\alpha_A = 2\alpha_l$ and $\alpha_V = 3\alpha_l$

Coefficients of Expansion at High Temperature

For a given solid, $\alpha_l$ , $\alpha_A$ and $\alpha_V$ are constant only at a given temperature.

When the temperature range changesTheir values change

At high temperatureThey become constant

1.12Anomalous Expansion of Water

Water behaves in an unusual (anomalous) way. Between 0 °C and 4 °C, it contracts on heating.
Cool some water below room temperature. Its volume decreases until it reaches 4 °C.
Below 4 °C, its volume increases. So its density decreases.
The second graph shows the density of water from 0 °C to 10 °C. The density is maximum at 4 °C. It falls on both sides of 4 °C.

Volume of 1 kg of water: smallest at 4 °C

Density of water from 0 °C to 10 °C

At 4 °C, water hasmaximum densityandminimum volume

Note

A Lake in Winter

In winter, a lake cools towards 4 °C. The surface water gives heat to the air. It becomes denser and sinks. The warmer, lighter water from the bottom rises up.

When the top water cools below 4 °C, it becomes less dense. So it stays at the surface and freezes there. The water below stays liquid. This lets water animals stay alive and move freely near the bottom.

1.13Expansion of Solids, Liquids and Gases

iSolids and Liquids

At ordinary temperature, they expand very little compared with gases.

iiLiquids

For a liquid, $\alpha_V$ is nearly independent of temperature.

Derivation

Volume Expansion of an Ideal Gas

Keep the pressure constant. Start from the ideal gas equation:

$pV = \mu RT \;\Rightarrow\; p\,\Delta V = \mu R\,\Delta T$
$\text{Dividing by } pV = \mu RT: \quad \dfrac{\Delta V}{V} = \dfrac{\Delta T}{T}$

For an Ideal Gas

\dfrac{\Delta V}{V} = \dfrac{\Delta T}{T} \quad\Rightarrow\quad \alpha_V = \dfrac{1}{T}

At 0 °C,

\alpha_V = 3.7 \times 10^{-3}\ \text{K}^{-1}

. This is far larger than for solids and liquids.Gas ≫ solid, liquid

A gas at room temperature and constant pressure

\alpha_V \approx 3300 \times 10^{-6}\ \text{K}^{-1}

1.14Thermal Strain and Thermal Stress

Fix both ends of a metal rod rigidly, so that it cannot expand or contract. Now change its temperature. Thermal strain and thermal stress develop in the rod.

Why? When the temperature rises, the rod tries to expand. But both its ends are fixed. So the rod pushes on the supports with a force.

Temperature rises→rod tries to expand→ends are fixed→force on supports

Derivation

Let a rod of length $l$ be heated through $\Delta T$ . If it were free, its length would increase by:

$\Delta l = l\,\alpha\,\Delta T$
The ends are fixed, so the rod cannot expand. A compressive thermal strain develops in it:
$\text{Thermal strain} = \dfrac{\Delta l}{l} = \alpha\,\Delta T$
This strain produces a thermal stress:
$\text{Thermal stress} = Y \times \text{thermal strain} = Y\alpha\,\Delta T$

Thermal (Compressive) Strain

\text{Strain} = \dfrac{\Delta l}{l} = \alpha\,\Delta T

where, $\alpha$ = coefficient of linear expansion of the rod's material

Thermal Stress

\text{Stress} = Y\alpha\,\Delta T

where, $Y$ = Young's modulus of the rod's material

Force exerted by the rod on the supports. Here

A

is the area of cross-section of the rod.

F = Y\alpha\,\Delta T\,A

Strain

= \alpha\Delta T

× Y →Stress

= Y\alpha\Delta T

× A →Force

= Y\alpha\Delta T A

Part 2

Calorimetry and Change of State

2.1Heat Capacity and Specific Heats

The heat needed to change the temperature of a substance depends on three things:

$m$ mass $\Delta T$ change in temperature natureof the substance

A substance absorbs or gives out heat when its temperature changes. A quantity called heat capacity describes this.

Definition

Heat Capacity

The heat capacity $(S)$ of a body is the amount of heat needed to change its temperature by one unit (1 °C or 1 K). Its SI unit is J K⁻¹.

Heat Capacity

S = \dfrac{\Delta Q}{\Delta T}

where, $\Delta Q$ = heat absorbed or given out by the body, $\Delta T$ = change in its temperature

Dimensional formula of heat capacity

[\mathrm{ML^2T^{-2}K^{-1}}]

Definition

Water Equivalent

The water equivalent of a body is the mass of water that has the same heat capacity as the body. Its unit is the gram.

Specific Heat Capacity

Definition

The specific heat capacity (or specific heat) $(s)$ of a substance is the heat required to increase the temperature of unit mass (1 kg) of it by one unit (1 K). Its SI unit is J kg⁻¹ K⁻¹.

Specific Heat Capacity

s = \dfrac{S}{m} = \dfrac{1}{m}\,\dfrac{\Delta Q}{\Delta T}

where, $m$ = mass of the substance, $S$ = its heat capacity, $\Delta Q$ = heat given, $\Delta T$ = rise in temperature

Note

Water Has a Very High Specific Heat

Water has the highest specific heat capacity among common substances, $4.18 \times 10^{3}\ \text{J kg}^{-1}\,{}^{\circ}\text{C}^{-1}$ . So water is used:

as a coolant in car radiators, and
as a heater in hot-water bags.

Molar Specific Heat Capacity

Definition

The molar specific heat capacity $(C)$ of a substance (gas) is the heat required to increase the temperature of 1 mole of it by one unit (1 K). Its SI unit is J mol⁻¹ K⁻¹.

Molar Specific Heat Capacity

C = \dfrac{S}{\mu} = \dfrac{\Delta Q}{\mu\,\Delta T}

where, $\mu$ = number of moles of the substance (gas)

Relation Between Specific Heat and Molar Specific Heat

Derivation

Number of moles $\mu = \dfrac{m}{M}$ . Here $m$ is the mass of the substance and $M$ is its molecular mass. So $m = \mu M$ .

Put $\mu = \dfrac{m}{M}$ in the formula for $C$ :
$C = \dfrac{1}{\mu}\left[\dfrac{\Delta Q}{\Delta T}\right] = M\left[\dfrac{\Delta Q}{m\,\Delta T}\right]$
But $\dfrac{\Delta Q}{m\,\Delta T} = s$ , the specific heat capacity. So:
$C = Ms$

Relation

C = Ms

So $C$ equals the molecular mass $M$ multiplied by the specific heat $s$ .

Types of Molar Specific Heat Capacity

iAt Constant Pressure,

C_p

The heat needed to raise the temperature of 1 mole of a gas by one unit, at constant pressure.

iiAt Constant Volume,

C_V

The heat needed to raise the temperature of 1 mole of a gas by 1 °C, at constant volume.

Note

Both kinds of heat capacity are very useful in calorimetry.

2.2Calorimetry and Calorimeter

Definition

Calorimetry is the branch of physics in which we measure heat.

Put a hot body in contact with a cold body. Heat flows from the hot body to the cold body. It keeps flowing until both bodies reach the same temperature.

Body at higher temperature→ heat →Body at lower temperature·until both temperatures are equal

Principle

Principle of Calorimetry

The total heat given out by the hotter body is equal to the total heat taken in by the colder body.

\text{Heat lost by the hotter body} = \text{Heat gained by the colder body}

Let body 1 (hotter) have mass $m_1$ and specific heat $s_1$ . Let body 2 (colder) have mass $m_2$ and specific heat $s_2$ . Then:

Principle of Calorimetry

m_1 s_1\, \Delta T_1 = m_2 s_2\, \Delta T_2

where, $\Delta T_1$ = fall in temperature of the hot body, $\Delta T_2$ = rise in temperature of the cold body

Note

Isolated System

A system is isolated when no heat is exchanged between the system and its surroundings. In calorimetry, we assume that no heat escapes to the surroundings.

Calorimeter

Definition

A calorimeter is a device used to measure quantities of heat.

Construction

It has a cylindrical copper vessel with a stirrer.
The vessel sits inside a wooden jacket. The gap between them is packed with a heat-insulating material such as glass wool. So the calorimeter is thermally isolated from the surroundings.
The lid has holes. A thermometer and the stirrer go into the vessel through them.

Note

Less Heat Lost by Radiation

Heat lost by radiation is cut down further by polishing two surfaces:

the outer surface of the calorimeter, and
the inner surface of the jacket.

Bodies at different temperatures are mixed in the calorimeter. Heat is exchanged among the bodies and also with the calorimeter. If no heat is lost to the surroundings, the principle of calorimetry gives:

A calorimeter

Heat gained by the cold bodies=Heat lost by the hot bodies

2.3Change of State

Definition

Change of state is the process in which a substance changes from one state to another.

Matter usually exists in three states:

iSolid

e.g. ice

iiLiquid

e.g. water

iiiGas

e.g. steam

A substance can change from one state to another by taking in heat or giving out heat. The common changes of state are:

(i) Solid to liquid, and backSolid ⇄ Liquid

(ii) Liquid to gas, and backLiquid ⇄ Gas

These changes happen when heat is exchanged between the substance and its surroundings.

Note

Plasma, the Fourth State of Matter

Besides solid, liquid and gas, there is a fourth state of matter. It is called plasma.

Heat a substance to a very high temperature. Many electrons of its atoms become free from the nucleus. The free electrons and the positive ions together form plasma.

Heat to a very high temperature→electrons set free→plasma

Effect of Heat on Ice

Experiment

Put a few ice cubes at 0 °C in a beaker. Heat them slowly on a source that gives heat at a constant rate.
Stir the mixture of ice and water all the time. Note the temperature every minute.

Heating ice at a constant rate

Observations

The temperature–time graph looks like the one shown.
Heat is supplied all the time. Yet the temperature does not change as long as there is ice in the beaker.
The temperature stays constant until all the ice has melted.
So, during melting, the solid and liquid states exist together in thermal equilibrium.

Change of state of ice on heating

After all the ice has melted, keep heating. The temperature now starts to rise. It rises until it reaches about 100 °C. Then it becomes steady again. This happens at time $t$ on the graph.
The heat now changes the water from liquid to vapour. The temperature stays constant until all the water has turned into vapour.

Ice (0 °C)→ melts →Water→ warms up →Water (100 °C)→ boils →Steam

Note

With a constant supply of heat, a change of state takes place with no change in temperature.

2.4Terms Related to Change of State

iMelting and Melting Point

The change from solid to liquid is called melting. The temperature at which a solid starts to turn into liquid is its melting point.

iiFusion and Freezing Point

The change from liquid to solid is called fusion (freezing). The temperature at which a liquid starts to freeze is its freezing point.

iiiVaporisation and Boiling Point

The change from liquid to vapour (gas) is called vaporisation. During this change, the temperature stays constant. The liquid and vapour exist together in thermal equilibrium. This temperature is the boiling point of the liquid.

ivSublimation

The change from solid directly to vapour is called sublimation. The substance does not pass through the liquid state. e.g. camphor, naphthalene balls.

Definition

Normal Melting Point

The normal melting point of a substance is its melting point at standard atmospheric pressure.

Effect of Pressure on the Boiling Point of a Liquid

When the pressure increases, the boiling point of a liquidIncreases ↑

Water at 1 atm

Boiling point = 100 °C

Water at 2 atm

Boiling point = 128 °C

2.5Triple Point

The temperature of a substance stays constant during a change of state (phase change).

Definition

Phase Diagram

A graph between the temperature and the pressure of a substance is called its phase diagram, or p–T diagram.

The phase diagrams of water and CO₂ divide the p–T plane into three regions: the solid region, the liquid region and the vapour region.

Three curves separate these regions: the sublimation curve, the fusion curve and the vaporisation curve. On these curves, two phases exist together: solid and vapour, solid and liquid, and liquid and vapour.

Definition

Triple Point

The triple point of a substance is the temperature and pressure at which all three phases (solid, liquid and vapour) exist together.

e.g. Triple point of water: temperature273.16 K

Triple point of water: pressure6.11 × 10² Pa (≈ 0.006 atm)

(a) Water: triple point

O

(b) CO₂: triple point

O

A point on the sublimation curve

BO

Solid + vapour exist together

A point on the vaporisation curve

OC

Liquid + vapour exist together

A point on the fusion curve

AO

Solid + liquid exist together

2.6Latent Heat

Definition

The latent heat of a substance is the amount of heat per unit mass transferred during a change of state, with no change in temperature.

Latent heat is written as $L$ . Its SI unit is J kg⁻¹. Latent heat depends on pressure. So its value is usually given at standard atmospheric pressure.

Let a mass $m$ of a substance change from one state to another. The heat needed is:

Heat Needed for a Change of State

Q = mL \qquad \text{or} \qquad L = \dfrac{Q}{m}

where, $Q$ = heat of transformation, $m$ = mass of the substance, $L$ = latent heat. So the heat needed depends on the mass and on the latent heat.

Types of Latent Heat

iLatent Heat of Fusion,

L_f

The latent heat for the change between solid and liquid. It is also called the heat of fusion.

L_f = \dfrac{Q}{m}

(SI unit J kg⁻¹)

iiLatent Heat of Vaporisation,

L_V

The latent heat for the change between liquid and gas. It is also called the heat of vaporisation.

L_V = \dfrac{Q}{m}

(SI unit J kg⁻¹)

Example

Water

For water, $L_f = 3.33 \times 10^{5}\ \text{J kg}^{-1}$ and $L_V = 22.6 \times 10^{5}\ \text{J kg}^{-1}$ . This means:

3.33 × 10⁵ J of heat melts 1 kg of ice at 0 °C.
22.6 × 10⁵ J of heat changes 1 kg of water at 100 °C into steam.

Temperature During a Change of State

The graph shows that the temperature stays constant while heat is added (or removed) during a change of state.
The sloping lines for ice, water and steam have different slopes. So the specific heats of the different states are not equal.

Temperature against heat for water at 1 atm (not to scale)

2.7Mechanical Equivalent of Heat

We can also raise the temperature of a body by doing mechanical work on it.

Definition

The mechanical equivalent of heat $(J)$ is the number of joules of mechanical work needed to raise the temperature of 1 g of water by 1 °C.

Its unit is J/cal. It is only a number that converts heat in calories into work in joules.

Mechanical Equivalent of Heat

W = JH

where, $W$ = work done, in joules (SI unit), $J$ = mechanical equivalent of heat (J/cal), $H$ = heat given or taken out, in calories (CGS unit)

Part 3

Heat Transfer

3.1Modes of Heat Transfer

Definition

Heat flows from one body to another because of a temperature difference. It can flow as radiation, through molecular vibrations or through molecular displacement. All these processes of heat flow together are called heat transfer.

There are three modes of heat transfer:

(i) Conductionmolecular vibrations

(ii) Convectionmolecular displacement

(iii) Radiationradiations

The three modes of heat transfer

3.2Conduction and Thermal Conductivity

Definition

Conduction

Conduction is the transfer of heat through molecular vibrations (molecular collisions). The average position of the molecules does not change. So there is no movement of matter as a whole. In solids, heat is usually transferred by conduction.

Thermal Conductivity

Definition

Thermal conductivity is the ability of a material to conduct heat through it. We measure it by the heat that flows per second through the material when a fixed temperature difference is kept across it.

Setup — Metal Rod

Length of the rod $= L$ . Area of cross-section $= A$ .
Temperatures of its two ends $= T_1$ and $T_2$ $(T_1 > T_2)$ .
The rod is in the steady state.

Heat conduction through a metal rod

In the steady state, the heat $Q$ that flows through any section of the rod is:

directly proportional to the temperature difference

\Delta T = T_1 - T_2

Q \propto \Delta T

directly proportional to the area of cross-section

Q \propto A

directly proportional to the time of flow

Q \propto t

inversely proportional to the length of the rod

Q \propto \dfrac{1}{L}

So the rate of heat flow $H$ does not depend on time. It is:

H = \dfrac{\Delta Q}{\Delta t} = \dfrac{KA\,(T_1 - T_2)}{L} \quad\text{i.e.}\quad H = KA\,\dfrac{\Delta T}{L}

Since $H = \dfrac{Q}{t}$ , the total heat that flows in time $t$ is:

Heat Conducted

Q = KA\,\dfrac{\Delta T}{L}\,t

where, $K$ = coefficient of thermal conductivity of the material. A larger $K$ means the material conducts heat faster.

SI unit of

K

J s⁻¹ m⁻¹ K⁻¹ or W m⁻¹ K⁻¹

Dimensions of

K

[\mathrm{MLT^{-3}K^{-1}}]

The ratio

\dfrac{\Delta T}{L}

is called thetemperature gradient

The thermal conductivity of a material changes slightly with temperature. But over the normal range of temperature, we can take it as constant.

Definition

Coefficient of Thermal Conductivity

Put $A = 1$ , $\Delta T = T_1 - T_2 = 1$ , $L = 1$ and $t = 1$ . Then $Q = K$ .

So the coefficient of thermal conductivity of a material is the heat that flows in unit time through a unit cube of it, when its opposite faces differ in temperature by one degree.

Steady stateThe temperature of each part of the body stays constant while heat flows through it

Note

Metals and Non-metals

Metals conduct heat much better than non-metals. Metals have a large number of free electrons. These electrons carry heat from the hotter part to the colder part.

3.3Heat Current and Thermal Resistance

Definition

Heat Current

The heat current $(H)$ is the rate of flow of heat. Its SI unit is J/s or watt (W). Its dimensions are $[\mathrm{ML^2T^{-3}}]$ .

H = \dfrac{\Delta Q}{\Delta t} = KA\,\dfrac{(T_1 - T_2)}{L} = \dfrac{T_1 - T_2}{\left(\dfrac{L}{KA}\right)}

Thermal Resistance

R = \dfrac{\Delta T}{H} = \dfrac{T_1 - T_2}{H} = \dfrac{L}{KA}

Comparison with Electric Current

iElectric Current

i = \dfrac{V_1 - V_2}{R}

$V_1 - V_2$ = potential difference, $R$ = electrical resistance

iiHeat Current

H = \dfrac{T_1 - T_2}{(L/KA)}

$T_1 - T_2$ = temperature difference, $\dfrac{L}{KA}$ = thermal resistance

So $\dfrac{T_1 - T_2}{(L/KA)}$ behaves like a thermal current (heat flow), and $\dfrac{L}{KA}$ behaves like a thermal resistance.

SI unit of thermal resistance

R

K s/J or K/W

Dimensions of thermal resistance

[\mathrm{M^{-1}L^{-2}T^{3}K}]

3.4Rods in Series and in Parallel

When two or more rods are joined together, the equivalent thermal conductivity of the whole set-up changes.

(a) Series Combination of Two Conductors

For resistances in series, $R_{eq} = R_1 + R_2$ . Use $R = \dfrac{L}{KA}$ for each rod:

\dfrac{L_1 + L_2}{K_{eq}A} = \dfrac{L_1}{K_1 A} + \dfrac{L_2}{K_2 A}

Two rods in series

If the rods have equal lengths, $L_1 = L_2 = L$ , then $\dfrac{2L}{K_{eq}A} = \dfrac{L}{K_1 A} + \dfrac{L}{K_2 A}$ . This gives:

Series — Equivalent Thermal Conductivity

K_{eq} = \dfrac{2K_1 K_2}{K_1 + K_2}

where, $K_{eq}$ = equivalent thermal conductivity of the two rods joined end to end

(b) Parallel Combination of Two Conductors

For resistances in parallel, $\dfrac{1}{R_{eq}} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$ . Take rods of equal length and equal area. Together they have area $2A$ :

\dfrac{1}{\left(\dfrac{L}{K_{eq}\,2A}\right)} = \dfrac{1}{\left(\dfrac{L}{K_1 A}\right)} + \dfrac{1}{\left(\dfrac{L}{K_2 A}\right)}

Two rods in parallel

\Rightarrow\ \dfrac{2A\,K_{eq}}{L} = \dfrac{K_1 A}{L} + \dfrac{K_2 A}{L}

Parallel — Equivalent Thermal Conductivity

K_{eq} = \dfrac{K_1 + K_2}{2}

for two rods of equal length and equal area, placed side by side

3.5Convection

Definition

Convection is the transfer of heat by the actual movement of matter from a region of higher temperature to a region of lower temperature. It takes place only in fluids, that is, in liquids and gases.

Convection is of two types:

iForced Convection

The heated material is forced to move, for example by a blower or a pump. It carries heat with it.

e.g. Heat transfer in the human body.

iiNatural (Free) Convection

The heated material moves because of a difference in density.

e.g. Heat transfer in water being heated.

Sea Breeze (During the Day)

In the day, the land heats up faster than the sea. Water has a greater specific heat. Mixing currents also spread the heat through its large volume.
The hot air above the land expands, becomes less dense and rises. Cooler air from the sea takes its place. The warm air from the land moves towards the sea and completes the cycle.
So a breeze blows from the sea to the land. It is called a sea breeze.

Land Breeze (During the Night)

At night, the opposite happens. The land cools faster than the sea.
The warm air above the sea rises. Cooler air from the land takes its place.
So a breeze blows from the land to the sea. It is called a land breeze.

Formation of Trade Winds

The equator and the poles of the Earth get unequal heat from the Sun. Air near the surface at the equator is hot. Air in the upper atmosphere at the poles is cool.
So a convection current forms between them. Air at the equator rises and moves out towards the poles. It then comes down and flows back towards the equator.
The rotation of the Earth changes this convection current. Air near the equator has an eastward speed of 1600 km/h. Close to the poles, this speed is zero.
So the air comes down not at the poles but at about 30° N latitude. It then returns to the equator. This is called the trade wind.

Equator: hot air rises→moves towards the poles→comes down at 30° N→returns to the equator

Definition

Trade Wind

The trade wind is the steady wind that blows near the Earth's surface from the north-east towards the equator. It is an example of natural convection.

3.6Radiation

Definition

Radiation is the transfer of heat from one point to another without warming the medium in between. The heat travels as thermal radiation, also called radiant energy or simply radiation.

The word radiation has two meanings:

iThe Process

A body gives out energy. The energy travels through space and falls on another body.

iiThe Energy

The energy itself that travels through space.

Example

From the Sun to the Earth

Heat from the Sun reaches the Earth by radiation. It crosses millions of kilometres of empty space, with no material medium in between.

Thermal Radiation

Definition

Thermal radiation is the electromagnetic radiation given out by a body because of its temperature.

e.g. Radiation from a red-hot iron. Light from a filament lamp.

Colour of a Body

Thermal radiation falling on a body is partly reflected and partly absorbed.

The heat a body absorbs by radiation depends on its colour.
A body has a particular colour because it reflects back radiation of that wavelength.

Black Body Radiation

Definition

Black Body

A black body is a body that absorbs all the radiation falling on it. The radiation given out by a black body is called black body radiation.

absorbsall the radiation falling on it emitsat the fastest rate also calledan ideal radiator

A body that absorbs 100% of the radiation is only an ideal idea. A perfect black body does not exist in the universe.Ideal only

Lamp black reflects only 1% of the radiation falling on it. So it is close to a black body.≈ Black body

3.7Absorptive Power and Emissivity

Definition

Absorptance (Absorptive Power)

The absorptance $(a)$ , or absorptive power, of a body is the ratio of the thermal radiation it absorbs in a given time to the total thermal radiation falling on it in the same time.

a = \dfrac{\text{Energy absorbed}}{\text{Energy incident}}

Definition

Emissive Power

The emissive power of a body is the radiant energy it gives out per unit time per unit surface area, within a unit wavelength range around a wavelength $\lambda$ . It is taken at a fixed temperature.

Definition

Emissivity

The emissivity $(\varepsilon)$ of a body is the ratio of its emissive power $(e)$ to the emissive power of a perfect black body $(E)$ at the same temperature.

Emissivity

\varepsilon = \dfrac{e}{E}

where, $e$ = emissive power of the body, $E$ = emissive power of a perfect black body at the same temperature

3.8Stefan–Boltzmann Law

Law

Stefan–Boltzmann Law

The total energy emitted per second per unit area of a perfect black body is directly proportional to the fourth power of its absolute temperature.

E \propto T^{4}

Stefan–Boltzmann Law

E = \sigma T^{4}

where, $E$ = total energy emitted per second per unit area, $T$ = absolute temperature, $\sigma$ = Stefan–Boltzmann constant (a universal constant) $= 5.67 \times 10^{-8}\ \text{W m}^{-2}\,\text{K}^{-4}$

Note

A Body That Is Not a Perfect Black Body

Suppose the body is not a perfect black body. Let its emissivity be $\varepsilon$ . Then the law becomes:

Perfect Black Body

E = \sigma T^{4}

Body with Emissivity ε

E = \varepsilon\,\sigma\,T^{4}

3.9Wien's Displacement Law

Law

Wien's Displacement Law

The wavelength $(\lambda_m)$ at which a black body emits the maximum energy is inversely proportional to its absolute temperature $(T)$ .

\lambda_m \propto \dfrac{1}{T}

Wien's Displacement Law

\lambda_m\, T = b

where, $b$ = Wien's constant $= 2.9 \times 10^{-3}\ \text{m K}$

Temperature

T

increases ↑

\lambda_m

decreases ↓

Temperature

T

decreases ↓

\lambda_m

increases ↑

Hotter body: peak at a shorter wavelength

3.10Newton's Law of Cooling

Law

Newton's Law of Cooling

The rate at which a body loses heat is directly proportional to the difference between its temperature and the temperature of its surroundings. This is true only when the difference is small.

Rate of loss of heat

\propto

Temperature difference between the body and its surroundings

-\dfrac{dQ}{dt} \propto (T - T_0)

Newton's Law of Cooling

-\dfrac{dQ}{dt} = k\,(T - T_0)

where, $-\dfrac{dQ}{dt}$ = rate of loss of heat, $T$ = temperature of the body, $T_0$ = temperature of the surroundings, $k$ = a constant

Straight line through the origin

End of Chapter 10 · notesshala.in