# Chapter – 14, Thermal Properties Of Matter

### Heat :

It is a form of energy. It flows from higher temperature region to lower temperature region. That it flows due to temperature difference of two objects. An object does not possess it. As for example, cheque is a form of money. It is not money but money can be transferred through it from one place to another place. In the same manner, Heat is not energy but it is a form of energy through which it can be transferred from one body to another body. In the above picture, we can see that the Plants takes the sun rays. That it gains heat from the sun to make his food through photosynthesis. That is the energy of the sun is transferred to the plant by the flow of heat. By rubbing the hands in the winter season, the hands get warmed. It is due to transferred of mechanical energy into heat energy.

### Temperature :

It is degree of hotness and coldness. It is the main factor on which the flow of heat depends.

The water flows due to height difference between two places. In the same manner the heat flow between two bodies due to temperature difference. Calorimeter : The heat measuring instrument is termed as Calorimeter.

Thermometer : The temperature measuring instrument is called as Thermometer.

Types of temperature :

2 ) Fahrenheit thermometer

3 ) Absolute thermometer ( on this scale negative temperature is not possible )

Important formulae related to conversion of temperature of one thermometer to another :

1 ) T = t + 273

2 ) t = T – 273

3 ) C = 5( F – 32 ) / 9

4 ) F = ( 9 / 5 )C + 32

Where T = Temperature in absolute scale

t = temperature in celcius scale = C

F = Temperature in Fahrenheit scale

5 ) Lower limit of Celcius scale = 0 degree

6 ) Lower limit of Fahrenheit scale = 32 degree

7 ) Lower limit of absolute scale = 273 Kelvin

# Thermal Expansion Of Solids :

When the heat is supplied to a solid, it expands in different ways.

### 1 ) Linear Expansion Of Solids :

If a solid body expands linearly on heating, then the expansion is Linear Expansion.

Let L = initial length of  a solid body at temperature t0 C

L2 = final length of the body at t2 0C

Change in length ( ΔL ) = L2 – L1

Change in temperature ( Δt ) = ( t2 – t1 ) 0C

The change in length is directly proportional to the original length L and change in temperature Δt

Δ L ∝ L1 Δt

ΔL = L1Δt

where ∝ is the proportionality constant and it is termed as the coefficient of the linear expansion.

Expression for the ∝ :

∝ = ΔL / L1Δt

Definition of the  ∝ :

∝ = ΔL / L1Δt

if Δt = 1 unit and L1 = 1 unit

then  ∝ = ΔL

Numerically, the coefficient of linear expansion is the increase in length per unit length per unit rise in temperature of a solid body.

Unit of ∝ :

Its units are 0 C-1, 0F-1 and K-1

### 2 ) Superficial Expansion :

A body is heated and if there is expansion in surface area, the expansion is called as superficial expansion.

Let S1 = Original surface area of the solid body

S2 = surface area of the body after raising temperature from t1 0C to t2 0C

Δt = ( t2 – t1 ) 0C = change in temperature

ΔS = S2 -S1 = change in surface area of the body

As we know that the change in surface area of the body is proportional to the original surface area and change in temperature.

Therefore, ΔS ∝ SΔt

ΔS = SβΔt

where β is the proportionality constant and it is known as coefficient of superficial expansion

Expression for β :

β = ΔS / SΔt

Definition of β :

Numerically, the coefficient of superficial expansion is defined as the increase in surface area per unit surface area per unit rise in temperature of a solid body.

Units of β :

Its units are 0 C-1, 0F-1 and K-1

### 3 ) Volume Expansion :

A solid is heated and if there is expansion in volume, the expansion is termed as volume expansion.

Let V1 = Original volume of the solid body

V2 = volume of the body after raising temperature from t1 0C to t2 0C

Δt = ( t2 – t1 ) 0C = change in temperature

ΔV = V2 -V1 = change in surface area of the body

As we know that the change in surface area of the body is proportional to the original surface area and change in temperature.

Therefore, ΔV ∝ VΔt

ΔV = VγΔt

where γ is the proportionality constant and it is known as coefficient of volume expansion

Expression for γ :

γ = ΔV / VΔt

Definition of γ :

If Δt = 1 unit and V = 1 unit then

γ = ΔV

Numerically, the coefficient of volume is defined as the Change in volume per unit volume per unit rise in temperature of a solid body.

Units of γ :

Its units are 0 C-1, 0F-1 and K-1

## Thermal Expansion Of Liquid

The liquid has no fixed shape as the force of interaction between its molecules is weaker than that of the solids. Therefore linear expansion and superficial expansion are meaningless in case of liquid substance. Its volume depends on the temperature. Pressure has negligible effect on the volume of the liquid. There is two different expansion in volume of the liquid. One is real expansion and other is apparent expansion.

#### 1 ) The Real Expansion Of Liquid : When a liquid is heated in a container, at first the container will expand due to increase in temperature then the liquid will expand. When the container expands the level of the liquid decreases from L1 to L2. After the expansion the of the container, the liquid starts to expand and its level increases to L3 from L2.

L1 = initial level of the liquid before heating

L2 = level of the liquid after the expansion of the container

L3 = level of the liquid after the the expansion of the liquid

The expansion of the liquid ( L2 – L3 ) is termed as the real expansion of the liquid.

Coefficient of real expansion is given by

γr = ΔV / ( V ΔT )

#### 2 ) Apparent Expansion Of Liquid : When a liquid is heated in a container, at first the container will expand due to increase in temperature then the liquid will expand. When the container expands the level of the liquid decreases from L1 to L2. After the expansion the of the container, the liquid starts to expand and its level increases to L3 from L2. But

L1 = initial level of the liquid before heating

L2 = level of the liquid after the expansion of the container

L3 = level of the liquid after the the expansion of the liquid

The expansion of the liquid ( L2 – L3 ) is termed as the real expansion of the liquid but ( L1 – L3 ) is seen like the expansion and this expansion is called as apparent expansion of the liquid.

Coefficient of apparent expansion is given by

γ’ = ΔV’ / ( V ΔT )

where  ΔV’ is the apparent change in volume of the liquid and V is the original volume of the liquid.

#### Relation Between The Real Expansion And Apparent Expansion Of A Liquid : L1 = initial level of the liquid before heating

L2 = level of the liquid after the expansion of the container

L3 = level of the liquid after the the expansion of the liquid

V1 = Volume of the liquid when its level is L1

V2 =Volume of the liquid when its level is L2

V3 = Volume of the liquid when its level is L3

Real expansion of the liquid ( ΔV ) = V3 – V2

Apparent expansion of the liquid ( ΔV’ ) = V3 – V1

We know that

Real expansion = apparent expansion + expansion of the container

ΔV = ΔV’ + ΔVc

dividing both sides by V1 ΔT, we get

ΔV / ( V1 ΔT ) = ΔV’ / ( V1 ΔT )  + ΔVc / ( V1 ΔT )

γr = γ’ + γc

#### Anomalous Expansion Of Water : We know that the volume of the liquid increase with the increase in temperature. But in case of water, its volume does not vary linearly with the temperature. At first, the volume of water decreases with the increase in temperature up to 4 0C then after reaching 4 0C, the volume of water increases with increment of temperature. It is termed as the anomalous expansion of water. Thus, water has minimum volume and maximum density at 4 0C. This anomalous behaviour affects the marine life.

## Thermal Expansion Of Gas

The gas has neither fixed shape nor fixed volume. Its volume changes with slight change in temperature and pressure.

The expansion of gases is explained in details in the chapter “Kinetic theory of gas “

#### Specific Heat Capacity :

Definition : The amount of heat required to raise of temperature of a body of unit mass by 1 0C, is called as the specific heat capacity. It depends on the nature of the material of the body.

Unit : Its SI unit is joule/kg/K and CGS unit is cal/g/0C.

Expression : c = Q / ( mdt )

where c = specific heat capacity or specific heat

Q = amount of heat

m = mass of the body

dt = change in temperature

#### Molar Heat capacity :

The amount of heat required to raise the temperature of 1 mole of substance through 1 0C, is called as molar heat capacity.

As the volume of the gas changes with slight change of temperature and pressure, it has two types of heat capacity. One is heat capacity at constant volume and heat capacity at constant pressure.

#### Thermal Heat Capacity :

The amount of heat required to raise the temperature of a substance through 1 0C, is called as thermal heat capacity. Its SI unit is J/K and CGS unit is cal / 0C.

#### Water Equivalent :

A substance is heated and its temperature increases to a certain value. If this amount of heat raise the temperature of the some amount of water through same change in temperature, then the this amount of water is called as the water equivalent.

Hence, the water equivalent is defined as the amount of water that absorbs or emits the same amount of heat as is done by the given body for same rise or fall in temperature. Its SI unit is Kg and CGS unit is gm.

#### For 1 marks

1 ) Define heat.

Ans : Heat is a form of energy.

2 ) What is meant by temperature ?

Ans : The degree of hotness and coldness of an object is termed as temperature.

3 ) Name the instrument that measures the heat of an object .

Ans : Calorimeter

4 ) Name the instrument that measures the temperature of an object.

Ans: Thermometer

5 ) What is the SI unit of heat ?

Ans : Joule

6 ) What is the CGS unit of heat ?

Ans : Calorie

7 ) What is the SI unit of temperature ?

Ans : Kelvin

8 ) Is it possible for a body to have negative temperature on a Kelvin scale ?

Ans : No, it is not possible for a body to have negative temperature on a Kelvin scale.

9 ) What is the unit of Coefficient of thermal expansion of solid ?

Ans : 0 C-1, 0F-1 and K-1

10 ) At what temperature will wood and iron appear equally hot or cold ?

Ans : At human body temperature, wood and iron will appear equally hot or cold.

11 ) Why telephone wires are often given snag ?

Ans : Telephone wires are often given snag to allow safe contraction in winter.

12 ) Will the coefficient of linear expansion depend on the length of the solid material ?

Ans : No, the coefficient of linear expansion does not depend on the length of the solid material.

13 ) Why there is gap between the railway tracks ?

Ans : There is gap between the railway tracks to allow safe expansion in summer.

14 ) On which factor, the coefficient of thermal expansion of a solid body depends ?

Ans : The coefficient of thermal expansion of a solid body depends on the nature of the material of the body.

15 ) When a boiling tea is poured in a thick walled glass tumbler, it cracks ——————————- why ?

Ans : When a boiling tea is poured in a thick walled glass tumbler, it cracks due to thermal expansion of the glass.

16 ) Give one use of bimetallic strip ?

Ans : It is used in making thermoswitch.

17 ) Is there any relationship between the geometrical shape of a solid body and its linear coefficient of expansion ?

Ans : No, there is no relationship between the geometrical shape of a solid body and its linear coefficient of expansion.

18 ) What is the relation between the linear coefficient and superficial expansion of a solid body ?

Ans : β = 2 ∝

where ∝ = coefficient of linear expansion

β = coefficient of superficial expansion

19 ) What is the relation between the coefficient of volume expansion and linear expansion?

Ans : γ = 3∝

∝ = coefficient of linear expansion

γ = coefficient of volume expansion

20 ) What is the relation between the three types of coefficient of thermal expansion in solid body ?

Ans : ∝ = β / 2 = γ / 3

21 ) A liquid of expansivity Y is heated in a vessel of linear expansivity Y/3. What would be the effect on the level of the liquid ?

Ans : There is no effect on the level of the liquid.

22 ) At which temperature, the water has minimum volume ?

Ans : 4 0C

23 ) At which temperature, the water has maximum density ?

Ans : 4 0C

24 ) Due to which property of water, the marine life is stable when the temperature goes to its negative value ?

Ans : Anomalous behaviour of expansion of water.

25 ) Which substance has the highest value of specific heat capacity ?

An s: Hydrogen has the highest value of specific heat capacity.

26 ) Name the liquid having the highest value of specific heat ?

Ans : Water has the highest value of specific heat.

27 ) Among copper, water and nitrogen gas which has the higher and lower value of the specific heat ?

Ans : Water has higher value and copper has lower value of the specific heat.

28 ) Write down the SI unit of specific heat .

Ans : joule/kg/ 0C

29 ) What is the main difference between the thermal heat capacity and specific heat capacity ?

Ans : The specific heat capacity of a substance is involved unit mass of the substance whereas the thermal capacity does not.

30 ) Write down one condition of principle of calorimetry .

Ans : The bodies should be in contact with each other.

#### For 2 or 3 marks

1 ) ” The coefficient of expansion of aluminium is 23 * 10-6 K-1 “. What does it mean ?

Ans : The given statement means that if aluminium of 1 m long is heated to raise its temperature by 1 K, then increase in its length is 23 * 10-6 m.

2 ) Show that the coefficient of area expansion of a rectangular sheet of solid is twice its linear expansivity .

Ans : Let L = Original length of the sheet at temperature t 0C

L’ = increased length of the sheet at temperature t’ 0C

S = original surface area of the sheet at temperature t 0C = L2

S’ = new surface area of the sheet at temperature t’ 0C = L’2

∝ = coefficient of linear expansion

β = coefficient of superficial expansion

γ = coefficient of volume expansion

ΔS = increase in surface area
ΔL = increase in length
ΔT = increase in temperature
Now,
∝ = ΔL / LΔT
L’ = L + LΔT
squaring both sides we get
L’2 = ( L + LΔT )2
L’2 = L2 + L22ΔT2 + 2L2ΔT
( neglecting the higher ordered term )
S’ = S + S * ( 2∝ ) *  ΔT
S’ – S =  S * ( 2∝ ) *  ΔT
ΔS = S * ( 2∝ ) *  ΔT
By comparing above equation with the following equation we get
ΔS = S β ΔT
β = 2 ∝
Hence, the coefficient of superficial  expansion of solid is twice the coefficient of expansivity.

3 ) Railway lines are laid with gaps to allow for expansion. If the gap between the rails 66 m long be 3.63 cm at 10 0C, then at what temperature will lines just touch ? Coefficient of linear expansion for steel = 11 * 10-6 per 0C.

Ans : Original length ( L ) = 66 m

increase in length ( ΔL ) = 3.63 cm

Initial temperature ( T ) = 10 0C

final length ( L ‘ ) = L + ΔL = ( 66 * 100 ) + 3.63 ) = 6603.63 cm

required temperature ( T’ ) = x 0C

Coefficient of linear expansion for steel (  ∝  )  = 11 * 10-6 per 0C.

⇒ ΔL / ( L ΔT ) = 11 * 10-6

⇒ ΔT = Δ L / ( L * 11 * 10-6 )

⇒ ΔT =  3.63 / ( 6603.63 *  11 * 10-6 )

⇒ ΔT = 49.97

⇒ T’ – T = 49.97

⇒ x = 49.97 + 10

⇒ x = 59.97

Required temperature is 59.97 0C.

4 ) Consider a steel bridge 200 m long in a locality where temperature varies from 243 K to 313 K. Find change in length of the bridge for seasonal variation in temperature. Coefficient of linear expansion of steel = 11 * 10-6 per 0C.

Ans : Original length ( L ) = 200 m

Initial temperature  ( T ) = 243 K = ( 243-273 )0C = -30 0C

final temperature ( T’ ) = 313 K = ( 313 – 273 ) 0C= 40 0C

Coefficient of linear expansion of steel ( ∝ )  = 11 * 10-6 per 0C.

Change in length = ΔL = ???

We know that

∝ =  ΔL / ( L  ΔT )

⇒ ΔL = ∝LΔT

⇒ ΔL = 11 * 10-6 * 200 * { 40 – ( -30 ) } = 0.154

⇒ΔL = 0.154 m

Therefore, the required change in length is 0.154 m

5 ) On which factors the coefficient of expansion depend ?

Ans : The coefficient of expansion depends on the nature of material and the unit of temperature.

6 ) Establish the relation between the coefficient of linear expansion and coefficient of volume expansion of a solid body.

Ans : The coefficient of linear expansion is given by

∝ =  ΔL / ( L  ΔT ) ———————————– ( 1 )

where  ΔL = L’ – L = change in length

L’ = final length

L = original length

ΔT = Change in temperature

Let V = original volume = L3

V’ = final volume = L’3

Now, from equation ( 1 ) we get

L’ = L + ∝LΔT

cubing both sides we get

L’3 = L3 + ( L∝ΔT )3 + 3L2 (  ∝LΔT ) + 3L (  ∝LΔT )2

As ΔT is very small so its cube or square is also negligible

V’ = V + 3 L3 ∝LΔT

V’ = V + V ( 3∝ ) ΔT

V’ – V = V ( 3∝ ) ΔT

ΔV = V ( 3∝ ) ΔT

by comparing above equation with the following equation we get

ΔV = V γ ΔT

γ = 3 ∝

The above expression is the required relationship between the coefficient of volume expansion and linear expansion.

7 ) How temperature affect the density of the material  of a solid body ?

Ans : We know that Volume expansion of a solid body is given by

ΔV = V γ ΔT ——————————- ( 1 )

where V is original volume , γ is coefficient of volume expansion and  ΔT is the change in temperature. Let us consider that the mass of the body is ‘ m ‘ and its density is ρ

From equation ( 1 ) we get

V’ – V = V γ ΔT

V’ = V + V γ ΔT

V’ = V ( 1 + γ ΔT )

Dividing both sides by ‘ m ‘ we get

V’ / m = ( V / m ) ( 1 + γ ΔT )

1 / ρ’ = ( 1 / ρ  ) ( 1 + γ ΔT )

ρ = ρ ‘ ( 1 + γ ΔT )

This is the required relation between the temperature and the density of the solid body where ρ’ is the new density at changed temperature.

8 ) Give the graphical representation of coefficient of volume expansion versus temperature.

9 ) The radius of a metallic sphere at room temperature T is R and the coefficient of linear expansion is A. The sphere is heated to a temperature ΔT, so that its new temperature becomes T + ΔT. What is the increase in volume of the metallic sphere ?

Ans : The radius of the metallic sphere = R

The coefficient of linear expansion = A

Change in temperature =  ΔT

Room temperature = T

New temperature = T +  ΔT

Original volume of the metallic sphere at room temperature ( V ) = 4 πR3  / 3

Let the change in volume = ΔV

According to problem

A =  ΔV / ( VΔT )

ΔV = AVΔT

ΔV = A ( 4 πR3  / 3 ) ΔT

ΔV = 4AπΔTR3  / 3

10 ) A sphere, a cube and a plate all of same material and same mass are initially heated to same high temperature. Which will cool fastest and slowest ?

Ans : As the surface area is larger than among the three given shapes, so the plate will cool fastest and the sphere has minimum surface area, so it will cool slowest.

11 ) Establish the relation between the real expansion and apparent expansion of a liquid.

Ans : L1 = initial level of the liquid before heating

L2 = level of the liquid after the expansion of the container

L3 = level of the liquid after the the expansion of the liquid

V1 = Volume of the liquid when its level is L1

V2 =Volume of the liquid when its level is L2

V3 = Volume of the liquid when its level is L3

Real expansion of the liquid ( ΔV ) = V3 – V2

Apparent expansion of the liquid ( ΔV’ ) = V3 – V1

We know that

Real expansion = apparent expansion + expansion of the container

ΔV = ΔV’ + ΔVc

dividing both sides by V1 ΔT, we get

ΔV / ( V1 ΔT ) = ΔV’ / ( V1 ΔT )  + ΔVc / ( V1 ΔT )

γr = γ’ + γc

12 ) Why there is not apparent expansion in case of thermal expansion of a gas ?

Ans : The gas has not fixed volume. It occupies the volume of the vessel in which it is kept. So, if the gas is heated in a vessel, at first the vessel will expand and with the expansion of the vessel the gas will expand. Therefore, there is no apparent expansion in case of gas expansion.

13 ) Define specific heat capacity.

Ans : The amount of heat required to raise of temperature of a body of unit mass by 1 0C, is called as the specific heat capacity. It depends on the nature of the material of the body.

14 ) Define thermal heat capacity.

Ans :The amount of heat required to raise the temperature of a substance through 1 0C, is called as thermal heat capacity. Its SI unit is J/K and CGS unit is cal / 0C.

15 ) Define water equivalent.

Ans : The water equivalent is defined as the amount of water that absorbs or emits the same amount of heat as is done by the given body for same rise or fall in temperature. Its SI unit is Kg and CGS unit is gm.

16 ) Define molar heat capacity.

Ans : The amount of heat required to raise the temperature of 1 mole of substance through 1 0C, is called as molar heat capacity.