This document discusses various topics related to thermal physics including: 1. Thermal expansion of solids and the coefficients of linear, areal, and volumetric expansion. 2. The ideal gas equation and how temperature can be measured using a gas in a constant volume gas thermometer. 3. Specific heat capacity and the differences between specific heat at constant volume and constant pressure.

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Thermal Expansion
Increase in the Dimensions of a substance on heating
is called its Thermal Expansion.
Expansion in Solids
Almost all the solids (except bismuth, ice and cast iron)
expand on heating. In
solids all the atoms and
molecules are closely bound to each other. So, they
exert strong electromagnetic forces on each other. Due
to this tight bonding between them, though they vibrate
about their mean positions, they cannot leave it
forever. When a solid is heated, the atoms and
molecules absorb thermal energy and start vibrating
with greater amplitude.
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So, they move away from each other or intermolecular
distance
between
them
increases.
This
causes
changes in different dimensions of the solid.
1. Linear Expansion
Increase in the length of the solid on heating is
called linear expansion.
Let,
o
L0 = length of the solid at 0 C.
o
Lt = length of the solid at t C.
t = rise in temperature of the solid.
∴ Lt – L0 = increase in the length.
It is found that
Lt – L0 α L0t
∴ Lt – L0 = α L0t .....(1) where α is coefficient of
linear expansion of the solid.
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L t L0
L0 t
If L0 = 1m or cm and t = 1℃, then
α = Lt – L0
Thus, temperature coefficient of linear expansion of
a solid is increase in its length per unit original
length at 0℃, per unit rise in its temperature. Its unit
is per ℃ (/℃) and values for solids are ranging
-6
-5
from 10 to 10 / ℃.
From equation (1)
Lt = L0 + L 0 α t
∴ Lt = L0 (1 + α t) ..... (2)
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Let, L1 = length of the solid at t1℃
L2 = length of the solid at t2℃
From equation (2),
L1 = L0 (1 + α t1) and
L2 = L0 (1 + α t2)
L2
L1
1
t2
1
t1
∴ L2 / L1 = (1 + α t2) (1 + α t1)
-1
-1
Binomial expansion of (1 + α t1) will be
1 – α t1 + terms containing higher powers of α.
As α is very small, its higher powers can be
neglected.
-1
∴ (1 + α t1) = 1 – α t1
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∴ L2/L1 = (1 + α t2) (1 – α t1)
2
∴ L2/L1 = 1 – α t1 + α t2 – α t1t2
∴ L2/L1 = 1 + α (t2 – t1)
∴ L2 = L1 + L1α (t2 – t1) ... (3)
∴ L2 – L1 = L1 α (t2 – t1)
L2 L1
L1 t2 t1
Thus, by knowing L1, L2 and rise in temperature
(t2 – t1), a can be calculated approximately.
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2. Areal or Superficial Expansion
Let, A0 = area of the solid at 0℃.
At = area of the solid at t℃.
t = rise in temperature of the solid.
∴ At – A0 = increase in the area.
It is found that At – A0 α A0t
∴ At – A0 = β A0t ..... (1) where β is coefficient of
areal expansion of the solid.
A t A0
A0 t
If A0 = 1m2 or cm2 and t = 1℃, then
β = At – A0
Thus, temperature coefficient of areal expansion of
a solid is increase in its area per unit original area
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at 0℃, per unit rise in its temperature. Its unit is per
℃(/ ℃).
From equation (1)
At = A0 + A0 βt
∴ At = A0 (1 + βt) ..... (2)
Every time it is not essential to measure area of the
solid at 0℃, to calculate β. It can be calculated at
any temperature with the formula :
A2 A1
A1 t2 t1
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3. Volumetric Expansion
Let,
V0 = volume of the solid at 0℃.
Vt = volume of the solid at t℃.
t = rise in temperature of the solid.
∴ Vt – V0 = increase in the volume.
It is found that Vt – V0 α V0t
∴ Vt – V0 = γ V0t ..... (1) where b is coefficient of
areal expansion of the solid.
V1 V0
V0 t
3
3
If V0 = 1m or cm and t = 1℃, then
γ = Vt – V0
Thus, temperature coefficient of volume expansion
of a solid is increase in its volume per unit original
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volume at 0℃, per unit rise in its temperature. Its
unit is per ℃ (/℃).
From equation (1)
Vt = V0 + V0 β t
∴ Vt = V0 (1 + γ t)..... (2)
Every time it is not essential to measure volume of
the solid at 0℃, to calculate γ. It can be calculated
at any temperature with the formula :
V2 V1
V1 t2 t1
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Relation between α, β and γ
Consider a thin square plate of a solid having side L0 at
0℃.
∴ Surface area at 0℃ = A0 =
2
L0
Let the plate be heated to t℃. Let,
Lt = length of the plate at t℃
∴ At = area of the plate at t℃ = Lt
2
For linear expansion, Lt = L0 (1 + α t)
2
∴ Lt =
2
L0
(1 + α t)
2
i.e. At = A0 (1 + α t)
2 2
∴ At = A0 (1 + 2αt + α t )
2 2
As α is very small α t can be neglected.
∴ At = A0 (1 + 2αt)
But, At = A0 (1 + β t)
∴ 2α = β
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Consider a uniform of a solid having side L0 at 0℃.
∴ Volume of the cube at 0℃ = V0 =
3
L0
Let the plate be heated to t℃. Let,
Lt = length of the plate at t℃
∴ Vt = volume of the plate at t℃ = Lt
3
For linear expansion, Lt = L0 (1 + α t)
3
∴ Lt =
3
L0
(1 + α t)
3
i.e. Vt = V0 (1 + α t)
2 2
3
3 3
∴ Vt = V0 (1 + 3αt + 3α t + α t )
2 2
3 3
As α is very small 3α t , a t can be neglected.
∴ Vt = V0 (1 + 3αt)
But, Vt = V0 (1 + γ t)
∴ 3α = γ
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∴ 6α = 3β = 2γ
This is the relation between the three expansion
coefficients.
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Boiling point / Steam point
The temperature at which pure water freezes at
standard atmospheric pressure is called as ice point or
freezing point. The temperature at which pure water
boils at standard atmospheric pressure is called as
boiling point or steam point.
1. Celsius scale
On this scale the ice point (melting point of pure
ice) is marked as 0ºC and the steam point (boiling
point of water) is marked as 100ºC, both taken at
normal atmospheric pressure. The interval between
these points is divided into 100 equal parts. Each of
these division is called as one degree Celsius and
is written as 1ºC.
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2. Fahrenheit scale
On this scale the ice point (melting point of pure
ice) is marked as 32ºF (boiling point of water)
steam point is marked as 212ºF. The interval
between these two reference points is divided into
180 equal parts. Each division is called as degree
Fahrenheit and is written as ºF.
The graph of Fahrenheit temperature (tf) versus
Celsius temperature (tc). It is a straight line whose
equation is,
tf
32
180
tc 0
100
t f 32
180
tc 0
100
.............(1 (a))
Tk
273.15
100
15
.........(1(b))

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Ideal gas equation
Boyle’s law states that at constant temperature, the
volume of a given mass of a gas is inversely
proportional to its pressure.
Mathematically this law may be expressed as,
V
1
, at constant temperature ......(2)
P
Charle’s law states that at constant pressure, volume
of a given mass of a gas is directly proportional to
its absolute temperature. Mathematically this law
may be expressed as,
V ∝ T, at constant pressure .... (3)
Combining (2) and (3) we have,
PV
= constant .....(4)
T
For one mole of a gas the constant of proportionality is
R
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PV
T
R .............(5)
PV = RT---This relation is called as ideal gas equation.
R is universal gas constant.= 8.31 JK-1 mol-1
PV = nRT ......... (6)
From equation (6) PV ∝ T.
This relationship allows a gas to be used to measure
temperature in a constant volume gas thermometer
keeping the volume of a gas constant, it gives P ∝ T.
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Specific heat Capacity
Specific heat of a substance is defined as the quantity
of heat required to rise the temperature of unit mass of
a substance through 1ºC (or 1 K).
Q ∝ M ∆T
Q ∝ cM ∆T ............. (7)
where c is called as specific heat (or specific heat
capacity) of a substance. From equation (8) we get
c
Q
.............. (9)
m T
If m = 1 kg and ∆T = 10C then c = Q.
SI unit of specific heat is Jkg-10C-1 or Jkg-1 K-1, CGS
Unit is cal g-1 K-1 or Jkg-1 K-1. The specific heat of water
is 4.2 Jkg-10C-1; it means that 4.2 Jkg-10C-1 of energy
must be added to 1g of water to rise its temperature by
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1ºC. In case of gas slight change in temperature is
accompanied with considerable changes in both i.e. in
volume and pressure.
If gas is heated at constant pressure, volume changes
and hence some work is done on surrounding in
expansion. Hence more heat is required. Therefore
specific heat at constant pressure is greater than
specific heat at constant volume.
1. Specific heat of a gas at constant volume (cv) is
defined as the quantity of heat required to rise the
temperature of unit mass of a gas through 1K (or
10C) when its volume is kept constant.
2. Specific heat of a gas at constant pressure (cp) is
defined as the quantity of heat required to rise the
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temperature of unit mass of a gas through 1K (or
10C) when its pressure is kept constant.
For unit mass of a gas, the specific heats are called
principal specific heats.
i.
Molar specific heat of a gas at constant volume
(Cv) is defined as the quantity of heat required
to rise the temperature of one mole of the gas
through 1 K (or 10C), when is volume is kept
constant.
ii. Molar specific heat of gas at constant pressure
(CP) is defined as the quantity of heat required
to rise the temperature of one mole of the gas
through 1 K (or 10C) when its pressure is kept
constant. The number of molecules in one mole
of a gas is given by Avogadro’s number
N = 6.025 × 1023 molecules per mole
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= 6.025 × 1026 molecules per kilomole. SI unit
of molar specific heat is J/K and mole K.
The molar specific heat = molecular weight
× principle specific heat.
Cp = M × cp and Cv = M cv .............. (10)
At standard pressure the temperature at which a
substance changes its state from solid to liquid is
called as its melting point. Melting point of water is 0ºC.
At standard pressure the temperature at which a
substance changes it state from liquid to gas is called a
boiling point. The boiling point of water is 100ºC.
The triple point of water is that point where water in a
solid, liquid and gas states coexists in equilibrium and
this occurs only at a unique temperature and a
pressure.
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Latent heat of a substance is the quantity of heat
required to change the state of unit mass of the
substance without changing its temperature.
Conduction
It is mode of transfer of heat through a medium, without
actual migration of the particles of the medium.
Temperature gradient
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It is the rate of change of temperature with distance for
a metal rod in steady state.
The figure shows, a section of a metal rod of length
∆x. In steady state, if the temperature of the left end is
θ + ∆θ and that of the right and is θ, the temperature
gradient is ∆θ / ∆x.
In steady state, the rod doesn’t absorb any heat. So,
the amount of heat entering the rod per second (Q) is
same as the amount of heat leaving the rod per
second.
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Experimentally it is found that, the heat flowing through
a conductor or the amount of heat conducted by a
conductor is directly proportional to area (A) of the
conductor, temperature gradient (∆θ / ∆x) and time t
for which the heat is flowing.
i.e. Q ∝ A (∆θ / ∆x) t
∴ Q = kA (∆θ / ∆x)t
where k is coefficient of
thermal conductivity.
Q
k
A
If A
1,
x
x
t
1 and t
1, then
k=Q
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So, coefficient of thermal conductivity is the amount of
heat flowing through a conductor, of unit area, in unit
time, when unit temperature gradient is maintained
along its length.
Its units are cal / cm ℃ s in C.G. S., kcal / m ℃ s in
M.K.S. and J / m ºK s in S.I.
1 1 -3
-1
Its dimensions are [M L T K ]
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Convection
Transfer of heat energy because of actual migration of
particles from 1 point to another point is known as
Convection
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Radiation
Transfer of heat energy through vacuum is known as
radiation
Newton’s law of cooling
The rate of fall of temperature of a body is directly
proportional to excess temperature of a body over the
surrounding (provided that the excess is small)
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