“&ana¸ iva&ana AaiNa sausaMskar yaaMsaazI iSaxaNap`saar”
- iSaxaNa mahYaI- Da^ baapUjaI saaLuMKo
EaI svaamaI ivavaokanaMd iSaxaNa saMsqaa kaolhapUr saMcalaIt
ivavaokanaMd ka^laoja¸kaolhapUr
XI SCIENCE
[PHYSICS – I]
THERMAL EXPANSSION
XI SCIENCE NOTES
Prof. R. S. Gade
January 31, 2013
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THERMAL EXPANSION
Temperature and Heat:
The temperature of a body is its degree of hotness or coldness. A high temperature
of body represents that the body is hot while low temperature of body means the body is
cold. However sensation of hot and cold are not reliable because if one body has the
temperature greater than another body then we treat it as hot but the temperature of
same body may less than some another body and hence with respect to that body this
body is cold. Temperature of the body is the result of heat. If heat is increased the
temperature of the body is also increases.
When two bodies having different temperatures are brought together, heat flows
from the body at higher temperature to the body at lower temperature till the two bodies
are at the same temperature. Hence, “Heat is the form of energy transferred between
two (or more) systems or a system and its surroundings by virtue of temperature
difference.
SI unit of heat energy is joule (J) and SI unit of temperature is Kelvin (K) or 0C.
Measurement of temperature:
An instrument designed to measure temperature is called as thermometer. There
are different types of the thermometers: (I) Liquid-in-glass thermometer, (II) The
constant volume gas thermometers and (III) The resistance thermometers. In all of the
thermometers some measurable property of a substance which is sensitive to
temperature change is used. These thermometers can be calibrated in different scales of
temperatures. These scales are :
(I) Celsius Scale: In this scale the ice point (melting point of pure ice) is marked as
00C and the steam point (B.P. of water) is marked as 1000C. Both these points are taken
at normal atmospheric pressure. The interval between these points is divided into 100
equal parts. Each of these divisions is called as one degree Celsius and written as 10C.
(II) Fahrenheit scale: In this scale the ice point (meting point of pure ice) is marked as
320F and steam point is marked as 2120F. The interval between these two points is
divided into 180 equal parts. Each division is called as degree Fahrenheit and is written
as 0F.
(III) Kelvin Scale (Absolute Scale): The scale of temperature that has its zero at –
273.150C and temperature intervals is same as that on the Celsius scale is called Kelvin
scale of temperature.
Relation between the temperatures on different scales of temperatures:
If Fahrenheit temperature is tf and Celsius temperature is tc and Kelvin temperature
is T then:
t f − 32
100
=
tc − 0
100
- - - - (i); T = tc + 273.15 - - - - - (ii);
t f − 32
100
=
tc − 0
100
=
T −273.15
100
- - - - - (iii)
Importance of the gas thermometer over liquid thermometers:
There are some advantages of gas thermometers over the liquid thermometers as:
With constant volume gas thermometers temperature is calibrated in terms of pressure.
At low densities and pressures, all gases behave in the same manner. Thus the gas
thermometer gives the same reading independent of the gas used also thermal capacity
of a gas is low as compared to liquids. Therefore small change of temperature can be
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recorded accurately. Gas thermometers are suitable to measure low and high
temperatures.
Short note on absolute scale of temperature:
Absolute scale of temperature: “The temperature scale that has its zero at –
273.150C and temperature intervals are same as that on the Celsius scale is called as
Kelvin scale or absolute scale.
Lord Kelvin suggested this scale of temperature. In this scale the temperature at
which every substance in nature has least molecular activity is taken as zero of the
scale. This temperature is called absolute zero temperature and its value is -273.150C.
Absolute scale starts from this temperature and the value of each degree is same as
degree Celsius. The temperature in Kelvin scale is represented as T. Hence the
temperature in Celsius scale can be converted into Kelvin scale temperature by adding
273.15 in it. For example if tc = 1000C then
T = tc + 273.15 = 373.15 K
For all calculations and practical applications the absolute zero temperature is taken as 273 0C only
∴ T = (tc + 273) K
Triple point of water: The triple point of the 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. This pressure is 4.58 mm of mercury and the temperature is
273.15 K or 0.01 0C.
Ideal gas equation/ Derivation of P V = n R T:
The relation between the three variables of a gas such as pressure, volume and
temperature is called ideal gas equation.
Derivation: According to Boyle’s law,” At constant temperature the volume of given
amount of gas is inversely proportional to its volume”
i. e. P V = Constant, at constant temperature. - - - - - (i)
According to Charles law, “At constant pressure, volume of a given amount of gas is
directly proportional to its absolute temperature.”
∴ V / T = constant at constant pressure. - - - - - - - (ii)
Combining (i) and (ii) we get, P V / T = constant - - - - - - (iii)
For one mole of a gas this constant is R called gas constant.
∴ P V / T = R or P V = R T - - - - - - (iv) This relation is called ideal gas equation
for one mole of any gas. If the given amount of gas consists n moles then above relation
can be written as: P V = n R T - - - - - - (v)
Expansion of solid:
Generally when the substance is heated it expands (Exceptions: water from 00C to
40C contracts, rubber on heating contracts.)
If a solid is heated, its molecules receive thermal energy and they begin to vibrate
with larger amplitude. Hence the distance between the two molecules increases, which
results into the expansion of that solid. The solid which expands equally in all possible
directions is called isotropic solids. All metals are isotropic substances.
During the expansion depending upon the increase in dimension there are three
types of the expansions for the solid as:
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1.
The Linear expansion: When the heat is supplied to isotropic solid, the increase in its
length is called linear expansion of that solid.
Let Lo be the length of the uniform rod at 00C. When the rod is heated its length
increases. Let Lt be the length of the rod at t0C then (Lt - Lo) is called linear expansion
of the rod.
Experimentally it is found that: (Lt -Lo) ∝ Lo and (Lt - Lo) ∝ t
∴ (Lt - Lo) ∝ Lo t = α Lo t ∴ Lt = Lo (1 + αt) ------ (i) where α is called the
coefficient of linear expansion of the material of the rod, this depends upon the material
of the solid. From equation (i) we get:
α=
*
*
Lt − L0
In this equation (ii) if L0 = 1 unit; t = 10C then α = (Lt - L0). Thus,
“Coefficient of linear expansion of the material of solid is defined as the increase
in length per unit original length at 00C, per a degree rise in temperature”.
SI unit for coefficient of linear expansion is per degree Kelvin (/0K)
For the practical determination of the coefficient of linear expansion we use:
α=
2.
*
L1 t2 − t1
where L1 and L2 are the length of the rod at t10C and t20C resp.
At − A0
A0t
---- (ii)
In this equation (ii) if A0 = 1 unit; t = 10C then β = (At - A0). Thus,
“Coefficient of areal expansion of the material of solid is defined as the increase in
surface area per unit original surface area at 00C, per a degree rise in temperature”.
SI unit for coefficient of areal expansion is per degree Kelvin (/0K)
For the practical determination of the coefficient of areal expansion we use:
β=
*
L2− L1
The Areal expansion: When the heat is supplied to isotropic solid, the increase in its
surface area is called areal expansion of that solid.
Let Ao be the area of the uniform solid at 00C. When the solid is heated its surface
area
increases. Let At be the area of the solid at t0C then (At - Ao) is called areal
expansion of the solid.
Experimentally it is found that: (At -Ao) ∝ Ao and (At - Ao) ∝ t
∴ (At - Ao) ∝ Ao t = β Ao t ∴ At = Ao (1 + β t) ------ (i) where β is called the
coefficient of areal expansion of the material of the solid, this depends upon the material
of the solid. From equation (i) we get:
β=
*
*
---- (ii)
L0t
A2− A1
A 1 t2 − t1
whereA1 and A2 are the surface area of the solid at t10C and t20C resp.
Coefficient of linear expansion, coefficient of superficial expansion and coefficient
of cubical expansion:
(i) Coefficient of linear expansion (α): “Coefficient of linear expansion of the material
of solid is defined as the increase in length per unit original length at 00C, per a degree
rise in temperature”.
SI unit for coefficient of linear expansion is per degree Kelvin (/0K)
(ii) Coefficient of superficial expansion (β): “Coefficient of superficial expansion or
areal expansion of the material of solid is defined as the increase in surface area per unit
original surface area at 00C, per a degree rise in temperature”.
SI unit for coefficient of areal expansion is per degree Kelvin (/0K)
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(iii) Coefficient of cubical expansion (γ): “Coefficient of cubical expansion of the
material of solid is defined as the increase in volume per unit original volume at 00C,
per a degree rise in temperature”.
SI unit for coefficient of cubical expansion is per degree Kelvin (/0K)
Relation between α and β:
Consider an isotropic solid in the form of a square plate having the length of each
side as L0 at 00C. Thus surface area of each face at 00C is A0 = L02. Let this plate is
heated to a temperature t0C such that length of each side of the plate becomes Lt, and
the area of each face becomes At = Lt2.
From the definition of linear expansion we have: Lt = L0 (1 + α t)
∴ Lt2 = L02 (1 + α t)2 ∴ At = A0 (1 + 2αt + α2 t2).
But generally α is very small, and hence its higher powers can be neglected.
∴ At = A0 (1 + 2αt) --- (i)
But from the definition of areal expansion we have, At = A0 (1 + β t) ---- (ii)
Comparing equations (i) and (ii) we get β = 2 α. This is the required relation.
Relation between α and γ:
Consider an isotropic solid in the form of a cube having the length of each side as
L0 at 00C. Thus volume of cube at 00C is V0 = L03. Let this cube is heated to a
temperature t0C such that length of each side of the cube becomes Lt, and volume of
cube becomes Vt = Lt3.
From the definition of linear expansion we have: Lt = L0 (1 + α t)
∴ Lt3 = L03 (1 + αt)3 ∴ Vt = V0 (1 + 2α3 t3 + 3 αt + 3 α2 t2).
But generally α is very small, and hence its higher powers can be neglected.
∴ Vt = V0 (1 + 3αt) --- (i)
But from the definition of volume expansion we have, Vt = V0 (1 + γ t) ---- (ii)
Comparing equations (i) and (ii) we get γ = 3 α. This is the required relation.
Relation between α, β and γ:
We have relation between α and β as: β = 2 α i. e. 3 β = 6 α ---- (i)
And relation between α and γ as: γ = 3 α i.e. 2γ = 6α ----- (ii)
Comparing equations (i) and (ii) we get: 6 α = 3 β = 2 γ.
Expansion of liquid:
As any liquid does not possess a definite length or definite surface area it is not
possible to define linear expansion and areal expansion for any liquid. But as liquid
possess definite volume we can study the volume or cubical expansion for liquid.
However as the liquid is always taken in the container, when the liquid is heated its
container also gets heated and therefore both liquid and container undergoes a change in
volume.
If we measure the increase in volume by considering the expansion of the
container then the increase in volume is called real expansion of the liquid. While if we
measure the increase in volume of the liquid without considering the expansion the
container then the increase in volume of the liquid is called the apparent expansion.
Real expansion the liquid is always greater than the apparent expansion of the
liquid as:
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Real expansion of liquid = (Apparent expansion of liquid) + (Cubical expansion of
container)
The volume or cubical expansion: When the heat is supplied to isotropic solid, the
increase in its volume is called cubical expansion of that solid.
Let Vo be the volume of the uniform solid at 00C. When the solid is heated its
volume increases. Let Vt be the volume of the solid at t0C then (Vt - Vo) is called
cubical expansion of the solid.
Experimentally it is found that: (Vt - Vo) ∝ Vo and (Vt - Vo) ∝ t
∴ (Vt - Vo) ∝ Vo t = γ Vo t ∴ Vt = Vo (1 + γ t) ------ (i) where γ is called the
coefficient of cubical expansion of the material of the solid, this depends upon the
material of the solid. From equation (i) we get:
γ=
*
*
Vt − V0
V0t
---- (ii)
“Coefficient of cubical expansion of the material of solid is defined as the increase
in volume per unit original volume at 00C, per a degree rise in temperature”.
SI unit for coefficient of cubical expansion is per degree Kelvin (/0K)
For the practical determination of the coefficient of cubical expansion we use:
γ=
*
In this equation (ii) if V0 = 1 unit; t = 10C then γ = (Vt - V0). Thus,
V2− V1
V 1 t2 − t1
where V1 and V2 are the volumes of the solid at t10C and t20C resp.
Specific heats of a gas:
Why do we generally consider two specific heats of a gas?
In case of solid and liquid a slight change in temperature produces negligible
changes in pressure and volume hence liquid and solid posses’ single value of specific
heat. But in case of any gas a slight change in temperature produces the considerable
changes in volume and pressure simultaneously. Hence specific heat measured by
keeping volume constant is different than the specific heat measured when pressure is
kept constant. Hence in case of all the gases it is necessary to consider two specific
heats i.e. specific heat at constant volume and specific heat at constant pressure.
From these two sp. heat specific heat at constant pressure is greater than specific heat at
constant volume.
There are two types of specific heat for any gas: principal and molar specific heats.
(A) The principal specific heats: The specific heat of a gas defined for unit mass i.e. 1
kg is called principal specific heat. There two types of principal specific heats as:
(i) Principal specific heat at constant volume (cv): The amount of heat required to
increase the temperature of unit mass of a gas through 1 K (or 10C) when its volume is
kept constant is called principal specific heat at constant volume.
(ii) Principal specific heat at constant pressure (cp): The amount of heat required to
increase the temperature of unit mass of a gas through 1 K (or 10C) when its volume is
kept constant is called principal specific heat at constant pressure.
(B) The molar specific heats: The specific heat of a gas defined for one mole
expressed in kg is called molar specific heat. There are two types of the molar specific
heats as:
(i) Molar specific heat at constant volume (Cv): The amount of heat required to
increase the temperature of unit mass of a gas through 1 K (or 10C) when its volume is
kept constant is called molar specific heat at constant volume.
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(ii) Molar specific heat at constant pressure (Cp): The amount of heat required to
increase the temperature of unit mass of a gas through 1 K (or 10C) when its volume is
kept constant is called molar specific heat at constant pressure.
It is therefore Cp = M cp and Cv = M cv where M is the molecular weight of gas.
Calorimetry: We know that when a body at higher temperature is brought in contact
with the body at lower temperature then heat is transferred from a body at higher
temperature to a body at lower temperature such that heat lost by the body at
temperature is equal to the heat gained by the body at lower temperature. This heat
transferred is measured by means of a device called calorimeter and the measurement of
heat is called Calorimetry.
Latent heats: The latent of a substance is the amount of heat required to change the
state of unit mass of that substance without change in its temperature. Depending upon
the change in state we can define following two latent heats.
(i) Latent heat of fusion: The quantity of heat required to convert unit mass of
substance from its solid state to the liquid state at its melting point without change in its
temperature is called latent heat of fusion.
(ii) Latent heat of vaporization: The quantity of heat required to convert unit mass of
substance from its liquid state to the gas (vapor) state at its boiling point without change
in its temperature is called latent heat of vaporization.
Heat transfer: There are three different modes of heat transfer as: (i) Conduction, (ii)
Convection and (ii) Radiation.
Thermal conduction:
Conduction of heat is the mode of transfer of heat in which heat is transferred
through the medium without actual migration of the particles of that medium from a
point at higher temperature to the point at lower temperature. Conduction is the slowest
process of heat transfer. The substance which conducts heat readily is called good
conductor of heat while the substance which do not conduct heat readily is called bad
conductor (insulator) of heat. e. g. all metals are good conductors of heat while rubber,
wood, glass etc are bad conductors of heat. That is why the boilers, cooking utensils,
calorimeters are made from metals while the handle of cooking utensils are made from
wood.
Temperature gradient:
When the heat is supplied to one end of a thick rod made from conducting material
then that heat is transferred to the other end of rod by the process of conduction. Thus
the temperature of different points along the length of the rod increases gradually. In
this state the received by any section of rod is used for two ways. Part of heat is
absorbed by the section so that temperature increases and the remaining part is
transferred to the next section. After some time it is found that temperature at all points
of the rod remains constant even the heat supplied is continued. This state of the rod is
called steady state. In this steady state there is a gradual fall of temperature along the
length of the rod.
“The rate of decrease in temperature with distance is called temperature gradient”
∴ Temperature gradient = dθ / dx where dθ is the decrease in temperature over a
distance dx along the length of the rod in steady state.
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Coefficient of thermal conductivity (K):
In the steady state of the rod the heat received by any section of the rod is
completely conducted to the next section without absorbing. this heat conducted (Q) is
found to be directly proportional to: (i) its area of cross-section (A), (ii) the temperature
gradient (dθ / dx) and (iii) time (t) for which heat is conducted.
∴ Q ∝ A (dθ/dt) t or
Q = K A (dθ/dt) t ----- (i) where K is a constant which
depends upon the material of the solid and is called coefficient of thermal conductivity
of the material of the solid.
In equation (i) if A = 1 unit, dθ/dt = 1 unit and t = 1 unit then Q = K. Thus,
“Coefficient of thermal conductivity of material can be defined as the quantity of
heat conducted in the steady state through a material per unit time, per unit area of
cross-section per unit temperature gradient”
CGS unit of thermal conductivity is cal/cm s 0C while MKS unit is kcal/m s K and
SI unit is J/m s K.
Transfer of heat by conduction through solid rod:
As any metal consists of large number of free electrons which are helping to
transfer the heat received at one point to another point as well as they are helpful for
transferring the changes and hence conducting the electric currents. Hence metals are
good conductors of heat and electricity.
When one end of the metal rod is placed in Bunsen flame then the molecules at
that end of the rod vibrates with more amplitude and collides with the neighboring
molecules. During these collisions heat is given to the next molecule and that molecule
start vibrating with more amplitude. In this way heat is transferred from particle to
particle without actual migration of particles. This process of heat transfer is called
conduction.
Steady state: During the conduction process, initially heat received by any section of
rod is partly transferred to next cross section and partly absorbed there so that
temperature at that point increases. But after some time it is found that heat received by
any section is transferred completed to the next section so that the temperature at that
point remains constant. This state of the body is known as steady state of the body. In
steady state the temperature of the body at different temperature may be different but it
is constant there.
Applications of thermal conductivity in everyday life:
In everyday life the use of thermal conductivity is made properly as follows:
(I) Cooling utensils are made of metals with handles of bad conductors i.e. having less
conductivity. Here heat can be easily conducted through metals as metals are good
conductors of heat. Bad conductors will not conduct heat from utensils to our hand and
are protected from burn.
(II) In winter, birds often swell their feathers. The air enclosed between their body and
feathers acts as a bad conductor of heat and prevents the flow of heat from the body of
the bird to the cold surroundings.
The significance of thermal conductivity: Thermal conductivity of solid is a measure
of ability of solid to conduct heat through it. The amount of heat transferred through the
solid depends on its thermal conductivity. If thermal conductivity is large then the solid
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is said to be good conductor of heat while if this conductivity is less then it is called bad
conductor of heat. Depending upon this the different metals are used for different
purposes. For examples all the utensils are made from good conductors while their
handles are made from the bad conductors. Hence depending on the thermal
conductivity the substance can used for convenient purpose.
Heat transfer by convection:
When the beaker containing water is placed on a burner the water at the bottom of
beaker get heated and expands so that its density decreases. This water thus becomes
lighter and moves upward and is replaced by the heavier water situated above and
around it. This water is in turn heated, expands so that its density decreases and it
moves upward. This cycle goes on repeating so long as the water is being heated. In this
way the heat is transported by the particles of the liquid by migrating from one place to
another place. This process of transfer of heat is called convection of heat. The heating
of gas and liquid takes place by convection. This process is faster as compared to
conduction.
Applications: (I) Land and sea breezes are formed as a result of convection currents in
air. During day time the land near the sea gets heated by the sun to a higher temperature
than the sea. Due to the greater heat capacity of water the temperature of the sea does
not rise as much as that of land. Also due to the mixing up of hot surface layers with the
colder layers below them the temperature of sea is lower than that of land. As a result of
this air above the land is heated, it becomes lighter and rises upwards. It is replaced by
the cooler air moving from sea towards land. This results in sea breeze. At night the
conditions are reversed and the hence land breeze is set up at night.
(II) The large scale wind currents over the earth, called trade winds, occur mainly due to
convection. When warm air from the equatorial regions rises up, cold air from the Polar
Regions moves towards the equatorial regions forming the trade winds.
Heat transfer by radiation:
The radiation of heat is defined as the process of transfer of heat in the form of
electromagnetic waves for which material medium is not necessary. Radiation is the
fastest process of heat transfer. If we sit near the fire we receive heat by radiation. We
receive heat from the sun in form of radiation only. The amount of heat that body can
absorb or radiate depends on the colour of the body.
Some reasons:
(i) Stainless steel cooking pan is fitted with copper bottom.
This is because copper is having less heat capacity but grater thermal conductivity.
Hence heat supped to the utensil goes to the food fast and without any loss of heat.
(ii) Two layers of cloth used to cover body retain body heat better than single layer of cloth
having double the thickness.
This is because if there are two layers of cloths, then there is a layer of air between
the two cloths. Now air is the bad conductor of heat and hence there is no heat exchange.
And hence body retains the heat. But if there is single layer of cloth it has some thermal
conductivity and hence heat exchange is possible and hence body does not retain the
heat.
(iii) Beakers crack when hot water is poured in it.
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When hot water is poured in a beaker of thick glass the inner surface of beaker
expands on heating. Since glass is a bad conductor of heat the heat inside does not reach
the outer surface quickly so outer surface doesn’t expand and glass cracks.
(iv) We can hold our fingers beside the candle without harm but not above the flame.
Due to flame of candle the air near and above the flame gets heated and it
becomes lighter due to decrease in density. And it moves above and not to side of flame.
Hence the temperature around the flame is less as compared above the flame.
(v) In summer one should wear white clothes and in winter one should wear black cloths.
The amount of heat absorbed of radiated by body depends on its colour. The black
body absorbs most of the heat incident on it while the white or well polished body
reflects most of the incident heat. Hence we wear dark cloths in winter because they
absorbs most of the radiant heat incident upon them and therefore we feel warm on the
other hand we wear white clothes in summer because they reflect most of the radiant
energy incident upon them.
Newton’s law of cooling:
Statement: “Newton’s law of cooling states that the rate of fall of temperature of a
body is directly proportional to the excess of temperature of the body over the
surroundings, provided this excess is small”
Verification: To verify the Newton’s law of cooling the experimental set is as shown in
figure (a). In this fill the calorimeter (c) up to two third of its capacity with boiling
water and cover it with holed lid (L). Fit the thermometer (T) through a hole in the lid
and adjust its position so that its bulb is fully immersed in the water. Keep the
calorimeter in constant temperature vessel (W).
Note the temperature of the surroundings as θ0 and then note the temperature of
water on the thermometer at every one minute interval until the temperature of water
decreases by about 250C. Plot a graph of temperature (θ) on Y-axis against the time (t)
on X-axis. This graph is called cooling curve as shown in fig(b). Draw tangents to this
curve at suitable points on the surve. The slope of each tangent (dθ/dt) gives the rate of
fall of temperature at that temperature. Taking (0, 0) as origin plot the graph of (dθ/dt)
against corresponding temperature excess (θ - θ0). This is a straight line passing
through origin as shown in fig (c). This straight line graph shows that:
(dθ/dt) ∝ (θ – θ0) . This verifies the Newton’s law of cooling.
T
S
L
•
(dθ/dt
θ
C
W
•
•
•
O
fig (a)
t
fig (b)
(0, 0)
(θ – θ0)
fig (c)
* * * * *
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SOLVED PROBLEMS FROM BOARD TEXT BOOK
1. The length of metal rod is 150 cm at 250C. Find its length when it is heated to 1500C. (α
= 2.2 x 10-5 /0C for steel)
Solution: α =
L2 − L1
L1 t2 − t1
⇒ L2 = L1 [1 + α (t2 − t1 )]
= 150 [1 + 2.2 x 10-5 x 125] = 150.4 cm
2. A metal rod, 1.8 m long increases in length by 1.4 mm when heated from 00C to 500C.
Find the coefficient of linear expansion of metal.
Solution: α =
L2 − L1
L1 t2 − t1
=
1.4 x 10−3
1.8 x 50
= 1.555 x 10-5 /0C
3. The difference between the length of a steel rod and brass rod is 0.6 m at all
temperatures. What are their lengths at 00C if α = 1.8 x 10-5 /0C for brass and α = 1.2 x
10-5 /0C for steel.
Solution: L0 – L0’ = 0.6 ---- (i) also Lt – Lt’ = 0.6 ----- (ii)
αbrass =
L′t
L′t
L′t − L′0
L′0 t
and αsteel =
Lt − L0
L0 t
L′0
∴
=
1 + αb t and Lt = L0 1 + αs t
∴
- Lt = L′0 + L′0 αb t - L0 - L0 αs t
∴
0.6 = 0.6 + t (L′0 αb - L0 αs )
∴
L′0 αb - L0 αs = 0 ----- (iii)
Solving equations (i) , (ii) and (iii) we get L0 = 1.8 m and L0’ = 1.2 m.
4. The surface area of the metal plate is 2.4 x 10-2 m2 at 200C. When the plate is heated to
1850C, its area increases by 0.8 cm2. Find the coefficient of areal expansion of metal.
Solution: β =
A2 − A1
A1 t2 − t1
=
0.8 x 10−4
2.4 x 10−2 x 165
= 0.002020 x 10-2 = 2.02 x 10-5 /0C
5. Calculate the difference in the temperature between the water at the top and bottom of
water fall 200 m high. Specific heat of water is 4200 J/kg 0C (Ans: 0.4670C)
Solution: The potential energy of water at height h is converted into heat and
temperature of water increases. Hence:
Potential energy = heat absorbed by water
m g h = m s dθ
∴ dθ = h g / s = 9.8 x 200 / 4200 = 9.8 / 21 = 0.4670C
6. A certain mass of a gas at 200C is heated until both its pressure and volume are doubled.
Calculate its final temperature. (Ans: 1172 K or 8990C).
Solution: According to ideal gas equation:
P1 V1
T1
=
P2 V2
T2
∴ T2 = P2 V2 T1 / P1 V1 = (2P1) (2V1) T1 / P1 V1
= 4 T1 = 4 (20 + 273) = 4 x 293 = 1172 K
* * * * *
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MHT CET KEY POINTS
*
*
*
*
*
In general the size of body increases with rise in temperature. (Exception: Rubber on
heating contracts; Water when heated from 00C to 40C contracts instead of expanding)
If the expansion in solid is same in all possible directions then it is called isotropic
solids e.g. metals, glass, rock salt etc. While if the expansion of solid is different in
different directions then the solid is called anisotropic solid, in such case if expansion is
in one direction there is contraction in the perpendicular direction.
Depending upon the type of change in dimensions there are three types of the
expansions as (i) Linear expansion, (ii) Areal expansion and (iii) Volume expansion.
Solids possess a definite length, surface area and volume. Therefore solid possess linear
expansion, areal expansion and volume expansion. While as liquid posses a definite
volume only, in case of liquid we can study volume expansion only which are two types
as: (i) Real volume expansion and (ii) Apparent volume expansion. In case of the gases,
volume not only depends upon temperature but also on pressure, hence in case of gases
we can consider volume expansions expansion at constant pressure.
Coefficient of linear expansion depends upon the nature of the material, original length
and rise in temperature and is given by: α = L 2 L1 = L t L 0 where L0 = length at 00C;
L1 t
L0 t
0
Lt = length at t C; t = Rise in temperature
In practice if L1 is length at t10C and L2 is the length at t20C resp. then α = L 2 L1
L1 t
*
Final length after the heating: Lt = L0 (1 + αt) or L2 = L1 (1 + αt).
Coefficient of linear
Coefficient of linear
Material
expansion
Material
expansion (/ 0C)
(/ 0C)
Steel
1.2 x 10-5
Brass
1.8 x 10-5
Gold
1.4 x 10-5
Silver
1.88 x 10-5
Copper
1.1 x 10-5
Aluminium
2.55 x 10-5
Nickel
1.3 x 10-5
Diamond
1.2 x 10-6
*
Coefficient of areal expansion depends upon the nature of the material, original surface
area and rise in temperature and is given by: β = A 2 A1 = A t A 0
A1 t
A0 t
where A0 = area
at 00C; At = area at t0C; t = Rise in temperature
In practice if A1 is area at t10C and A2 is the area at t20C resp. then: β = A 2 A1
A1 t
*
Final area after the heating: At = A0 (1 + β t) or A2 = A1 (1 + β t).
Coefficient of volume expansion depends upon the nature of the material, original
volume and rise in temperature and is given by: γ = V 2 V1 = V t V 0
V1 t
V0 t
0
where V0 = volume at 0 C; Vt = volume at t0C; t = Rise in
temperature
[XI/PHY/RSG/THERMAL EXPANSSION]
Page 12
In practice if V1 is volume at t10C and V2 is the volume at t20C resp. then: γ = V 2 V1
V1 t
*
*
Final volume after the heating: Vt = V0 (1 + γ t) or V2 = V1 (1 + γ t).
The relation between α, β is β = 2α; Between α and γ is γ = 3 α and that between all the
coefficient is:
6α = 3β = 2γ
In case of liquid the coefficient of expansion measured with considering the expansion
of the container then it is called the coefficient of real expansion (γr) while if we
measure the volume expansion of liquid without considering the expansion of the
container then the coefficient is known as coefficient of apparent expansion (γa).
γr = V r V 0
V0 t
*
*
*
*
*
*
*
*
*
and
γa = V a V 0
V0 t
and
γg = V r V 0
Va t
Where γg is the cubical (volume) expansion of the glass container.
It is found that γr = γa + γg
There are three different modes of transfer of heat energy as (i) conduction, (ii)
convection and (iii) radiation.
Conduction is the slowest process of heat transfer in which heat is transferred with the
help of vibrational motion of particles of medium without actual migration of these
particles. Compared to conduction convection is faster process of heat transfer in which
heat is transferred actual migration of particles of the medium. While radiation is the
fastest of the three modes of heat transfer in which heat is transferred in the form of
electromagnetic waves. Radiation does not require any material medium to heat
transfer.
Conduction generally takes place in solids. It does not take place in vacuum. Metals are
good conductors of heat.
Practical evidences of conduction: (i) If we put a copper vessel on stove parts of the
vessel not in contact with the flame also get hot rather quickly, (ii) On winter night a
metallic handle feels colder than a wooden door, (iii) In summer water in walls is cooler
and in winter it is warm.
If in case of a body surrounded by a non conducting material is heated continuously
then state in which heat received by any section of rod is completely transferred to the
next section of rod without absorbing so that temperatures of different points of the
body may be different but it is constant. Such a state is called steady state of the body. I
this state as we go in the direction of heat transfer away from source the temperature
goes on decreasing though it is constant there.
In the steady state of body the rate of fall of temperature with respect to distance in the
direction of flow of heat is called temperature gradient. i.e Temp. gradient = (dθ / dx).
SI unit for temp. gradient is K/m or 0C/m.
In the steady state, the quantity of heat flowing through the material, per unit area, per
unit temperature gradient is called the coefficient of thermal conductivity (K).
K=
Q
t A Δθ
Δx
 
the SI unit for K is J/s m K or W/m K or Kcal/s m K and
[K] = [M1L1T-3K-1]
[XI/PHY/RSG/THERMAL EXPANSSION]
Page 13
*
*
*
*
*
Thermal conductivities of some material:
Thermal conductivity
Thermal conductivity
Substance
Substance
0
cal/s cm C W/m K
cal/s cm 0C
W/m K
Copper
0.92
385
Ice
0.004
1.7
Aluminium
0.49
205
Glass
0.002
0.8
Brass
0.25
109
Oxygen
0.023
0.000056
Steel
0.12
50.2
Air
0.000057
0.024
Lead
0.083
34.7
Helium
0.00034
0.14
Mercury
0.020
8.3
Hydrogen
0.14
0.00013
Ideal gas equation: P V / T = Constant = R
∴ P V = R T for one mole; P V = n R T for n moles of a gas
Specific heat, S = Q / m dθ
Thermal stress = F / A = Y A dθ
Triple point of water is at the pressure 4.58 mm of mercury & at temperature 273.16 K.
* * * * *
[XI/PHY/RSG/THERMAL EXPANSSION]
Page 14
MULTIPLE CHOICE QUESTIONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
When water is heated from 00C to 100C, its volume:
(a) Decreases
(b) Increases
(c) Remains same
(d) First (a) & then (b)
For the given material which one of the following is maximum:
(a) Coefficient of linear expansion
(b) Coefficient of areal expansion
(c) Coefficient of cubical expansion.
(d) All have equal values
The coefficient of volume expansion of a solid is x times the coefficient of areal
expansion then x is:
(a) 1.5.
(b) 2.5
(c) 2
(d) 3
If the end of a metal rod is heated, the rate of flow of heat does not depend upon:
(a) Area of end of rod
(b) Time
(c) Mass of the rod.
(d)Temp. gradient
The fastest mode of heat transfer is:
(a) Conduction
(b) Convection
(c) Radiation.
(d) All have same speed
Under steady state the temperature of body:
(a) Increases with time
(b) Decreases with time
(c) Does not change with time and is same at all points of the body
(d) Does not change with time and can be different at different points of the body.
The SI unit of thermal conductivity is:
(a) J/s m 0C.
(b) Js/m0C
(c) J0C /s m
(d) Jm/s0C
The rate of flow of heat through a metal bar of area of cross section 1 m2 when
temperature gradient is unit under steady state is:
(a) Thermal resistance
(b) Thermal conductivity.
(c) Diffusivity
(d) Resistivity
If temperature difference between the two sides of a wall is doubled, its thermal
conductivity:
(a) Remains unchanged.
(b) Is halved
(c) Is doubled
(d) Becomes four times
In order to conduct the heat from one part of a solid to another part, what is required?
(a) Uniform density
(b) Uniform temperature
(c) Temperature gradient.
(d) Density gradient
The coefficient of thermal conductivity depends upon:
(a) Temperature difference of two surfaces (b) Area of the plate
(c) Thickness of plate
(d) Material of the plate.
In variable state, the rate of flow of heat is controlled by:
(a) Density of material
(b) Thermal conductivity
(c) Specific heat
(d) All of the above
Heat is transmitted from higher to lower temperature through molecular collision in:
(a) Conduction.
(b) Convection
(d) Radiation
(d) Heating
On heating one end of a rod, the temperature of whole rod will be uniform when:
[XI/PHY/RSG/THERMAL EXPANSSION]
Page 15
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
(a) K = 0
(b) K = 1
(c) K = 100
(d) K = ∞.
While measuring the thermal conductivity of a liquid we keep the upper part hot and
lower part cool, so that:
(a) Convection may be stopped.
(b) Heat conduction is easier downward
(c) Radiation may be stopped
(d) It is easier and more convenient to do so
For proper ventilation of building, windows must be open near the bottom and top of
the walls so as to let pass:
(a) In more air
(b) In cool air near the bottom and hot air out near the roof.
(c) In hot air near the roof and cool air out near the bottom
(d) Out hot air near the roof
In which process the rate of transfer of heat is maximum?
(a) Conduction
(b) Convection
(c) Radiation.
(d) In all same speed
Heat travels through vacuum by:
(a) Conduction
(b) Convection
(c) Radiation.
(d) Both (a) and (b)
Two bars of copper having same length but unequal diameter are heated to the same
temperature. The change in length will be:
(a) More in thinner bar
(b) More in thicker bar
(c) Same in both bars.
(d) Depends on ration of length to diameter
Coefficient of cubical expansion is negative between 00C and:
(a) 00C
(b) 40C.
(c) 15.50C
(d) 1000C
If temperature difference on the two sides of a wall increases from 100C to 200C, its
thermal conductivity:
(a) Remains unchanged.
(b) Is halved
(c) Is doubled
(d) Becomes four times.
A liquid having coefficient of cubical expansion γ is filled in the container having
coefficient of linear expansion α. If on heating the liquid overflows, then which of the
following relation is correct?
(a) γ = 2α
(b) γ > 3 α.
(c) γ < 3α
(d) 2γ = 3α
In heat transfer which method is based on gravitation?
(a) Natural convection.
(b) Radiation
(c) Conduction
(d) Stirring of liquids
Real expansion the liquid does not depend upon:
(a) Nature of liquid
(b) Temperature gradient
(c) Initial volume of the liquid
(d) Pressure exerted by liquid on container
The value of thermal conductivity is higher in case of:
(a) Insulator
(b) Good conductor. (c) Thick rod
(d) Thin body
The volume of a block of metal changes by 0.12% when it is heated through 200C. The
coefficient of linear expansion of the metal is:
(a) 2 x 10-5 /0C.
(b) 4 x 10-5 /0C
(c) 6 x 10-5 /0C
(d) 8 x 10-5/0C
The coefficient of linear expansion of iron is 1.1 x 10-5 /K. An iron rod is 10 m at 270C.
The length of the rod will be decreased by 1.1 mm when the temperature of the rod
changes to:
(a) 00C
(b) 170C.
(c) 100C
(d) 200C
[XI/PHY/RSG/THERMAL EXPANSSION]
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28. Two rods of lengths l1 and l2 are made of material whose coefficient of linear expansion
are α1 and α2 resp. The difference between their lengths will be independent of
temperature if l1 / l2 is equal to:
(a) α1 / α2
(b) α2 / α1 .
(c) (α1 / α2)1/2
(d) (α2 /α1)1/2
29. The length of an iron rail is 100 m at 200C. Its length at 400C is: (α = 1.2 x 10-5 /C)
(a) 0.024 m
(b) 200 m
(c) 99.98 m
(d) 100.02 m.
30. Four rods of the same material but of different radii (r) and length (l) are used to
connect two reservoirs of heat at different temperatures. The one which will conduct
most heat is:
(a) r = 2 cm, l = 0.5 m.
(b) r = 1 cm, l = 1 m
(c) r = 2 cm, l = 2 m
(d) r = 0.5 cm, l = 0.5 m
31. An iron plate has circular hole of diameter 10 cm. The diameter of the hole, when the
plate is uniformly heated from 100C to 900C is (α = 1.2 x 10-5 /C):
(a) 8.0096 cm
(b) 10.0096 cm.
(c) 9.0096 cm
(d) 11.0096 cm
0
0
32. A rectangular block is heated from 0 C to 100 C. The percentage increase in its length
is 0.1 %. What will be the percentage increase in its volume?
(a) 0.03 %
(b) 0.3 %.
(c) 0.10 %
(d) 3 %
33. Two rods of the same material have diameters in the ratio 1 : 2 and lengths in the ratio 2
: 1. If the temperature difference between their ends is the same, the ratio of heats
conducted by them in a given time is:
(a) 1 : 4
(b) 1 : 8.
(c) 4 : 1
(d) 8 : 1
34. One end of a metal rod of length 1 m and area of cross-section 100 cm2 is maintained at
1000C. If the other end of the rod is maintained at 00C, the quantity of heat transmitted
through the rod per minute will be: (K for material of rod = 100 W/m K)
(a) 3 x 103 J
(b) 9 x 103 J
(c) 6 x 103 J.
(d) 12 x 103 J
35. Two vessels of different materials are similar in size in every respect. The same quantity
of the ice filled in them gets melted in 20 minutes and 30 minutes. The ratio of their
coefficient of thermal conductivities will be:
(a) 2 / 3
(b) 1.5 .
(c) 1
(d) 4
36. The thickness of a metallic plate is 0.4 cm. The temperature between its two surfaces is
200C. The quantity of heat flowing per second is 50 cal from 5 cm2 area. In CGS
system, the coefficient of thermal conductivity will be:
(a) 0.2.
(b) 0.5
(c) 0.4
(d) 0.6
37. Two identical vessels are filled with equal amount of ice. The vessels are made from
different materials. If the ice melts in two vessels in times t1 and t2 resp. then their
coefficients of thermal conductivities are in the ratio:
(a) t2 : t1.
(b) t22 : t12
(c) t1 : t2
(d) t12 : t22
38. Wire A and B have identical lengths and have circular cross-sections. The radius of A is
twice the radius of B. For given temperatures difference between the two end, both
wires conduct heat at the same rate. The relation between the coefficients of thermal
conductivities is given by:
(a) KA = 4 KB
(b) KA = 2 KB
(c) KA = KB / 2
(d) KA = KB / 4.
[XI/PHY/RSG/THERMAL EXPANSSION]
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39. If the coefficient of thermal conductivity of aluminum is 0.5 cal/cm s 0C, then in order
to conduct 10 cal/scm2 in the steady state, the temperature gradient in aluminium must
be:
(a) 50C/cm
(b) 10.50C/cm
(c) 10 0C/cm
(d) 20 0C / cm.
40. A metallic bar is heated from 00C to 1000C. The coefficient of linear expansion is 10-5 /
K. What will be the percentage increase in length?
(a) 0.01 %
(b) 1 %
(c) 0.1 %.
(d) 10 %
0
41. A rod of 40 cm in length and temperature difference of 80 C at its two ends. Another
rod B of length 60 cm and of temperature difference 900C, having the same area of
cross-section. IF the rate of flow of heat is the same, then the ratio of their coefficient of
thermal conductivities will be:
(a) 1 : 2
(b) 3 : 4.
(c) 2 : 1
(d) 4 : 3
42. A brass rod of length 50 cm and diameter 3 mm is joined to a steel rod of the same
length and diameter at 400C. What is the change in length of the combined rod at
2500C? (αBrass = 2 x 10-5 / 0C; αSteel = 1.2 x 10-5 /0C)
(a) 0.0034 .
(b) 0.34
(c) 0.034
(d) 0.43
0
43. What should be the lengths of a steel and copper rod at 0 C, so that the steel rod is 5 cm
longer than the copper rod at all temperature?
(a) Ls = 14.17 cm, Lc = 9.17 cm.
(b) Ls = 19.17 cm, Lc = 13.17 cm
(c) Ls = 19.17 cm, Lc = 14.17 cm
(d) Ls = 13.17 cm, Lc = 18.17 cm
0
44. A steel rod of length 4 cm at 0 C is heated to 1000C. The length of the rod at 1000C is:
(a) 3.0048 m
(b) 5.0048 m
(c) 4.0048 m.
(d) 6.0048 m
0
3
45. The volume of a certain mass of liquid at 0 C is 10 cm and that at 1000C is 10.25 cm3.
The coefficient of cubical expansion of the liquid is:
(a) 1.1 x 10-4 /0C
(b) 2 x 10-4 /0C
(c) 1.7 x 10-4 0C
(d) 2.5 x 10-4 /0C.
46. The superficial expansivity is 1/x times the cubical expansivity. The value of x is:
(a) 2/3
(b) 2
(c) 3/2 .
(d) 3
-5 0
47. A metal rod having linear coefficient of expansion 2 x 10 / C and has length 1 m at
200C, the temperature at which it is shortened by 1 mm is:
(a) 300C
(b) – 300C .
(c) 500C
(d) – 500C
48. Solids expand on heating because:
(a) The K.E. of atoms increases
(b) P.E. of atom increases
(c) T.E. of atoms increases
(d) Interatomic separation increases.
49. Sun’s heat reaches the earth by:
(a) Conduction
(b) Convection
(c) Radiation.
(d) Scattering
0
50. The increase in length per unit original length at 0 C per degree rise in temperature is:
(a) Coefficient of linear expansion.
(b) Coefficient of superficial expansion
(c) Coefficient of cubical expansion
(d) Coefficient expansion of the solid
51. What should be the gap between two adjacent rails, each of length 2 m at a place where
the temperature varies from 100C to 480C? (α = 1.1 x 10-5 /0C)
(a) 4.8 mm
(b) 8.4 mm.
(c) 2.8 mm
(d) 6.4 mm
[XI/PHY/RSG/THERMAL EXPANSSION]
Page 18
52. A rod of length 3 m increases in length by 0.9 mm, when the temperature changes from
200C to 800C. The increase in length for rod of the same material and heated through the
same temperature difference would be 1.2 mm if the length of the rod at 200C is:
(a) 2 m
(b) 6 m
(c) 8 m
(d) 4 m.
2
0
0
53. A disc has an area of 0.32 m at 20 C, what will be its area at 100 C? (α = 2 x 10-6 /0C):
(a) 0.12 m2
(b) 0.51 m2
(c) 0.32 m2.
(d) 0.71 m2
54. A brass rod (α = 2 x 10-5 /K) is 0.70 m long at 400C. Its linear expansion when the
temperature rises to 500C is:
(a) 0.14 mm.
(b) 1.4 mm
(c) 0.7 mm
(d) 7 mm
55. The volume of metal block changes by 0.18% when it is heated through 200C then its
coefficient of cubical expansion will be:
(a) 9 x 10-5 /0C.
(b) 18 x 10-5/ 0C
(c) 3 x 10-5 /0C
(d) 36 x 10-5 /0C
56. Two metal rods has length in the ratio 3 : 2 and coefficients of linear expansions are in
the ratio 2 : 3. If they are heated from 350C to 950C then ratio of their linear expansions
is:
(a) 1 : 2
(b) 1 : 1.
(c) 2 : 1
(d) 1 : 4
-5
57. The coefficient of linear expansion of iron rod is 1.1 x 10 /K. An iron rod is 10 m long
at 270C. the length of the rod will be decreased by 1.1 mm when the temperature of the
rod changes to:
(a) 00C
(b) 170C.
(c) 100C
(d) 200C
58. In a lake when it cools to the point where it is about to freeze at 40C water settles to the
bottom because it is:
(a) Less dense
(b) Very cold
(c) More dense.
(d) Very hot
59. An iron ball is heated. Then percentage increase will be largest in:
(a) Diameter
(b) Volume.
(c) Surface area
(d) Density
60. The length of an aluminum rod is 2 m at room temperature. When the rod is heated to
1250C, its length increases by 0.5 cm. What is the room temperature if the coefficient of
linear expansion of aluminum is 25 x 10-6 /0C?
(a) 250C.
(b) 150C
(c) 500C
(D) 50C
61. One end of a copper rod is in contact with water at 1000C and the other end in contact
with ice at 00C . The length of the rod is 100 cm. At a point which is at a distance of 35
cm from the cold end temperature is (Assuming steady state heat flow):
(a) 350C
(b) 560C
(c) 650C.
(d) 530C
62. If the real and apparent cubical expansivity of glycerin are 4.9 x 10-4 W/m K and 4.63 x
10-4 W/mK, the linear expansivity of glass is:
(a) 9 x 10-5 W/m K (b) 0.9 x 10-5 W/m K. (c) 0.7 x 10-5 W/m K (d) 7 x 10-4 W/mK
63. Iron sheet 50 cm x 20 cm is heated through 1000C. If α = 12 x 10-6 /0C the change in
area is:
(a) 2.4 cm2.
(b) 4.2 cm2
(c) 3.4 cm2
(d) 4.3 cm2
64. The amount of heat flowing in 10 seconds through a copper rod of length 50 cm and
area 15 cm2 when the ends are at 1000C and 00C is (K = 380 W/mK):
(a) 1140 J.
(b) 4011 J
(c) 1260 J
(d) 1410 J
[XI/PHY/RSG/THERMAL EXPANSSION]
Page 19
65. If α, β and γ are coefficients of linear, superficial and volume expansions of solid then:
(a) α : β : γ = 1 : 2 : 3.
(b) α : β : γ = 2 : 3 : 1
(c) α : β : γ = 3 : 2 : 1
(d) α : β : γ: = 3 : 1 : 2
66. Liquid filled in flask up to 2/3 of its capacity. When it is heated the level of liquid:
(a) Initially increased and then decreases (b) Initially decreases and then increases.
(c) Increases abruptly
(d) Changes slowly
67. A clock with iron pendulum keeps correct time at 200C. If γ of iron is 36 x 10-6 /0C and
temperature of the room is 400C, the clock will lose or gain:
(a) 10.368 s per day. (b) 20.736 s per day (c) 5.368 s per day
(d) 31.104 s per day
68. How much heat will be conducted in one hour through a layer of ice 5 cm thick,
covering a pool of area 20 m2, if the water below is at 00C and the air above is at –
100C? (K for ice = 0.005 cal/cm s K)
(a) 72 kcal
(b) 7200 kcal
(c) 720 kcal.
(d) 2700 kcal
3
0
3
0
69. The volume of liquid is 830 m at 30 C and 850 m at 90 C. The coefficient of volume
expansion of liquid is:
(a) 2 x 10-4 /0C
(b) 4 x 10-4 / 0C .
(c) 8 x 10-4 /0C
(d) 2.5 x 10-4 /0C
70. The density of a liquid at 00C is 13.6 g/cm3 & its coefficient of real expansion is 18 x
10-5 / 0C. If the temperature is increased to 2000C, the percentage change in ts density is:
(a) 0.36%
(b) 18%
(c) 36%.
(d) 54%
71. A copper wire of length l increases in length by 0.2% on heating from 20 0C to 400C.
Then percentage change in area of copper plate of dimensions 31 x 21 on heating from
200C to 400C is:
(a) 0.1 %
(b) 0.2 %.
(c) 0.05 %
(d) 0.6%
0
72. What would be length of steel and copper rods at 0 C so that the length of steel rod is 5
cm longer than copper rod at all temperatures? (α for copper = 1.7 x 10-5 /0C):
(a) 7.14 cm, 8.42 cm
(b) 5.14 cm, 10.14 cm
(c) 12.14 cm, 7.14 cm.
(d) 10.14 cm, 5 cm
* * * * *
[XI/PHY/RSG/THERMAL EXPANSSION]
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