17Thermal expansion

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THERMAL EXPANSION
▢ Introduction:
❖ Definition: Increase in length, area or volume of body on heating
❖ General Rule:
  • •
    Usually body expands on heating
  • •
    Usually body contracts on cooling
❖ Cause: Increase in intermolecular separation against intermolecular force
❖ Order: Solids < Liquids < Gases
❖ Reason: Intermolecular force: maximum in solids, less in liquids, least in gases

Table 1: Types of Expansion

Substance
Expansion
Solids
Linear, superficial, cubical
Liquids
Only volume expansion
Gases
Only volume expansion
▢ Expansion of Solids:

Table 1: Linear, Superficial and Cubical Expansion

Feature
Linear expansion
Superficial expansion
Cubical / volume expansion
Meaning
Increase in length
Increase in area
Increase in volume
Dimension
1-D
2-D
3-D
Final value
\(L_2=L_1(1+\alpha\Delta\theta)\)
\(A_2=A_1(1+\beta\Delta\theta)\)
\(V_2=V_1(1+\gamma\Delta\theta)\)
Coefficient
\(\alpha\) = coefficient of linear expansion
\(\beta\) = coefficient of superficial expansion
\(\gamma\) = coefficient of cubical expansion
Change
\(\Delta L=L_1\alpha\Delta\theta\)
\(\Delta A=A_1\beta\Delta\theta\)
\(\Delta V=V_1\gamma\Delta\theta\)
% change
\(\frac{\Delta L}{L_1}\times100=\alpha\Delta\theta\times100\)
\(\frac{\Delta A}{A_1}\times100=\beta\Delta\theta\times100\)
\(\frac{\Delta V}{V_1}\times100=\gamma\Delta\theta\times100\)
❖ Important Points:
  • •
    \(\alpha,\beta,\gamma\) are independent of original dimensions
  • •
    \(\alpha,\beta,\gamma\) depend on nature of material
  • •
    For perfectly rigid body: \(\alpha=\beta=\gamma=0\)
▢ Relation Between Coefficients:

Table 1: Isotropic vs Anisotropic Body

Feature
Isotropic body
Anisotropic body
Linear expansion
\(\alpha_1=\alpha_2=\alpha_3=\alpha\)
\(\alpha_1\neq\alpha_2\neq\alpha_3\)
Cubical expansion
\(\gamma=\alpha+\alpha+\alpha=3\alpha\)
\(\gamma=\alpha_1+\alpha_2+\alpha_3\)
Superficial expansion
\(\beta=\alpha+\alpha=2\alpha\)
\(\beta=\alpha_1+\alpha_2\) or \(\alpha_2+\alpha_3\) or \(\alpha_3+\alpha_1\)
Relation
\(\gamma=\frac{3}{2}\beta\)
Depends on direction
Ratio
\(\alpha:\beta:\gamma=1:2:3\)
No fixed ratio
❖ Special Rod Condition: If two rods of lengths \(l_1,l_2\) and expansivities \(\alpha_1,\alpha_2\) have same difference in length at all temperatures, then \(\alpha_1l_1=\alpha_2l_2\)
▢ Effect of Temperature on Simple Pendulum:
❖ Basic Formula: \(T=2\pi\sqrt{\frac{l}{g}}\)
❖ Relation: \(T\propto\sqrt l\)

Table 1: Temperature Correction in Pendulum

Quantity
Formula
Fractional change in period
\(\frac{\Delta T}{T}=\frac{1}{2}\frac{\Delta l}{l}\)
Using linear expansivity
\(\frac{\Delta T}{T}=\frac{1}{2}\alpha\Delta\theta\)
% change in time period
\(\frac{\Delta T}{T}\times100=\frac{1}{2}\alpha\Delta\theta\times100\)
Change in time period
\(\Delta T=\frac{1}{2}\alpha\Delta\theta\times T\)
❖ Clock Effect:
  • •
    Temperature increases → length increases → time period increases → clock runs slow / loses time
  • •
    Temperature decreases → length decreases → time period decreases → clock runs fast / gains time
  • •
    Summer → pendulum clock runs slower
  • •
    Winter → pendulum clock runs faster
▢ Expansion of Liquids:
❖ Key Point: Liquid is heated in a vessel; vessel also expands
❖ Types of Expansion:

Table 1: Real and Apparent Expansion of Liquid

Type
Meaning
Formula
Real expansion
Actual change in volume of liquid only
\(\Delta V=V_0\gamma_r\Delta\theta\)
Apparent expansion
Observed change considering expansion of vessel
\(\Delta V=V_0\gamma_a\Delta\theta\)
Relation
Real expansion = apparent expansion + expansion of vessel
\(\gamma_r=\gamma_a+\gamma_s\)
❖ Density Change:
◉ Formula: \(\rho=\frac{\rho_0}{1+\gamma\Delta\theta}\approx\rho_0(1-\gamma\Delta\theta)\)
◉ Point: On heating, volume increases and density decreases
▢ Correction of Barometer Reading:
❖ Condition: Mercury column height \(h\) at temperature \(\theta\)
❖ Formula: \(h_0=h[1-(\gamma-\alpha)\theta]\)
❖ Symbols:
  • •
    \(h_0\) = corrected height
  • •
    \(h\) = observed height
  • •
    \(\alpha\) = linear expansivity of scale material
  • •
    \(\gamma\) = cubical expansivity of mercury
▢ Anomalous Expansion of Water:
❖ Definition: Peculiar behaviour of water in which it contracts on heating from 0°C to 4°C and expands beyond 4°C

Table 1: Water Expansion

Temperature range
Behaviour
0°C to 4°C
Water contracts on heating
At 4°C
Volume minimum; density maximum
Above 4°C
Water expands on heating
Density at 4°C
\(1\ g/cc\)
❖ Importance: Aquatic life survives in cold regions because water at bottom of lakes/ponds remains liquid near 4°C even when surface freezes
❖ Coefficient of Cubical Expansion of Water:

Table 1: Cubical Expansivity of Water

Range
\(\gamma\)
Below 4°C
Negative
At 4°C
Zero
Above 4°C
Positive
▢ Bimetallic Strip:
❖ Definition: Two different metal strips joined together to form a single strip
❖ Uses:
  • •
    Thermometer
  • •
    Thermostat
  • •
    Automatic breaking of electric circuits
  • •
    Fire alarm

Table 1: Bimetallic Strip Bending

Condition
Direction of bending
Concave side
Convex side
Heated
Bends towards metal with smaller \(\alpha\)
Smaller \(\alpha\)
Larger \(\alpha\)
Cooled
Bends towards metal with larger \(\alpha\)
Larger \(\alpha\)
Smaller \(\alpha\)
❖ Example: If \(\alpha_A>\alpha_B\), on heating strip bends towards B
▢ Expansion of Gases:
❖ Key Points:
  • •
    Gases have only volume expansion
  • •
    Coefficient of expansion of gases is greater than liquids and solids
  • •
    Volume expansivity is defined at constant pressure
  • •
    Pressure coefficient is defined at constant volume

Table 1: Expansion Coefficients of Gas

Coefficient
Definition
Formula
Value
Volume expansivity
Fractional increase in volume per unit rise of temperature at constant pressure
\(\gamma_v=\left(\frac{\Delta V}{V_0\Delta\theta}\right)_P\)
\(\frac{1}{273}\ ^\circ C^{-1}\)
Pressure coefficient
Fractional increase in pressure per unit rise of temperature at constant volume
\(\gamma_p=\left(\frac{\Delta P}{P_0\Delta\theta}\right)_V\)
\(\frac{1}{273}\ ^\circ C^{-1}\)
Relation
For ideal gas
\(\gamma_v=\gamma_p\)
\(\frac{1}{273}\ ^\circ C^{-1}\)
▢ Thermal Stress in Rigidly Fixed Rod:
❖ Condition: Metallic rod of length \(L\), area \(A\), linear expansivity \(\alpha\), Young's modulus \(Y\), clamped between rigid supports and heated by \(\Delta\theta\)

Table 1: Thermal Stress Formulae

Quantity
Formula
Thermal strain
\(\frac{\Delta L}{L}=\alpha\Delta\theta\)
Thermal stress
\(Y\alpha\Delta\theta\)
Thermal force
\(F=AY\alpha\Delta\theta\)
Thermal strain energy
\(U=\frac{1}{2}YAL\alpha^2\Delta\theta^2\)
Strain energy per unit volume
\(\frac{U}{V}=\frac{1}{2}Y\alpha^2\Delta\theta^2\)
▢ Heating of Holes and Cavities:

Table 1: Expansion of Holes

Case
Result
Metal disc with hole heated
Size of hole increases
Solid ball with concentric spherical cavity heated
Volume of cavity increases
Reason
Hole/cavity expands as if filled with same material
▢ Liquid in Container:
❖ Condition: Liquid with cubical expansivity \(\gamma\) in container of material with linear expansivity \(\alpha\)

Table 1: Level of Liquid on Heating

Condition
Result
\(\gamma>3\alpha\)
Liquid level rises / overflows if completely filled
\(\gamma=3\alpha\)
Liquid level remains stationary
\(\gamma<3\alpha\)
Liquid level falls
❖ Initial Fall Then Rise: If mercury is heated in vessel, liquid column first descends due to vessel expansion and then ascends due to mercury expansion
▢ Apparent Weight with Temperature:

Table 1: Apparent Weight in Liquid

Condition
Formula / Result
Solid cubical expansivity
\(\gamma_s\)
Liquid cubical expansivity
\(\gamma_l\)
Temperature change
\(\Delta\theta=\theta_2-\theta_1\)
Change in apparent weight
\(\Delta W=W_2-W_1=W(\gamma_l-\gamma_s)\Delta\theta\)
\(\gamma_l>\gamma_s\)
\(W_2>W_1\)
\(\gamma_l=\gamma_s\)
\(W_2=W_1\)
\(\gamma_l<\gamma_s\)
\(W_2
❖ Note: Cases above are for heating; converse is true for cooling
❖ General Point: A metallic piece weighed in a liquid whose temperature is raised continuously has increasing apparent weight
▢ Empty Space in Measuring Cylinder:
❖ Condition: Measuring cylinder capacity \(V_0\), cubical expansivity \(\gamma_0\), contains liquid volume \(V\), cubical expansivity \(\gamma\)
❖ Formula: \(\gamma_0V_0=\gamma V\)
❖ Meaning: Condition for empty space to remain unchanged on heating or cooling
▢ Invar:
❖ Property: Very low coefficient of linear expansion
❖ Uses:
  • •
    Metal scales
  • •
    Pendulum clocks
▢ Read and Digest:

Table 1: Important Thermal Expansion Points

Fact
Answer
Water heated from 0°C to 10°C
Volume first decreases then increases
Temperature of liquid increases
Volume increases; density decreases
Copper sphere heated
% increase maximum for volume
Reason
\(\gamma=3\alpha\), \(\beta=2\alpha\)
Expansion on heating
Density decreases
Water density
Maximum at 4°C
Water volume
Minimum at 4°C
Water from 0°C to 4°C
\(C_P
Water at 4°C
\(C_P=C_V\)
Beaker completely filled with water at 4°C
Overflows on heating as well as cooling
Glass jar full of water in freezing mixture
Breaks because water expands from 4°C to 0°C
Two rods same length difference at all temperature
\(L_1\alpha_1=L_2\alpha_2\)
Water bottle at 0°C opened on Moon
Water boils because pressure is nearly zero
Identical hollow and solid spheres heated by equal heat
Hollow sphere expands more due to less mass
Metal disc with hole heated
Hole increases
Spherical cavity heated
Cavity volume increases
Fire alarm
Works on bending of bimetallic strip
Metal floating in mercury heated
Floats at lower level
Same volume of benzene
Weighs less in summer than winter
Same mass of benzene
Occupies more volume in summer than winter
50 g benzene
Weighs same in summer and winter
\(\alpha,\beta,\gamma\)
Depend on unit of temperature
Invar
Used in metal scales and pendulum clocks
▢ High-Yield Recall:

Table 1: Thermal Expansion One-Liners

Fact
Answer
Thermal expansion
Increase in dimensions on heating
Cause
Increase in intermolecular separation
Expansion order
Solid < Liquid < Gas
Solids expansion
Linear, superficial, cubical
Liquids and gases
Only volume expansion
Linear expansion
\(L_2=L_1(1+\alpha\Delta\theta)\)
Superficial expansion
\(A_2=A_1(1+\beta\Delta\theta)\)
Cubical expansion
\(V_2=V_1(1+\gamma\Delta\theta)\)
Isotropic relation
\(\alpha:\beta:\gamma=1:2:3\)
Cubical coefficient
\(\gamma=3\alpha\)
Superficial coefficient
\(\beta=2\alpha\)
Pendulum temperature correction
\(\frac{\Delta T}{T}=\frac{1}{2}\alpha\Delta\theta\)
Summer pendulum clock
Runs slow
Winter pendulum clock
Runs fast
Real expansion of liquid
\(\Delta V=V_0\gamma_r\Delta\theta\)
Apparent expansion of liquid
\(\Delta V=V_0\gamma_a\Delta\theta\)
Real-apparent relation
\(\gamma_r=\gamma_a+\gamma_s\)
Density on heating
\(\rho\approx\rho_0(1-\gamma\Delta\theta)\)
Barometer correction
\(h_0=h[1-(\gamma-\alpha)\theta]\)
Anomalous expansion of water
Contracts from 0°C to 4°C; expands above 4°C
Maximum density of water
At 4°C
Minimum volume of water
At 4°C
Bimetallic strip heated
Bends towards smaller \(\alpha\)
Bimetallic strip cooled
Bends towards larger \(\alpha\)
Gas expansivity
\(\gamma_v=\gamma_p=\frac{1}{273}\ ^\circ C^{-1}\)
Thermal stress
\(Y\alpha\Delta\theta\)
Thermal force
\(AY\alpha\Delta\theta\)
Thermal strain energy
\(\frac{1}{2}YAL\alpha^2\Delta\theta^2\)
Hole in metal disc on heating
Increases
Invar use
Metal scales and pendulum clocks
Q1.
A liquid is placed in a graduated glass cylinder. The coefficient of real expansion of the liquid is thrice the coefficient of linear expansion of glass. On heating, the level of liquid will
📅IOM 2013
Q2.
The pendulum of a clock is made of brass. If the clock keeps correct time at 20°C, how many seconds per day will it lose at 35°C? Given αbrass = 2 × 10^-5 °C^-1
📅IOM 2010
Q3.
C and S are coefficients of apparent expansion of a liquid in copper and silver vessels respectively. If coefficient of linear expansion of copper is A, then coefficient of linear expansion of silver is
📅IOM 2009
Q4.
A thin copper wire of length L at 0°C is heated to t°C and its length increases by 0.1%. If a thin copper plate of dimensions L × 2L is heated through the same range, its surface area will increase by
📅IE 2010
Q5.
When a liquid kept in a copper vessel has apparent expansion 6 × 10^-6 °C^-1, and in a steel vessel has apparent expansion 24 × 10^-6 °C^-1. If αcopper = 18 × 10^-6 °C^-1, find αsteel.
📅IE 2013
Q6.
The resistance of a conductor is 15 Ω at 60°C and 20 Ω at 100°C. The resistance at 10°C is
📅MOE 2012
Q7.
The resistance of a material at 20°C is 1.72 Ω. At 100°C its resistance becomes nearly, given α = 0.00393 K^-1
📅KU 2010
Q8.
A zinc rod has length 1 m at 0°C. Find its length at 50°C. Given αzinc = 26 × 10^-6 °C^-1
📅MOE 2014
Q9.
Two rods of lengths L1 and L2 and linear expansivities α1 and α2 have the same difference in length at any temperature. Then
📅MOE 2014
Q10.
A brass rod and lead rod are each 80 cm long at 0°C and clamped together at one end. If heated in steam bath, the difference in their lengths is nearly
Q11.
Two spheres, one solid and one hollow, are made of the same material and same radius. If both are heated to the same temperature, expansion will be more in
Q12.
Two spheres, one solid and one hollow, are made of the same material and same radius. If the same heat is given to both spheres, expansion will be more in
Q13.
A disc has a hole of diameter 1.5 cm at 20°C. If it is heated to 150°C and α = 1.9 × 10^-5 °C^-1, the hole diameter will
Q14.
A uniform rod is heated from 0°C to 20°C. If α = 12 × 10^-6 °C^-1 and Young’s modulus = 1 × 10^11 N/m², energy stored per unit volume is
Q15.
The coefficients of cubical expansion of brass and iron are 54 × 10^-6 °C^-1 and 36 × 10^-6 °C^-1 respectively. If brass and iron rods show same difference of length at all temperatures, their lengths are in the ratio
Q16.
On heating a liquid of cubical expansion γ in a container of linear expansion α, the level of liquid will
Q17.
An iron cube floats in mercury at 20°C. If temperature is increased by 100°C, the cube will float
Q18.
The real coefficient of volume expansion of glycerin is 0.000597 °C^-1 and linear coefficient of glass is 0.000009 °C^-1. Apparent coefficient of glycerin in glass is
Q19.
Steel rails 40 m long are laid at -10°C. The gap to allow expansion up to 40°C is, if α = 12 × 10^-6 °C^-1
Q20.
A steel tape gives correct measurement at 20°C. A piece of wood measured with it at 0°C reads 25 cm. The real length of wood is
Q21.
Density of mercury is 13.6 × 10^3 kg/m³ at 0°C. If real expansion of mercury is 18.0 × 10^-5 °C^-1, its density at 50°C is
Q22.
A thin copper wire increases in length by 1% when heated from T1 to T2. What is the percentage change in area of a thin copper plate heated through the same range?
Q23.
The moment of inertia of a body is I and coefficient of linear expansion is α. If temperature rises by small amount Δθ, the change in moment of inertia is nearly
Q24.
An iron tyre is to be fitted onto a wooden wheel 1.0 m in diameter. The tyre diameter is 6 mm smaller. If cubical expansion of iron is 3.6 × 10^-5 °C^-1, the tyre should be heated by
Q25.
Coefficient of linear expansion is α per degree Celsius. If temperature is measured on Fahrenheit scale, coefficient of expansion will be
Q26.
A glass vessel just holds 50 g of a liquid at 0°C. If coefficient of linear expansion of glass is 8 × 10^-6 °C^-1, the mass it holds at 80°C is approximately
Q27.
A brass disc fits tightly in a hole in a steel plate. To loosen the disc from the hole, we should
Q28.
A metal ball immersed in alcohol weighs W1 at 0°C and W2 at 50°C. If cubical expansion of metal is less than that of alcohol, then
Q29.
A metallic piece is weighed in a liquid whose temperature is raised continuously. The apparent weight of the metallic piece generally
Q30.
A liquid with coefficient of volume expansion γ is filled in a container with coefficient of linear expansion α. If liquid overflows on heating, then
Q31.
The loss in weight of a solid when immersed in a liquid at 0°C is W0 and at t°C is Wt. If cubical expansion coefficients of solid and liquid are γs and γl, then Wt is
Q32.
A glass flask of volume 1 litre is completely filled with mercury at 0°C. γHg = 1.82 × 10^-4 °C^-1 and αglass = 10 × 10^-6 °C^-1. On heating to 100°C, mercury spilled is
Q33.
A clock keeps correct time at 20°C. Its metallic pendulum has α = 2 × 10^-5 °C^-1. If temperature falls to 10°C, the clock will
Q34.
A pendulum clock gains 5 seconds per day at 16°C and loses 15 seconds per day at 40°C. It keeps correct time at
Q35.
If a bimetallic strip is heated, it will
Q36.
Density of a liquid decreases by 0.1%. If temperature increase is 100°C, the linear expansivity is
📅IOM 1997
Q37.
Two rods A and B have same length. Linear expansivity of A is 12 × 10^-6 K^-1 and cubical expansivity of B is 3 × 10^-5 K^-1. If both are heated to 80°C, rod A will be
📅MOE 2058
Q38.
What happens when water at 4°C is heated further?
📅IE 2006
Q39.
When water is heated from 0°C to 100°C, its volume
Q40.
A vessel of volume V and linear coefficient of expansion α contains a liquid. The level of liquid does not change on heating. The volume coefficient of real expansion of the liquid is
📅KU 2015