Thermal and Mechanical Properties of Matter: Key Concepts, Formulas and Applications

When a substance is heated, cooled or subjected to an external force, its physical properties can change. It may expand, contract, transfer heat or experience stress. Understanding these changes helps students solve many questions in Physics, especially in competitive exams.

In this topic, you will learn about thermal expansion, calorimetry, heat transfer, thermal radiation and mechanical effects of temperature changes. Important laws and formulas, along with their applications, can also make numerical problems easier to understand and solve.

From the expansion of a metal rod to the formation of ice on a lake, these concepts explain many situations that we see in everyday life as well as in Physics problems.

Thermal Expansion

Thermal expansion describes the tendency of matter to change in volume in response to a change in temperature.

1. Linear Expansion

When a rod of initial length L₀ is heated by ΔT, its length increases by ΔL.

• ΔL = α L₀ ΔT
  Where α is the coefficient of linear expansion.

• The final length (L_final) is L_final = L₀ (1 + α ΔT).

2. Area Expansion (Superficial Expansion)

For an object with initial area A₀ heated by ΔT, the change in area ΔA is:

• ΔA = β A₀ ΔT
  Where β is the coefficient of area expansion.

• The final area (A_final) is A_final = A₀ (1 + β ΔT).

• For an isotropic material (uniform expansion in all directions), β = 2α.

3. Volume Expansion (Cubical Expansion)

For a substance with initial volume V₀ heated by ΔT, the change in volume ΔV is:

• ΔV = γ V₀ ΔT
  Where γ is the coefficient of volume expansion.

• The final volume (V_final) is V_final = V₀ (1 + γ ΔT).

• For an isotropic material, γ = 3α.

Relationship between α, β, and γ

For an isotropic material, the ratio of coefficients is α : β : γ = 1 : 2 : 3.

Variation of Density with Temperature

When temperature changes, mass remains constant but volume changes, affecting density.

• ρ_final = ρ₀ / (1 + γ ΔT)

• Using binomial approximation (for small γ ΔT): ρ_final ≈ ρ₀ (1 - γ ΔT).

Expansion of Holes in a Sheet

When a sheet with a hole is heated, all dimensions expand proportionally, including the hole, as if it were made of the same material.

Applications of Thermal Expansion

1. Condition for Constant Length Difference of Two Rods

For two rods with initial lengths L₁, L₂ and coefficients α₁, α₂, their length difference remains constant if L₁ α₁ = L₂ α₂.

2. Thermal Stress and Force by Walls on a Rod

When a rod is fixed between rigid walls and heated, it attempts to expand. This causes thermal stress and a restoring force exerted by the walls.

• Hypothetical expansion: ΔL = L α ΔT.

• Force F exerted by walls: F = A Y α ΔT
  Where A is cross-sectional area, Y is Young's Modulus, α is coefficient of linear expansion, and ΔT is change in temperature.

• Thermal Stress = F/A = Y α ΔT.

• Thermal Strain = ΔL/L = α ΔT.
  (Memory Tip: To derive F = A Y α ΔT, combine ΔL = LαΔT with Y = (F/A)/(ΔL/L) by substituting ΔL/L = αΔT into the Young's Modulus expression.)

3. Effect of Temperature on a Simple Pendulum

The time period T = 2π √(L/g). If temperature increases, L increases, leading to an increase in T (pendulum swings slower, clock loses time).

• Time Lost per Second = (1/2) α ΔT_temp.

4. Effect of Temperature on Moment of Inertia

For an object like a rod, its moment of inertia I changes with length.

• ΔI / I = 2 α ΔT_temp

• Percentage Increase in Moment of Inertia = 2 α ΔT_temp × 100%.

5. Expansion of Liquid in a Beaker and Overflow

When a beaker with liquid is heated, both expand.

• If γ_liquid < γ_beaker, liquid level falls.

• If γ_liquid = γ_beaker, liquid level rises, but no overflow occurs.

• If γ_liquid > γ_beaker, liquid overflows.

• Overflow Volume = V₀ (γ_liquid - 3α_beaker) ΔT.

Temperature Scales

Temperature scales (Celsius, Kelvin, Fahrenheit) are interconvertible.

• C / 100 = (F - 32) / 180 = (K - 273.15) / 100

• Simplified: C / 5 = (F - 32) / 9 = (K - 273.15) / 5

• K = C + 273.15 (approximated as K = C + 273).

• F = (9/5)C + 32.
  (Memory Tip: Remembering ice and steam points for each scale helps derive conversions: Celsius (0°C, 100°C), Fahrenheit (32°F, 212°F), Kelvin (273.15 K, 373.15 K).)

Calorimetry

Calorimetry involves measuring heat transfer, guided by two fundamental formulas:

1. Heat Transfer without Phase Change

When a substance changes temperature but not phase:

• ΔQ = m s ΔT
  Where m is mass, s is specific heat capacity, and ΔT is temperature change.

• Specific Heat Capacities to Remember:

• Water (s_water): 1 cal/g°C.

• Ice (s_ice): 0.5 cal/g°C.

• Steam (s_steam): 0.5 cal/g°C.

• Heat Capacity (C) = m s. So, ΔQ = C ΔT.

2. Heat Transfer during Phase Change

When a substance undergoes a phase change at a constant temperature:

• ΔQ = m L
  Where m is mass undergoing phase change, and L is latent heat.

• Latent Heats to Remember:

• Latent Heat of Fusion (L_f) for ice to water: 80 cal/g.

• Latent Heat of Vaporization (L_v) for water to steam: 540 cal/g.
  (Memory Tip: Phase changes occur at constant temperature, so ΔT is zero, making ΔQ = msΔT inappropriate. Use ΔQ = mL.)

Specific Heat as a Function of Temperature

If specific heat s varies with temperature (e.g., S = A + BT), total heat Q is found by integration:

• Q = ∫_(T₁)^(T₂) m (A + BT) dT = m [A(T₂ - T₁) + (1/2)B(T₂² - T₁²)].

Heat Exchange with Beaker and Water Equivalent

When heating a liquid in a container, heat is absorbed by both.

• ΔQ_total = (m_liquid * s_liquid * ΔT) + (m_beaker * s_beaker * ΔT).

• The product (m_beaker * s_beaker) is the Water Equivalent (W) of the beaker.

• W = m_beaker * s_beaker.

• Application and Limitation: Water Equivalent is useful when no phase change occurs. Do NOT use by direct addition of masses if a phase change is involved for the primary substance. (Memory Tip: Water Equivalent is the mass of water that would absorb the same heat as the beaker for the same temperature change.)

Principle of Calorimetry

For an isolated system, Heat Given = Heat Taken. This is a statement of the Law of Conservation of Energy for heat transfer.

• In mixing problems, hotter substances give heat, colder substances take heat.

• The final equilibrium temperature applies to all components of the mixture.

Predicting Final State in Ice-Water Mixtures

To predict the final state and temperature of an ice-water mixture:

1. Bring all components to 0°C water (a reference state).

2. Calculate heat released (by hot components to reach 0°C water) and heat absorbed (by cold components to reach 0°C water).

3. Calculate Net Heat = (Heat Released) - (Heat Absorbed).

4. If Net Heat > 0: All ice melts, and the mixture becomes water above 0°C.

5. If Net Heat < 0: Not enough heat to melt all ice; final state is ice, or ice and water at 0°C.

6. If Net Heat = 0: Final state is all water at 0°C.
   (Memory Tip: Think of heat released as 'money earned' and heat absorbed as 'debt'. A positive net means money left over to raise temperature.)

Thermal Radiation

Thermal radiation is heat transfer via electromagnetic waves, requiring no medium. All objects above 0 Kelvin emit radiation.

Stefan-Boltzmann Law

Quantifies the rate of heat radiated or absorbed.

• Rate of Heat Emitted (dq/dt_emitted) = e * σ * A * T⁴

• Rate of Heat Absorbed (dq/dt_absorbed) = e * σ * A * T_s⁴

• Net Rate of Heat Transfer (dq/dt_net) = e * σ * A * (T⁴ - T_s⁴)
  Where e is emissivity (1 for black body), σ is Stefan-Boltzmann constant, A is surface area, T is body temperature, and T_s is surrounding temperature (all in Kelvin).

Rate of Cooling (Rate of Fall of Temperature)

• dT/dt = − eσA(T⁴ − Tₛ⁴)/(mc)
  This formula is crucial for problems involving cooling. Always convert all temperatures from Celsius to Kelvin.

Definitions in Thermal Radiation

• Reflectivity (r): Fraction of incident radiation reflected.

• Transmissivity (t): Fraction of incident radiation transmitted.

• Absorptivity (a): Fraction of incident radiation absorbed.

• r + t + a = 1.

• Emissive Power (E): Total radiant energy emitted per unit surface area per unit time. E = e * σ * T⁴.

• Emissivity (e): Ratio of emissive power of a body to that of a black body at the same temperature (0 ≤ e ≤ 1).

Kirchhoff's Law of Radiation

"Good emitters are good absorbers, and poor emitters are poor absorbers."

The ratio of emissive power to absorptivity is the same for all bodies at a given temperature, equal to that of a black body.

Black Body Radiation

An idealized body that absorbs all incident radiation (a = 1) and is also a perfect emitter (e = 1). A small hole in a hollow sphere is a practical model.

Spectral Emissive Power and Wien's Displacement Law

Spectral Emissive Power (Eλ)

Energy emitted per unit time per unit surface area per unit wavelength range. λm is the wavelength with maximum spectral emissive power.

Wien's Displacement Law

Relates the absolute temperature (T) of a black body to the wavelength of maximum emission (λm):

• λm T = b (constant)
  Where b is Wien's displacement constant (approx. 2.9 x 10^-3 m·K).
  (Memory Tip: An increase in temperature causes the Eλ vs. λ graph to shift left (shorter λm) and up (higher total emitted energy).). This law explains why objects glow different colors as they heat up: from red to yellow to white (or even blue).

Newton's Law of Cooling

The rate of heat loss from a body is directly proportional to the temperature difference between the body and its surroundings, provided the temperature difference (ΔT) is small.

• dT/dt = -k(T - Ts)

• Integrated form: ln((T_final - Ts) / (T_initial - Ts)) = -kt.

• For small temperature changes, an approximation can be used: (ΔT_body) / Δt = -k * (T_average - Ts).
  

Mechanical Properties of Matter

Mechanical properties describe how a substance behaves when an external force is applied to it. These properties help explain how materials deform, stretch, compress or flow under different forces. Important concepts include stress, strain, elasticity, elastic moduli, pressure, viscosity, surface tension and fluid flow.

Stress

Stress is the internal restoring force developed per unit area when an external force acts on a body.

Stress = F/A

Where:

• F = applied force

• A = cross-sectional area

The SI unit of stress is pascal (Pa) or N/m².

Types of Stress

• Longitudinal stress: Produced when the force acts along the length of a body. It may be tensile or compressive.

• Normal stress: Acts perpendicular to the surface.

• Shearing stress: Produced when the force acts tangentially to the surface.

Strain

Strain is the deformation produced per unit original dimension of a body.

For longitudinal deformation:

Strain = ΔL/L

Strain has no unit or dimension because it is a ratio of two lengths.

Hooke's Law

Within the elastic limit, stress is directly proportional to strain.

Stress ∝ Strain

or

Stress = Elastic Modulus × Strain

Hooke's law is applicable only when the material remains within its elastic limit.

Young's Modulus

Young's modulus measures the resistance of a material to a change in its length.

Y = Longitudinal stress / Longitudinal strain

Therefore,

Y = (F/A)/(ΔL/L)

or

Y = FL/(AΔL)

A material with a higher Young's modulus is generally more resistant to stretching.

Bulk Modulus

Bulk modulus measures the resistance of a substance to a change in its volume.

K = -ΔP/(ΔV/V)

Where:

• ΔP = change in pressure

• ΔV = change in volume

• V = original volume

The negative sign indicates that an increase in pressure generally causes a decrease in volume.

Shear Modulus

Shear modulus measures the resistance of a material to a change in its shape due to a tangential force.

G = Shearing stress / Shearing strain

It is also called the modulus of rigidity.

Elastic Potential Energy

When a body is deformed within its elastic limit, work is done on it and stored as elastic potential energy.

For a stretched wire:

U = 1/2 FΔL

The energy stored per unit volume is:

Energy density = 1/2 × Stress × Strain

Stress-Strain Curve

The stress-strain curve shows how a material deforms as the applied stress increases.

Important points include:

• Elastic limit: Maximum stress up to which the material returns to its original shape after removing the force.

• Yield point: The material begins to undergo significant permanent deformation.

• Plastic region: Permanent deformation occurs.

• Breaking point: The material finally breaks.

Materials can be broadly described as ductile or brittle based on their behaviour under stress.

Pressure

Pressure is the normal force acting per unit area.

P = F/A

The SI unit of pressure is pascal (Pa).

For a liquid at depth h:

P = P₀ + ρgh

Where:

• P₀ = atmospheric pressure at the surface

• ρ = density of the liquid

• g = acceleration due to gravity

• h = depth below the surface

Pascal's Law

Pascal's law states that pressure applied to an enclosed fluid is transmitted equally and undiminished in all directions.

It is used in devices such as:

• Hydraulic lifts

• Hydraulic brakes

• Hydraulic presses

For a hydraulic machine:

F₁/A₁ = F₂/A₂

Archimedes' Principle

Archimedes' principle states that when a body is partially or completely immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by it.

Buoyant force = ρVg

Where:

• ρ = density of the fluid

• V = volume of displaced fluid

• g = acceleration due to gravity

This principle explains why objects float or sink in fluids.

Equation of Continuity

For an incompressible fluid flowing through a pipe:

A₁v₁ = A₂v₂

Where:

• A = cross-sectional area

• v = speed of fluid

This means that when the area of a pipe decreases, the speed of the fluid increases.

Bernoulli's Principle

For an ideal fluid in steady flow, the total mechanical energy per unit volume remains constant along a streamline.

P + 1/2 ρv² + ρgh = constant

Where:

• P = pressure

• ρ = density

• v = fluid velocity

• h = height

Applications include:

• Aeroplane wings

• Venturimeters

• Atomisers and sprayers

• Flow of fluids through pipes

Viscosity

Viscosity is the property of a fluid that opposes the relative motion between its different layers.

For a fluid:

F = ηA(dv/dx)

Where η is the coefficient of viscosity.

The SI unit of viscosity is Pa·s.

Generally, the viscosity of liquids decreases with temperature, while the viscosity of gases increases with temperature.

Stokes' Law

When a small spherical body moves slowly through a viscous fluid, the viscous force acting on it is:

F = 6πηrv

Where:

• η = coefficient of viscosity

• r = radius of the sphere

• v = velocity of the sphere

Terminal Velocity

When a body falls through a viscous fluid, its speed eventually becomes constant. This constant speed is called terminal velocity.

For a small spherical body:

vₜ = 2r²(ρₛ - ρ)g / 9η

Where:

• r = radius of the sphere

• ρₛ = density of the sphere

• ρ = density of the fluid

• η = viscosity of the fluid

Surface Tension

Surface tension is the property of a liquid surface due to which it tends to minimise its surface area.

T = F/L

Where:

• F = force acting tangentially

• L = length over which the force acts

The SI unit of surface tension is N/m.

Surface tension explains phenomena such as:

• Formation of spherical water droplets

• Floating of small insects on water

• Rise of liquid in a capillary tube

Capillary Rise

When a liquid rises or falls in a narrow tube, the phenomenon is called capillarity.

The height of capillary rise is:

h = 2T cosθ/(ρgr)

Where:

• T = surface tension

• θ = angle of contact

• ρ = density of liquid

• r = radius of the capillary tube

For water in a clean glass tube, the liquid rises because the angle of contact is less than 90°.

Excess Pressure

The pressure inside a liquid drop is greater than the pressure outside it.

For a liquid drop:

ΔP = 2T/R

For a soap bubble:

ΔP = 4T/R

The difference occurs because a soap bubble has two surfaces, while a liquid drop has only one surface.

Thermal and Mechanical Properties of Matter connects the behaviour of matter with changes in temperature and the application of external forces. Thermal expansion, calorimetry and radiation explain heat-related phenomena, while elasticity, fluid mechanics, viscosity and surface tension help explain mechanical behaviour. Understanding the concepts and practising their formulas can help students solve numerical and conceptual questions in NEET Physics.