Thermal Properties of Matter — Class 11 Physics

1. Temperature and Heat

Temperature indicates the thermal state of a body and determines the direction of spontaneous heat transfer. Heat is energy transferred between bodies because of a temperature difference. Heat is not something a body simply “contains”; it is energy in transit.

Temperature scales commonly used are Celsius, Fahrenheit and Kelvin. The absolute temperature scale is Kelvin: T(K)=t(°C)+273.15.

2. Thermal Expansion

Most substances expand when heated and contract when cooled, although water has an important anomaly near 0–4°C. For a solid rod, linear expansion is described by ΔL=αL₀ΔT, where α is the coefficient of linear expansion.

For area expansion, ΔA≈βA₀ΔT, and for volume expansion, ΔV≈γV₀ΔT. For an isotropic solid undergoing small expansion, β≈2α and γ≈3α.

3. Thermal Expansion of Liquids

Liquids have no fixed shape, so volume expansion is normally considered. When a liquid is heated in a vessel, both the liquid and vessel expand. The observed or apparent expansion of the liquid is therefore different from its real expansion.

4. Anomalous Expansion of Water

Between 0°C and 4°C, water contracts when heated and expands when cooled. Water has maximum density at approximately 4°C. This unusual behaviour is important for aquatic life in cold climates because surface water can freeze while deeper water remains near 4°C.

5. Heat Capacity and Specific Heat Capacity

Heat capacity is the heat required to raise the temperature of a body by one kelvin: C=Q/ΔT. Specific heat capacity is heat capacity per unit mass: c=Q/(mΔT), so Q=mcΔT.

A material with a high specific heat capacity requires more heat for the same mass and temperature rise.

6. Calorimetry

In an isolated calorimetry problem, the basic principle is conservation of energy: heat lost = heat gained, provided external heat exchange is negligible. Phase changes must be included separately using latent heat.

7. Change of State and Latent Heat

During melting or boiling of a pure substance at constant pressure, temperature remains constant while energy is used to change the phase. The heat involved is Q=mL, where L is the specific latent heat. For melting, L is latent heat of fusion; for boiling, it is latent heat of vaporisation.

8. Heat Transfer

Heat can be transferred by conduction, convection and radiation.

• Conduction: energy transfer through a material without bulk motion of the material.
• Convection: heat transfer involving bulk movement of a fluid.
• Radiation: transfer by electromagnetic waves and does not require a material medium.

9. Thermal Conductivity

For one-dimensional steady conduction through a uniform slab, Q/t = kA(T₁−T₂)/L, where k is thermal conductivity. A high k indicates that the material conducts heat readily under comparable conditions.

10. Newton’s Law of Cooling

For a body cooling through a small temperature difference from its surroundings, the rate of loss of heat is approximately proportional to the temperature difference: dQ/dt ∝ −(T−T_s). The simple law works as an approximation under suitable conditions.

11. Stefan–Boltzmann Law

An ideal blackbody emits radiant power per unit area proportional to the fourth power of absolute temperature: P/A=σT⁴. For a real surface, emissivity modifies the expression. Net radiation exchange with surroundings depends on the temperatures of both the body and surroundings.

12. Wien’s Displacement Law

For a blackbody spectrum, the wavelength at which emission is maximum satisfies λ_maxT=b. Therefore hotter objects have their peak thermal radiation at shorter wavelengths.

Worked Numerical 1 — Celsius to Kelvin

Question: Convert 27°C to Kelvin.

T=27+273.15=300.15 K, approximately 300 K.

Worked Numerical 2 — Linear Expansion

Question: A 2 m metal rod has α=1.2×10⁻⁵ K⁻¹. Find its increase in length for a temperature rise of 50 K.

ΔL=αL₀ΔT=(1.2×10⁻⁵)(2)(50)=1.2×10⁻³ m = 1.2 mm.

Worked Numerical 3 — Heat Required

Question: How much heat is required to raise 2 kg of water by 10°C? Take c=4200 J kg⁻¹ K⁻¹.

Q=mcΔT=2×4200×10=84,000 J.

Worked Numerical 4 — Calorimetry

Question: 0.2 kg of water at 80°C is mixed with 0.3 kg of water at 20°C in an insulated container. Find the final temperature, neglecting heat capacity of the container.

Because both substances are water, c cancels: 0.2(80−T)=0.3(T−20).

16−0.2T=0.3T−6, so 22=0.5T and T=44°C.

Worked Numerical 5 — Latent Heat

Question: Find the heat required to melt 0.5 kg of ice at 0°C if the latent heat of fusion is 3.34×10⁵ J/kg.

Q=mL=0.5×3.34×10⁵=1.67×10⁵ J.

Worked Numerical 6 — Thermal Conduction

Question: A slab has k=0.8 W m⁻¹ K⁻¹, area 2 m², thickness 0.1 m and temperature difference 20 K. Find the steady heat-transfer rate.

Q/t=kAΔT/L=(0.8)(2)(20)/0.1=320 W.

Worked Numerical 7 — Wien’s Law

Question: A blackbody has peak wavelength 1.0 μm. Taking Wien’s constant b=2.9×10⁻³ m K, estimate its temperature.

T=b/λ_max=(2.9×10⁻³)/(1.0×10⁻⁶)=2900 K.

Worked Numerical 8 — Stefan–Boltzmann Scaling

Question: If the absolute temperature of an ideal blackbody doubles, by what factor does its emitted power per unit area change?

Since P/A∝T⁴, the factor is 2⁴=16.

Important Derivation — Calorimetry Mixing Equation

For two bodies exchanging heat in an insulated system, conservation of energy gives heat lost by the hotter body plus heat gained by the colder body equal to zero. For bodies without phase change, m₁c₁(T₁−T_f)=m₂c₂(T_f−T₂). This relation provides the final equilibrium temperature.

Important Derivation — Linear Expansion

The coefficient of linear expansion is defined as the fractional increase in length per unit temperature rise: α=(ΔL/L₀)/ΔT. Rearranging gives ΔL=αL₀ΔT. For small temperature changes and ordinary isotropic solids, this linear approximation is highly useful.

Explained MCQs

1. Heat is best described as:
   Answer: Energy transferred because of a temperature difference. Heat is a process of energy transfer rather than a state variable stored in a body.
2. What is the SI unit of specific heat capacity?
   Answer: J kg⁻¹ K⁻¹.
3. During melting of pure ice at its melting point, supplied heat mainly:
   Answer: Changes the phase. The temperature remains approximately constant while latent heat is absorbed.
4. Why are gaps provided between sections of railway tracks or bridges?
   Answer: To allow thermal expansion. Without allowance, large thermal stresses can develop.
5. If the absolute temperature of a blackbody doubles, its ideal emitted power per unit area becomes:
   Answer: 16 times. Stefan–Boltzmann law gives P/A∝T⁴.
6. Which mode of heat transfer can occur through empty space?
   Answer: Radiation. Electromagnetic radiation does not require a material medium.

Competency Question

A metal bridge is exposed to large temperature changes between summer and winter. Explain why expansion joints are provided and what could happen if the bridge were rigidly constrained.

HOTS

Two equal masses of water and another liquid receive the same amount of heat. The water shows a smaller temperature rise. Explain what this indicates about the relative specific heat capacities of the substances.

Common Mistakes

• Confusing heat with temperature.
• Using Celsius directly in formulas that require absolute temperature, such as Stefan–Boltzmann or Wien’s law.
• Forgetting latent heat during a phase-change problem.
• Mixing units such as grams with kg or mm with m.
• Assuming all heat transfer is conduction; convection and radiation have different mechanisms.
• Applying Newton’s law of cooling without considering the assumption of a relatively small temperature difference and suitable conditions.

Quick Revision

• T(K)=t(°C)+273.15
• ΔL=αL₀ΔT
• ΔA≈βA₀ΔT
• ΔV≈γV₀ΔT
• Q=mcΔT
• Q=mL during phase change
• Q/t=kAΔT/L
• dQ/dt∝−(T−T_s) for Newtonian cooling approximation
• P/A=σT⁴ for an ideal blackbody
• λ_maxT=b