Chapter 11 — Thermal Properties Of Matter

Fig 1 — Educational Diagram: Heat and Temperature

Overview: Heat and temperature are central concepts in Thermal Properties of Matter. Temperature is a measure of the average kinetic energy (or degree of hotness) of particles in a body. Heat is energy transferred between systems due to a temperature difference.

Temperature (T): A scalar quantity that indicates how hot or cold a body is. SI unit: kelvin (K). Related to average microscopic kinetic energy: for an ideal monatomic gas, <KE> = (3/2)k_B T per particle (k_B = Boltzmann constant). Temperature scales: Celsius (°C), Kelvin (K), Fahrenheit (°F). Conversion: T(K) = T(°C) + 273.15.

Heat (Q): Energy in transit due to temperature difference. Heat is not a property of a body (it depends on process). SI unit: joule (J). Heat can increase internal energy and/or do work on surroundings.

Key distinctions (Heat vs Temperature):

• Temperature measures average kinetic energy; heat is energy transfer.
• Temperature is an intrinsic property; heat depends on a process (flow).
• Two bodies at same temperature exchange no net heat (thermal equilibrium).

Thermal equilibrium and Zeroth Law: If A is in thermal equilibrium with B, and B with C, then A is in equilibrium with C. This allows definition of temperature and use of thermometers.

Internal energy (U): Sum of microscopic kinetic and potential energies of particles. For an ideal monatomic gas, U = (3/2) nRT. Change in internal energy (first law): ΔU = Q - W, where W is work done by the system.

Specific heat and Heat capacity: Heat capacity (C) is amount of heat required to raise temperature of a body by 1 K: C = dQ/dT. Specific heat (c) is heat capacity per unit mass: c = (1/m)(dQ/dT). For processes at constant volume or pressure, use molar/ specific heat at constant volume (C_v) or constant pressure (C_p). For ideal gases: C_p - C_v = R (molar values).

Latent heat: Heat absorbed or released during a phase change at constant temperature. Q = mL, where L is latent heat (fusion, vaporization).

Modes of heat transfer:

• Conduction: transfer through direct contact. Fourier’s law: dQ/dt = -k A (dT/dx) (k = thermal conductivity).
• Convection: transfer by bulk motion of fluid. Newton's law of cooling (empirical): rate ∝ (T - T_env); dT/dt = -k(T - T_env) or heat flux ≈ h A (T - T_env).
• Radiation: emission of electromagnetic waves. Stefan–Boltzmann law: P = εσA T^4 (ε = emissivity, σ = Stefan–Boltzmann constant).

Calorimetry: Experimental branch to measure heat capacities and latent heats. Principle: conservation of energy—heat lost by hot bodies = heat gained by cold bodies (neglecting losses).

Practical notes and examples: Heating water on a stove (Q = mcΔT), melting ice at 0°C (latent heat), thermal expansion of solids with temperature rise (important in engineering), thermometer reading and body fever detection, cooling of hot coffee (Newton’s law), refrigerators and heat engines (use transfer of heat for work).

Summary: Temperature quantifies thermal state; heat quantifies energy transfer due to temperature difference. Understanding heat, temperature, specific/latent heat, and modes of transfer is essential for solving calorimetry problems and analyzing thermal processes.

Fig 2.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

Thermal equilibrium is the state in which two or more bodies in thermal contact exchange no net heat energy; their macroscopic temperatures are equal and there is no further spontaneous heat flow between them. Thermal contact means they can exchange energy (typically as heat) through a common boundary.

Zeroth law of thermodynamics states: If two systems A and B are each in thermal equilibrium with a third system C, then A and B are in thermal equilibrium with each other. Symbolically, if T(A) = T(C) and T(B) = T(C), then T(A) = T(B). The law establishes temperature as a fundamental, transitive property that can be used to compare thermal states and to define thermometers.

Key concepts and consequences:

• Temperature is the property that is equal for systems in thermal equilibrium; it is a measurable scalar that indicates the direction of spontaneous heat flow (from higher T to lower T).
• At thermal equilibrium the net heat flow between systems is zero. In other words, dQ/dt = 0 (no net rate of heat transfer).
• The Zeroth law justifies the use of a thermometer: a thermometer (system C) brought into thermal contact with any system A will come to the same temperature as A, allowing temperature measurement.
• For isolated systems, thermal equilibrium corresponds to a maximum of total entropy; this gives a statistical justification: ∂S_total/∂E_A = ∂S_total/∂E_B implies 1/T_A = 1/T_B and hence T_A = T_B.

How equilibrium is reached: When two objects at different temperatures are put in thermal contact, heat flows from the hotter to the colder object. Over time, the temperatures change until they become equal. The rate at which temperature approaches equilibrium often follows Newton's law of cooling for convective heat transfer, giving an exponential approach to the environment temperature in many practical cases.

Fig 3 — Educational Diagram: Measurement of Temperature and Thermometers

What is temperature? Temperature is a measure of the hotness or coldness of a body related to the average kinetic energy of its molecules. Two bodies in thermal equilibrium have the same temperature.

Zeroth law of thermodynamics: If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then A is in thermal equilibrium with C. This law justifies the use of thermometers (it lets us assign a number to temperature).

Thermometric property and thermometer: A thermometric property X is any physical quantity of a substance that changes predictably with temperature (example: length of a mercury column, pressure of gas at constant volume, electrical resistance). A thermometer uses such a property and a chosen scale to measure temperature.

Fixed points and temperature scales:

• Fixed points are reproducible temperatures used to define scales: common ones are the ice point (pure water at 1 atm, 0 °C) and steam point (pure water vapor at 1 atm, 100 °C).
• Celsius (centigrade) scale: 0 °C and 100 °C are defined as above.
• Kelvin (absolute) scale: T(K) = t(°C) + 273.15. Zero of Kelvin scale (0 K) is absolute zero.
• Fahrenheit: t(°F) = (9/5)t(°C) + 32.

Properties of a good thermometer:

• Well-defined and reproducible thermometric property.
• High sensitivity (large change in property per unit temperature).
• Linearity (property varies linearly with temperature over range) or known calibration curve.
• Small thermal mass and fast response time.
• Wide useful range and stability (no hysteresis).

Common types of thermometers (principles & brief notes):

• Liquid-in-glass (mercury or alcohol): volume expansion of liquid raises the column. Simple, fairly linear over a moderate range. Not used for very low temperatures (mercury freezes) or where mercury toxicity is a concern.
• Constant-volume gas thermometer: at constant volume, gas pressure p ∝ absolute temperature T (Kelvin). Very accurate and used as a laboratory standard. Relation: T = T0 (p/p0) at constant V.
• Thermocouple: two different metals joined produce a thermoelectric emf proportional to temperature difference (Seebeck effect). Robust, wide range, used in furnaces and engines. E ≈ S ΔT for small ranges (S: Seebeck coefficient).
• Resistance thermometer (RTD): electrical resistance of a metal (usually platinum) increases approximately linearly with temperature. R = R0[1 + α (T − T0)]. Precise and stable for industrial calibration.
• Thermistor: semiconductor whose resistance falls rapidly with temperature. High sensitivity over limited range; used in digital thermometers and sensors.
• Bimetallic strip: two metals with different expansion coefficients bend with temperature; used in thermostats and mechanical thermometers.
• Infrared (pyrometer): measures thermal radiation emitted by a body; useful for non-contact measurement (e.g., moving objects, human forehead thermometers).

Calibration and linearity: A thermometer must be calibrated by relating its property X to known fixed-point temperatures (e.g., ice and steam points). If X varies linearly with temperature, one calibration straight line suffices; otherwise a calibration curve or table is needed.

Errors and limits:

• Systematic errors: poor calibration, non-linearity, self-heating (electrical sensors), heat losses.
• Random errors: fluctuations, sensor noise. Response time and thermal contact affect readings.

Practical notes: For clinical use, digital thermometers (thermistor or RTD based) are common because of safety and speed. Industrial processes often use thermocouples or RTDs depending on required range and accuracy.

Fig 4 — Educational Diagram: Thermal Expansion of Solids

Definition: Thermal expansion of solids is the increase in linear dimensions, area or volume of a solid when its temperature is raised. It occurs because atomic vibrations increase with temperature and, owing to the anharmonicity of interatomic potential, the average separation between atoms increases.

Linear expansion: For a rod of initial length L₀, when its temperature changes by ΔT, the change in length ΔL is given (for small ΔT) by

ΔL = α L₀ ΔT

Here α is the coefficient of linear expansion (units: K^-1). Physically α = (1/L)(dL/dT) (for infinitesimal changes).

Area and volume expansion (isotropic solids): If the linear coefficient is α, then for small temperature change ΔT

• Change in area: ΔA = 2 α A₀ ΔT
• Change in volume: ΔV = 3 α V₀ ΔT

So the coefficient of volume expansion β (sometimes written γ) is β = 3 α for isotropic solids.

Exact (integral) form: If α varies with temperature, dL/dT = α(T) L, so

L(T) = L₀ exp(∫_T0^T α(T') dT')

For approximately constant &alpha:, L(T) ≈ L₀(1 + αΔT).

Thermal stress (constrained expansion): If a body that tends to expand is prevented from doing so, it develops thermal stress. For a rod rigidly clamped at both ends, the induced stress is approximately

σ = Y α ΔT

where Y is Young's modulus. The corresponding thermal strain = αΔT.

Physical origin: The interatomic potential is not symmetric; with increasing thermal vibrations the average atomic separation increases, giving macroscopic expansion. Some materials have very small or even negative α over certain temperature ranges (e.g., Invar alloy has very low α).

Important remarks: For large ΔT higher order terms may be relevant. Coefficients differ for different materials and depend weakly on temperature. Expansion is anisotropic in crystals with low symmetry.

Fig 5 — Educational Diagram: Area and Volume Expansion

Overview
When the temperature of a solid is raised, its dimensions increase. For isotropic (same in all directions) solids the change in length, area and volume are related by coefficients of linear, area and volume expansion. These describe how a physical quantity changes per unit original quantity per unit temperature rise.

Linear expansion (recap)
If a rod of initial length L₀ is heated by ΔT, its length becomes L = L₀(1 + αΔT). Here α is the coefficient of linear expansion (units K⁻¹ or °C⁻¹).

Area expansion — derivation
Consider a square plate with side L₀ and area A₀ = L₀². After heating by ΔT each side becomes L = L₀(1 + αΔT). New area:

A = L² = L₀²(1 + αΔT)² = A₀[1 + 2αΔT + (αΔT)²].

For typical materials αΔT ≪ 1, so the (αΔT)² term is negligible and we get the approximate relation

ΔA = A − A₀ ≈ 2α A₀ ΔT.

Thus the coefficient of area expansion ≈ 2α. Exactly, A = A₀(1 + αΔT)² so the fractional change is (1 + αΔT)² − 1.

Volume expansion — derivation
For a cube of edge L₀ and volume V₀ = L₀³, after heating:

V = L³ = L₀³(1 + αΔT)³ = V₀[1 + 3αΔT + 3(αΔT)² + (αΔT)³].

Neglecting higher-order small terms,

ΔV ≈ 3α V₀ ΔT.

So the coefficient of volume expansion β ≈ 3α. Exactly, fractional change is (1 + αΔT)³ − 1.

Notes and special points

• Units: α, β are in K⁻¹ (or °C⁻¹); β ≈ 3α for isotropic solids.
• For anisotropic materials (different expansion along axes) area/volume changes add the relevant linear coefficients; e.g., for a rectangle with α_x, α_y, ΔA/A ≈ (α_x + α_y)ΔT.
• For large ΔT or materials with large α one should use exact expressions: A = A₀(1 + αΔT)², V = V₀(1 + αΔT)³.
• If thermal expansion is prevented (constrained), it produces thermal stress. For a rod fully constrained in length, induced stress σ ≈ EαΔT (E = Young's modulus).

Typical magnitudes
For metals α is typically of order 10⁻⁵ K⁻¹; for glass it is much smaller, and for polymers it can be larger and more temperature-dependent.

Fig 6 — Educational Diagram: Relation between Expansion Coefficients

Definition: Thermal expansion coefficients measure fractional change of dimensions per unit temperature rise.

Linear expansion coefficient (α): α = (1/L)(dL/dT) ≈ ΔL/(L ΔT). It gives fractional change in length.

Area expansion coefficient (β): β = (1/A)(dA/dT) ≈ ΔA/(A ΔT). It gives fractional change in area.

Volume expansion coefficient (γ): γ = (1/V)(dV/dT) ≈ ΔV/(V ΔT). It gives fractional change in volume.

Derivation and relations (isotropic solid):

Consider a cuboid with initial sides l, w, h. After temperature rise ΔT each side becomes l(1+αΔT), w(1+αΔT), h(1+αΔT) (assuming same α in all directions).

New area of one face (l×w): A' = l w (1+αΔT)^2 = A[1 + 2αΔT + (αΔT)^2]. Therefore
β = (ΔA)/(A ΔT) = [A' - A]/(A ΔT) = 2α + α^2 ΔT. For small ΔT (neglect α^2ΔT) we get β ≈ 2α.

New volume: V' = l w h (1+αΔT)^3 = V[1 + 3αΔT + 3(αΔT)^2 + (αΔT)^3]. Hence
γ = (ΔV)/(V ΔT) = 3α + 3α^2 ΔT + α^3 ΔT^2. For small ΔT we get γ ≈ 3α.

General (possibly anisotropic) case: If linear coefficients along three orthogonal directions are αx, αy, αz, then
β = αx + αy (for area in the xy-plane),
γ = αx + αy + αz (for volume). For isotropic materials αx = αy = αz = α, so β = 2α and γ = 3α.

Approximate relations used commonly:
ΔL ≈ α L ΔT,
ΔA ≈ β A ΔT ≈ 2α A ΔT,
ΔV ≈ γ V ΔT ≈ 3α V ΔT.

Notes: The approximations β ≈ 2α and γ ≈ 3α are valid when αΔT << 1. For large temperature changes or materials with large α, keep higher-order terms or use exact expressions above. Liquids and glasses are usually treated isotropically for volume expansion, but solids can be anisotropic (different α along different crystal axes).

Fig 7 — Educational Diagram: Anomalous Expansion of Water

Definition: The anomalous expansion of water is the unusual behaviour that water contracts on heating between 0 °C and 4 °C (i.e., its volume decreases and density increases with temperature in that range). Water attains its maximum density at 4 °C. Above 4 °C it behaves normally and expands on heating; below 0 °C it becomes ice with a much larger, open structure.

Molecular explanation: Water molecules form hydrogen bonds that favour an open tetrahedral arrangement (as in ice). Near 0 °C many molecules are arranged in this open structure, giving a relatively large volume. As the temperature rises to 4 °C, some of these open structures collapse and molecules pack more closely, so volume decreases and density increases. Above 4 °C, thermal motion dominates and the usual thermal expansion (increase of volume with temperature) takes over.

Key consequences:

• Ice is less dense than liquid water and floats — lakes and rivers freeze from the top downward, insulating aquatic life below.
• Seasonal convection in lakes: water at 4 °C sinks, helping mixing and oxygenation.
• Practical effects on engineering (pipes, reservoirs) and on biological habitats in cold climates.

When linear expansion formulas fail: The simple linear expansion relation (ΔV = V0 β ΔT) still applies as a definition, but here the volume expansion coefficient β becomes negative between 0 and 4 °C. Thus one must use measured β(T) or density data rather than assume a constant positive β.

Experimental observation: The anomalous behaviour is commonly demonstrated using a dilatometer (a flask with a narrow capillary). By cooling/heating water and recording the liquid level in the capillary versus temperature, one observes the level fall as temperature increases from 0 to 4 °C (volume decreases) and then rise for T > 4 °C.

Fig 8 — Educational Diagram: Applications of Thermal Expansion

Thermal expansion is the change in size (length, area or volume) of materials when their temperature changes. The phenomenon is quantified by coefficients of linear (α), area and volume expansion (β, γ). Practical applications either use expansion intentionally (to produce motion or fit parts) or must allow for it (to prevent damage). Key ideas:

• Controlled use: Devices that convert temperature change into mechanical motion (e.g., bimetallic strips in thermostats) exploit different expansions of two metals.
• Allowing for expansion: Large structures such as railway tracks and bridges include gaps or expansion joints to avoid buckling or fracture when parts expand.
• Thermal fitting (shrink/heat fit): Components are intentionally heated/cooled so their size changes to create tight assemblies (e.g., fitting a metal ring onto a shaft).
• Measurement: Liquid-in-glass thermometers use the volume expansion of liquids (mercury/alcohol) to measure temperature changes.
• Stress due to restraint: When thermal expansion is prevented, large stresses develop (thermal stress), which must be accounted for in design.

Each application is based on the basic expansion formulas (linear, area, volume) and on the behavior of materials when they are free to expand or are constrained. Designers either provide freedom (gaps, joints, flexible hangers) or use the expansion to perform useful work (thermostats, shrink fits).

Fig 9 — Educational Diagram: Heat Capacity and Specific Heat

Definitions: Heat capacity (C) of a body is the amount of heat required to raise its temperature by 1 K (or 1 °C). It is an extensive property. Specific heat (specific heat capacity, c) is the heat required to raise the temperature of unit mass of a substance by 1 K (or 1 °C). It is an intensive property.

Symbols and SI units: C (J K^-1), c (J kg^-1 K^-1), temperature change ΔT in K (or °C).

Basic relations: If a heat Q is supplied and the body temperature changes by ΔT, then for small temperature changes (or assuming c constant over the range):

Q = C ΔT = mc ΔT

So C = mc and c = C/m. More generally, when C depends on T, Q = ∫ C(T) dT.

Molar heat capacity: Heat capacity per mole, C_m = C/n (n = number of moles). For a substance with molar mass M, C_m = M c. SI unit: J mol^-1 K^-1.

Gases (important relations): For ideal gases one distinguishes heat capacity at constant volume (C_V or c_v per unit mass) and at constant pressure (C_P or c_p per unit mass). Per mole: C_p - C_v = R (universal gas constant). Per unit mass: c_p - c_v = R_specific = R/M.

Experimental measurement (method of mixtures / calorimetry): Using a calorimeter (with heat capacity C_cal), when a hot body (mass m_h, heat capacity c_h) at temperature T_h is put into cold water (mass m_w, specific heat c_w) at T_w, final equilibrium temperature T_f satisfies:

m_h c_h (T_h - T_f) = m_w c_w (T_f - T_w) + C_cal (T_f - T_w)

One can rearrange this to determine an unknown c_h or the calorimeter's heat capacity. The water-equivalent of a calorimeter is W = C_cal / c_water.

Conceptual notes: - Heat capacity is extensive (depends on amount of substance); specific and molar heat capacities are intensive. - For most solids and liquids mechanical work during heating is negligible so Cp ≈ Cv; for gases the difference is significant. - Some substances have temperature-dependent specific heats; for many solids at room temperature c is roughly constant (Dulong–Petit law: molar heat capacity of many solid metals ≈ 3R ≈ 25 J mol^-1 K^-1).

Physical meaning: A large specific heat means the substance requires more heat to change its temperature (has higher thermal inertia). This explains why water warms/cools slowly compared to metals or sand.

Fig 10 — Educational Diagram: Calorimetry

Definition: Calorimetry is the experimental study of heat transfer between bodies and the measurement of heat changes accompanying physical and chemical processes. A calorimeter is the device used to measure heat exchanged.

Basic ideas: Heat (Q) is energy transferred due to temperature difference. Temperature (T) is a measure of the average kinetic energy of particles. In calorimetry we use the principle of conservation of energy: heat lost by hot parts = heat gained by cold parts (if no heat is lost to surroundings).

Key quantities:

• Specific heat capacity (c): amount of heat required to raise 1 kg of a substance by 1 K (SI unit: J kg⁻¹ K⁻¹). For water c ≈ 4186 J kg⁻¹ K⁻¹ (≈ 4.18 J g⁻¹ K⁻¹).
• Latent heat (L): heat absorbed or released during a phase change at constant temperature (J kg⁻¹). Examples: latent heat of fusion (melting) Lf, latent heat of vaporization (boiling) Lv.
• Heat capacity (C): heat required to raise temperature of an object by 1 K (C = mc for a homogeneous body).

Principle of calorimetry (method of mixtures): If a hot object (mass m1, specific heat c1, initial temperature T1) is placed in a colder fluid (mass m2, specific heat c2, initial temperature T2) inside a calorimeter (heat capacity Ccal), the final equilibrium temperature Tf satisfies energy balance:

heat lost = heat gained, i.e.

m1 c1 (T1 − Tf) = m2 c2 (Tf − T2) + Ccal (Tf − Tcal)

Often Ccal is combined with the water term as an effective water equivalent. In ideal (no loss) cases ΣQ = 0. For phase changes include latent heat terms: Q = mL for melting/boiling.

Experimental notes & assumptions: Assume thermal equilibrium is reached, negligible heat exchange with surroundings (or correct for it), include heat capacity of thermometer/calorimeter if significant, and use consistent units (K or °C increments are equivalent for ΔT).

Fig 11 — Educational Diagram: Latent Heat

Definition: Latent heat is the amount of heat energy absorbed or released by a substance during a phase change (solid↔liquid or liquid↔gas) at constant temperature and pressure, without any change in temperature of the substance.

When heat is supplied during a phase change, it changes the internal energy (mainly the potential energy associated with intermolecular forces) rather than the kinetic energy of molecules, so the measured temperature remains constant while the phase change proceeds.

Types:

• Latent heat of fusion (L_f): heat required per unit mass to change a solid into a liquid at its melting point (or released when liquid freezes).
• Latent heat of vaporization (L_v): heat required per unit mass to change a liquid into a gas at its boiling point (or released when gas condenses).
• Latent heat of sublimation (L_s): heat required per unit mass to change a solid directly into a gas (L_s = L_f + L_v for a substance undergoing solid→liquid→gas).

Physical explanation: During melting or boiling the energy supplied is used to overcome intermolecular attractions and increase the potential energy of the system. Since average molecular kinetic energy (and thus temperature) does not change during the phase transition, the temperature remains constant until the entire sample has changed phase.

Dependence: Latent heat values depend on the substance and on pressure (especially for boiling). For most substances L_v > L_f because converting liquid to gas requires breaking more intermolecular bonds than solid→liquid.

Fig 12 — Educational Diagram: Internal Energy (Qualitative)

What is internal energy?
Internal energy (U) of a system is the total microscopic energy contained in it — the sum of kinetic energies of all molecules/atoms (translational, rotational, vibrational) and the potential energies arising from intermolecular forces and chemical bonds. It is a property of the state of the system and is extensive (depends on amount of substance).

Microscopic picture (qualitative)
• Kinetic part: motion of particles. Faster motion (higher average speed) → larger kinetic energy → higher internal energy.
• Potential part: interactions between particles. When particles move apart (e.g. melting/evaporation) the potential energy typically increases even if temperature does not change.

Dependence on temperature and state
• For an ideal gas, internal energy depends only on temperature (not on volume or pressure). Increasing T increases particle kinetic energy and hence U.
• For liquids and solids, U depends on temperature and also on molecular arrangement; during phase changes temperature can stay constant while U changes (energy goes into changing potential energy — latent heat).

Changes in internal energy
You cannot measure the absolute value of U directly; only changes ΔU are meaningful. According to the first law of thermodynamics (qualitatively), the change in internal energy equals energy added as heat minus work done by the system. Heat input can increase kinetic energy (raise temperature) or increase potential energy (cause phase change).

Key qualitative examples:
• Heating a gas in a closed container: particles move faster → U increases.
• Melting ice at 0°C: temperature stays ~constant while energy increases the potential energy (separating molecules) → U increases.
• Compressing a gas rapidly (adiabatically): work done on gas increases kinetic energy → temperature and U increase.

Important conceptual points
• Internal energy is a state function: ΔU depends only on initial and final states, not the path.
• For an ideal gas, isothermal processes (constant T) imply ΔU ≈ 0.
• The choice of zero of internal energy is arbitrary; only changes are physically measurable.

Fig 13 — Educational Diagram: Heat Transfer — Conduction

What is conduction? Conduction is the mode of heat transfer within a body or between bodies in direct physical contact, due to temperature differences. Heat flows from the hotter region to the colder region by microscopic collisions and energy exchange between particles (in solids mainly by lattice vibrations—phonons—and, in metals, also by free electrons).

Macroscopic description (Fourier's law). In one dimension, the rate of heat transfer (heat current) through an area A with a temperature gradient dT/dx is given by Fourier's law: q_x = dQ/dt = -k A (dT/dx). The negative sign indicates heat flows from high to low temperature. The constant k is the thermal conductivity of the material (units: W·m⁻¹·K⁻¹).

Steady vs transient conduction. In steady-state conduction the temperature at each point does not change with time; the temperature profile is time-independent and the heat current is constant. In transient (non‑steady) conduction the temperature changes with time and is described by the heat diffusion equation: ∂T/∂t = α ∇²T, where α = k/(ρ c) is thermal diffusivity, ρ is density and c is specific heat.

Common geometries and useful results. For a plane slab of thickness L with faces at temperatures T1 and T2 (steady state, uniform k): Q/t = k A (T1 - T2)/L. For composite media (layers in series) thermal resistances add: R_total = Σ(L_i/(k_i A)). For cylindrical and spherical shells there are analogous forms (see formulas list).

Important points. Heat conduction is most effective in metals (high k) because free electrons carry energy. Insulators (like wood, air, vacuum) have low k. Thermal contact resistance at interfaces can reduce heat flow. The idea of thermal resistance and conductance (G = kA/L) is useful for solving practical problems.

Fig 14 — Educational Diagram: Heat Transfer — Convection

What is convection? Convection is the mode of heat transfer that occurs in fluids (liquids and gases) due to bulk motion of the fluid. Heat is carried from one place to another by the movement of warmer and cooler portions of the fluid.

How it works (mechanism): When a part of a fluid is heated, its temperature increases, its density usually decreases and it becomes buoyant. The warmer (lighter) fluid rises while the cooler (denser) fluid moves down to replace it. This continuous motion sets up convection currents that transfer heat. In forced convection, an external agent (fan, pump) causes the fluid motion; in natural (free) convection, fluid motion is caused by buoyancy differences due to temperature gradients.

Boundary layer and heat exchange: Near a solid surface, a thin layer of fluid (the boundary layer) develops where velocity and temperature change from the surface values to the free-stream values. The thickness and properties of this layer strongly influence the rate of convective heat transfer.

Comparison with other modes: Unlike conduction (heat transfer through direct molecular contact without bulk motion) and radiation (energy transfer by electromagnetic waves), convection requires a moving fluid and combines bulk transport with conduction inside the fluid.

Fig 15 — Educational Diagram: Heat Transfer — Radiation

What is thermal radiation?
Thermal radiation is the transfer of heat in the form of electromagnetic waves (mainly infrared) emitted by all bodies because of the thermal motion of their charged particles. Radiation does not require any material medium and can occur through vacuum (for example, Sun to Earth).

Key characteristics

• No medium required — can travel through vacuum.
• All bodies emit and absorb radiation; the amount and spectral distribution depend on temperature and surface properties.
• Directionality, wavelength distribution and intensity depend on temperature; hotter bodies emit shorter-wavelength, higher-intensity radiation.

Black body and grey/real bodies
A black body is an idealized object that absorbs all incident radiation (absorptivity α = 1) and emits the maximum possible radiation for a given temperature. Real bodies are grey bodies with emissivity 0 < ε < 1 (ε = 1 for a perfect black body).

Emissivity and absorptivity
Emissivity (ε) is the ratio of radiation emitted by a body to that emitted by a black body at the same temperature. Absorptivity (α) is the fraction of incident radiation absorbed. For an object in thermal equilibrium and for each wavelength, Kirchhoff's law states ε(λ,T) = α(λ,T). For opaque surfaces (no transmission), α + ρ = 1 where ρ is reflectivity.

Stefan–Boltzmann law
The total power per unit area emitted by a black body is proportional to the fourth power of its absolute temperature:

P/A = σ T^4

For a real surface with emissivity ε, the emitted power per unit area is εσT^4.

Net radiative exchange with surroundings
If an object of area A and emissivity ε at temperature T is inside a large surrounding at temperature T_s, the net radiative power lost (or gained) by the object is approximately:

P_net = ε σ A (T^4 - T_s^4)

Spectral distribution (qualitative)
A black body emits radiation over a continuous spectrum. As T increases, the peak wavelength shifts to shorter values (objects glow from red to white). Wien's displacement law gives the peak wavelength.

Practical points

• Shiny surfaces (low emissivity) radiate and absorb less than dark/matte surfaces (high emissivity) — e.g., a shiny kettle cools slower by radiation than a black-painted one.
• Radiation is dominant when conduction and convection are small (vacuum, large temperature differences, high temperatures).
• Greenhouse effect: the atmosphere is more transparent to visible solar radiation than to outgoing terrestrial infrared, causing warming—an application of radiative transfer principles.

Fig 16.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

Statement: Newton's law of cooling states that the rate of change of temperature of a body is proportional to the difference between its temperature and the temperature of the surrounding medium, provided the temperature difference is not too large and the body is approximately uniform in temperature.

Mathematical form: Let T(t) be the temperature of the object at time t and T_e be the ambient (surrounding) temperature. Define the temperature difference θ(t)=T(t)−T_e. Then

dT/dt = −k (T − T_e) or dθ/dt = −k θ,

where k (>0) is the cooling constant (s−1) that depends on the object and the environment.

Solution (derivation sketch): Solve dθ/dt = −k θ by separation of variables:

θ(t) = θ(0) e^{−k t} ⇒ T(t) = T_e + (T_0 − T_e) e^{−k t},

where T_0 is the initial temperature at t = 0. The temperature approaches T_e exponentially as t → ∞.

Relation to heat transfer properties: For the lumped-capacitance model (object of uniform temperature), k = hA/(m c), where h is the convective heat-transfer coefficient, A is surface area, m is mass, and c is specific heat. This model is valid when the Biot number Bi = h L_c / k_material << 1 (usually Bi < 0.1).

Important derived quantities: time constant τ = 1/k (time for θ to fall to 1/e of initial value). ‘‘Half-time’’ for the temperature difference t_{1/2} = (ln 2)/k (time for θ to reduce to half).

How to determine k from measurements: If temperatures T_1 and T_2 are recorded at times t_1 and t_2, then

k = (1/(t_2 − t_1)) ln[(T_1 − T_e)/(T_2 − T_e)].

Assumptions and limitations: constant ambient temperature, object nearly uniform in temperature (lumped model), linear dependence of heat flux on temperature difference (valid when convection dominates and ΔT is not too large). For large ΔT, radiative heat transfer (proportional to T^4) becomes significant and the simple law fails. Also k may vary with time if flow conditions or properties change.

Applications (brief): cooling of hot drinks, temperature decay of heated metal pieces, cooling of electronic components, and approximate forensic estimation of time of death (using body cooling).

Fig 17 — Educational Diagram: Problems and Numerical Techniques

Overview

In the chapter Thermal Properties of Matter, many questions require setting up and solving numerical problems that combine concepts: heat, temperature change, phase change, thermal expansion, and heat transfer. Solving these reliably needs both the right formulas and sound numerical techniques (unit consistency, approximations, error checking).

Typical problem types

• Calorimetry and mixing problems (use conservation of energy: heat lost = heat gained).
• Heating with phase change (include latent heat terms where temperature remains constant).
• Linear and volume expansion problems (find change in length/volume, combined expansions).
• Steady-state conduction (use Fourier's law, series/parallel thermal resistance).
• Cooling and heating dynamics (Newton's law of cooling, exponential approach to ambient temperature).
• Thermal stress in constrained objects (use elastic modulus and thermal strain).

Problem-solving strategy

1. Read carefully and list knowns and unknowns with units.
2. Decide physics principle (energy conservation, expansion relation, heat conduction, etc.).
3. Write down corresponding formula(s) and substitute known values. For multi-stage problems (heating then melting then heating) write energy balance for each stage and sum Q.
4. Pay attention to sign convention (heat gained positive, lost negative) and to whether temperatures are absolute (Kelvin) when needed (e.g., radiation) or relative (Celsius for ΔT).
5. Solve algebraically, check units, estimate magnitude, and state answer with appropriate significant figures and units.

Numerical techniques and tips

• Unit consistency: convert masses to kg, lengths to m, temperatures to °C or K consistently (ΔT is same in °C and K).
• Significant figures: keep 3 significant figures during calculation and round final answer appropriately.
• Dimensional check: ensure both sides of equation have same dimensions.
• Approximation: use β ≈ 3α for volume expansion of isotropic solids; neglect small terms (e.g., product of two small expansions) when justified.
• Composite conduction: treat layers as thermal resistances in series: R = L/(kA); total heat flow = ΔT / sum(R).
• Linearize for small changes: when ΔT is small, use linear relations for expansion and ignore higher-order terms.
• Graphical and iterative methods: plot temperature vs time to find cooling constant or use least-squares for best-fit; for transcendental equations use simple iteration or numerical root-finding (Newton–Raphson) if needed.
• Error propagation: when result depends on measured quantities, estimate fractional uncertainties by combining relative errors (for multiplication/division) and absolute errors (for addition/subtraction).

Worked-logic templates

• Calorimetry: m1 c1 (Tfinal - T1) + m2 c2 (Tfinal - T2) + ... = 0 (include latent heat terms mL where phase change occurs).
• Expansion: ΔL = αL0ΔT, ΔV = βV0ΔT (β ≈ 3α for solids), effective expansion for composite objects found by combining contributions.
• Conduction steady-state: dQ/dt = kA(ΔT)/L. For layers: dQ/dt = ΔT / [L1/(k1A) + L2/(k2A) + ...].

Using these structured steps and numerical techniques will make solving Class 11 thermal problems systematic and reliable.

Fig 18 — Educational Diagram: Units, Dimensions and Important Constants

Overview
This topic gives the units and dimensional formulas of physical quantities used in thermal physics (temperature, heat, specific heat, thermal conductivity, coefficients of expansion, etc.) and lists important constants used in Class 11 thermal physics and kinetic theory.

Base SI units and fundamental dimensions
SI base units relevant here: metre (m) for length, kilogram (kg) for mass, second (s) for time, kelvin (K) for thermodynamic temperature, mole (mol) for amount of substance. Standard dimension symbols used below: M (mass), L (length), T (time), Θ (thermodynamic temperature), N (amount of substance, sometimes written as mol).

Common thermal quantities — definitions, SI units and dimensions

• Temperature (T): SI unit kelvin (K). Dimension: Θ.
• Heat (thermal energy) Q or ΔE: SI unit joule (J = kg·m²·s⁻²). Dimension: M L² T⁻².
• Heat capacity (C): amount of heat required to raise temperature by 1 K. Unit: J·K⁻¹. Dimension: M L² T⁻² Θ⁻¹.
• Specific heat capacity (c): heat per unit mass per K; unit: J·kg⁻¹·K⁻¹. Dimension: L² T⁻² Θ⁻¹.
• Latent heat (L, specific latent heat): energy per unit mass for phase change; unit: J·kg⁻¹. Dimension: L² T⁻².
• Coefficient of linear expansion (α): fractional change in length per K; unit: K⁻¹. Dimension: Θ⁻¹.
• Coefficient of area expansion (β) ≈ 2α, coefficient of volume expansion (γ) ≈ 3α: unit K⁻¹, dimension Θ⁻¹.
• Thermal conductivity (k): rate of heat conduction per unit area per unit temp. gradient; unit: W·m⁻¹·K⁻¹ (J·s⁻¹·m⁻¹·K⁻¹). Dimension: M L T⁻³ Θ⁻¹.
• Emissivity (ε): dimensionless (0–1).

Why dimensions matter
Dimensional formulas check equation consistency and help derive relations (up to dimensionless constants). For example, the dimension of specific heat c must make Q = mcΔT dimensionally correct.

Important relations (explained below)

• Heat required to change temperature: Q = m c ΔT
• Latent heat for phase change: Q = m L
• Linear thermal expansion: ΔL = α L ΔT
• Area and volume expansion: ΔA = β A ΔT (β ≈ 2α), ΔV = γ V ΔT (γ ≈ 3α)
• Steady conduction (one-dimensional): Q/t = (k A ΔT)/L
• Newton’s law of cooling (convective): dT/dt ∝ −(T − T_env)
• Stefan–Boltzmann law (radiation): Power radiated = ε σ A T⁴ (net exchange uses T⁴ − T_env⁴)

Practical notes
Always use absolute temperature (K) in formulas involving thermal energy (e.g., Stefan–Boltzmann, kinetic theory). For linear expansion and coefficients, ΔT in K or °C is fine because sizes of degree are equal; convert Celsius to Kelvin when needed for absolute relationships.

Important constants (values in SI units)

• Boltzmann constant, k_B = 1.380649×10⁻²³ J·K⁻¹ (dimension: M L² T⁻² Θ⁻¹). Used in kinetic theory: average kinetic energy per molecule = (3/2) k_B T.
• Universal gas constant, R = 8.314462618 J·mol⁻¹·K⁻¹ (dimension: M L² T⁻² N⁻¹ Θ⁻¹). PV = nRT.
• Avogadro’s number, N_A = 6.02214076×10²³ mol⁻¹ (dimension: N⁻¹ or mol⁻¹). Links per-molecule and per-mole quantities: R = N_A k_B.
• Stefan–Boltzmann constant, σ = 5.670374419×10⁻⁸ W·m⁻²·K⁻⁴ (dimension: M T⁻³ Θ⁻⁴). Used for blackbody radiation.
• Specific heat of water (approx.): c_water ≈ 4186 J·kg⁻¹·K⁻¹.
• Latent heat of fusion of ice: L_f ≈ 3.34×10⁵ J·kg⁻¹; latent heat of vaporization of water: L_v ≈ 2.26×10⁶ J·kg⁻¹.
• Typical thermal conductivities (approx.): copper ≈ 400 W·m⁻¹·K⁻¹, air ≈ 0.025 W·m⁻¹·K⁻¹, glass ≈ 1 W·m⁻¹·K⁻¹.

Dimension formulas (summary)

• Heat Q: [Q] = M L² T⁻²
• Specific heat c: [c] = L² T⁻² Θ⁻¹
• Heat capacity C: [C] = M L² T⁻² Θ⁻¹
• Latent heat (specific) L: [L] = L² T⁻²
• Linear expansion α: [α] = Θ⁻¹
• Thermal conductivity k: [k] = M L T⁻³ Θ⁻¹
• Stefan–Boltzmann σ: [σ] = M T⁻³ Θ⁻⁴

How to use these in problem solving
1) Check units/dimensions on both sides of an equation. 2) Use Q = mcΔT and Q = mL for calorimetry. 3) For conduction problems, use Q/t = kAΔT/L. 4) For radiative heat transfer at high temperatures, use σT⁴ relations. 5) Use k_B and N_A to move between molecular and molar descriptions (e.g., average kinetic energy per molecule vs per mole).