Thermal Properties

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Thermal Properties — Key Facts Heat: $Q = mc\Delta T$ (sensible heat); latent heat $L = Q/m$ (phase change at constant temperature) Ideal gas equation: $PV = nRT$; for air at STP: $P = 1$ atm, $T = 273$ K, $V = 22.4$ L/mol Conduction: $\frac{dQ}{dt} = -kA\frac{dT}{dx}$; thermal conductivity $k$ (W/m·K): copper $\approx 400$, glass $\approx 1$, air $\approx 0.02$ Stefan-Boltzmann: $P = \varepsilon\sigma AT^4$ ($\sigma = 5.67 \times 10^{-8}$ W/m²K⁴); $\varepsilon$ = emissivity (0 to 1) ⚡ Exam tip: In phase change problems, temperature doesn’t change during the phase change — all energy goes into latent heat

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of heat transfer and thermodynamics.

Thermal Properties — CUET Physics Study Guide

Heat is energy transferred due to temperature difference. The three mechanisms are conduction, convection, and radiation. Conduction occurs through a material without bulk motion — microscopically, it involves lattice vibration and free electron transport. Metals are good conductors because of free electrons; glass and plastics are poor conductors (insulators).

Heat Conduction: The rate of heat flow (heat current) is $H = \frac{dQ}{dt} = -kA\frac{dT}{dx}$, where $k$ is thermal conductivity, $A$ is cross-sectional area, and $\frac{dT}{dx}$ is temperature gradient. The negative sign indicates heat flows from hot to cold. For a rod of length $L$ with ends at $T_1$ and $T_2$: $H = kA\frac{T_1 - T_2}{L}$.

Convection: Occurs in fluids involving bulk motion of the medium. Natural convection: density differences drive the flow (hot air rises). Forced convection: a fan or pump moves the fluid. The heat transfer rate is $Q = hA\Delta T$, where $h$ is the convective heat transfer coefficient.

Radiation: All bodies emit electromagnetic radiation. The power radiated is $P = \varepsilon\sigma AT^4$ (Stefan-Boltzmann law). A perfect black body has $\varepsilon = 1$. The Sun radiates approximately as a black body at 5800 K. The peak wavelength follows Wien’s displacement law: $\lambda_{\max}T = b = 2.9 \times 10^{-3}$ m·K.

Newton’s Law of Cooling: Rate of cooling $\frac{dT}{dt} = -k(T - T_{\text{surr}})$. The solution is $T(t) = T_{\text{surr}} + (T_0 - T_{\text{surr}})e^{-kt}$. This applies for small temperature differences ($< 30°C$). For larger differences, the cooling is faster than exponential.

Thermal Expansion: Most materials expand when heated. Linear expansion: $\Delta L = \alpha L_0 \Delta T$. Area expansion: $\Delta A = 2\alpha A_0 \Delta T$. Volume expansion: $\Delta V = \gamma V_0 \Delta T$, where $\gamma = 3\alpha$ for isotropic materials. Water is anomalous — it contracts from 0°C to 4°C, then expands.

Example: A copper rod 2 m long at 20°C is heated to 100°C. Find elongation. $\Delta L = \alpha L_0 \Delta T = 17 \times 10^{-6} \times 2 \times 80 = 2.72 \times 10^{-3}$ m = 2.72 mm.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Thermal Properties — Complete CUET Physics Notes

Compound Bar and Bimetallic Strip: Two metals with different $\alpha$ joined together. On heating, the bar bends with the higher-$\alpha$ metal on the outer (larger radius) curve. This principle is used in thermostats — the bending contacts close or open an electrical circuit. The radius of curvature $R = \frac{d}{\Delta\alpha \cdot \Delta T \cdot (m+1)}$, where $d$ is thickness and $m$ is the ratio of widths.

Heat Conduction Through Composite Wall: Consider three walls in series with thermal resistances $R_1 = \frac{d_1}{k_1 A}$, $R_2 = \frac{d_2}{k_2 A}$, $R_3 = \frac{d_3}{k_3 A}$. Total resistance $R = R_1 + R_2 + R_3$. Heat current $H = \frac{\Delta T}{R}$. For walls in parallel, the total area is summed.

Critical Radius of Insulation: For a cylindrical pipe of radius $r_i$ covered with insulation of outer radius $r_o$, there is a critical radius $r_c = \frac{k_{\text{ins}}}{h}$ (for convection boundary). If $r_o < r_c$, adding insulation increases heat loss (reduces resistance more than it reduces temperature gradient). If $r_o > r_c$, further insulation reduces heat loss.

Black Body Radiation: Planck’s law reduces to Rayleigh-Jeans law at long wavelengths and Wien’s law at short wavelengths. The energy density of black body radiation: $u = aT^4$ where $a = \frac{4\sigma}{c}$ is the radiation constant. The radiation pressure on a perfectly reflecting surface is $p = \frac{2u}{3} = \frac{4\sigma T^4}{3c}$.

Entropy: Change in entropy $\Delta S = \int \frac{dQ_{\text{rev}}}{T}$. For a reversible process, $\Delta S = 0$ (isentropic). The second law states $\Delta S_{\text{universe}} \geq 0$. For an isothermal expansion of an ideal gas: $\Delta S = \frac{Q}{T} = nR\ln\frac{V_f}{V_i}$.

Carnot Refrigerator: Coefficient of Performance $\text{COP} = \frac{T_c}{T_h - T_c}$, where $T_c$ is cold reservoir temperature and $T_h$ is hot reservoir temperature. For a refrigerator, work input $W = \frac{Q_c}{\text{COP}} = Q_c \times \frac{T_h - T_c}{T_c}$. Real refrigerators have COP of 2–4; ideal (Carnot) would be higher.

Convection Without a Medium — Radiation Heat Transfer: Unlike conduction and convection, radiation requires no medium. It can transfer heat through vacuum. The absorptivity equals emissivity at each wavelength (Kirchhoff’s law). A perfect black body absorbs all radiation ($\varepsilon = 1$, absorptivity $= 1$). A white polished surface reflects most radiation ($\varepsilon \approx 0$).

CUET Exam Patterns (2022–2024):

• Stefan’s law and Newton’s cooling law appear frequently (1–2 marks)
• Composite wall heat conduction tested in 2023 (2 marks)
• Thermal expansion numerical problems are common
• Anomalous behaviour of water (4°C) is a CUET favourite
• Common mistakes: confusing $Q = mc\Delta T$ with $Q = mL$ (latent heat), wrong units for $k$

⚡ Key insight: When solving heat transfer problems with multiple mechanisms, draw the temperature gradient. In a composite wall, temperature drop across each layer is proportional to that layer’s thermal resistance. Always check whether phase change is occurring — during melting/freezing or boiling/condensation, temperature stays constant.

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