Thermal Properties: Expansion, Calorimetry & Heat Transfer

Thermal Expansion

When a body is heated, its dimensions increase due to increased molecular vibrations. Three types of expansion:

Linear Expansion: $\Delta L = L_0 \alpha \Delta T \quad \Rightarrow \quad L = L_0(1 + \alpha \Delta T)$

• $\alpha$: coefficient of linear expansion (K$^{-1}$)
• Typical values: steel $\approx 12 \times 10^{-6}$ K$^{-1}$, aluminium $\approx 23 \times 10^{-6}$ K$^{-1}$

Area Expansion: $\Delta A = A_0 \beta \Delta T \quad \Rightarrow \quad A = A_0(1 + \beta \Delta T)$

• $\beta$: coefficient of area expansion; $\beta = 2\alpha$

Volume Expansion: $\Delta V = V_0 \gamma \Delta T \quad \Rightarrow \quad V = V_0(1 + \gamma \Delta T)$

• $\gamma$: coefficient of volume expansion; $\gamma = 3\alpha$

Relation: $\alpha : \beta : \gamma = 1 : 2 : 3$

Expansion of Holes and Cavities

A hole or cavity in a solid expands exactly as if it were filled with the same material. A metal ring heated gets a larger inner diameter, not smaller.

Thermal Stress

When expansion is constrained: $\sigma = Y\alpha\Delta T$, $F = YA\alpha\Delta T$ (covered in JME-08 but relevant here).

Anomalous Expansion of Water

Water has maximum density at 4 degrees C. Between 0 degrees C and 4 degrees C, water contracts on heating (anomalous). Above 4 degrees C, it expands normally. This is crucial for aquatic life — lakes freeze from the top while the bottom stays at 4 degrees C.

Calorimetry

Heat capacity: $C = \frac{\Delta Q}{\Delta T}$ (J K$^{-1}$)

Specific heat capacity: $c = \frac{\Delta Q}{m \Delta T}$ (J kg$^{-1}$ K$^{-1}$)

Molar heat capacity: $C_m = \frac{\Delta Q}{n \Delta T}$ (J mol$^{-1}$ K$^{-1}$)

Principle of Calorimetry (conservation of energy): $\text{Heat lost by hot body} = \text{Heat gained by cold body}$ $m_1 c_1 (T_1 - T_f) = m_2 c_2 (T_f - T_2)$

Latent Heat

Latent heat of fusion: $L_f$ — heat required to change unit mass from solid to liquid at constant temperature. $Q = mL_f$

• Water: $L_f = 3.34 \times 10^5$ J kg$^{-1}$ = 80 cal g$^{-1}$

Latent heat of vaporization: $L_v$ — heat required to change unit mass from liquid to gas at constant temperature. $Q = mL_v$

• Water: $L_v = 2.26 \times 10^6$ J kg$^{-1}$ = 540 cal g$^{-1}$

Heat Transfer Mechanisms

1. Conduction: $\frac{dQ}{dt} = \frac{kA(T_1 - T_2)}{L}$

• $k$: thermal conductivity (W m$^{-1}$ K$^{-1}$)
• $A$: cross-sectional area
• $L$: length of conductor
• Dimensional formula of $k$: [M L T$^{-3}$ K$^{-1}$]

Thermal resistance: $R = L/(kA)$ (analogous to electrical resistance)

For slabs in series: $R_{\text{total}} = R_1 + R_2 + ...$ For slabs in parallel: $1/R_{\text{total}} = 1/R_1 + 1/R_2 + ...$

2. Convection: Heat transfer by bulk movement of fluid. Natural convection (density-driven) and forced convection (fan/pump driven). Not easily quantified by simple formulas at the JEE level.

3. Radiation: Stefan-Boltzmann Law: $E = \sigma T^4 \quad \text{(power per unit area for a blackbody)}$ $P = \sigma A T^4 \quad \text{(total power radiated)}$

• $\sigma = 5.67 \times 10^{-8}$ W m$^{-2}$ K$^{-4}$ (Stefan-Boltzmann constant)

For a non-blackbody: $P = e\sigma A T^4$ (where $e$ is emissivity, $0 \leq e \leq 1$)

Newton's Law of Cooling (for small temperature excess): $\frac{dT}{dt} = -bK(T - T_s)$ where $T_s$ is surrounding temperature. Approximately: $\frac{T_1 - T_2}{t} = k\left(\frac{T_1 + T_2}{2} - T_s\right)$

Wien's Displacement Law: $\lambda_{\max} T = b = 2.898 \times 10^{-3} \text{ m K}$ Higher temperature shifts peak wavelength to shorter (bluer) wavelengths.

---