JEE Physics Thermal Properties Of Matter Notes Set 05

Modes of Heat Transfer

Heat is energy in transit because of a temperature difference, always from the hotter to the colder body. It travels by conduction (through matter, with no bulk movement of the particles), convection (by bulk movement of a fluid) and radiation (electromagnetic waves, needing no medium). Convection is outside the JEE syllabus; conduction and radiation are the main topics.

Conduction

In conduction the particles vibrate about fixed positions and pass the energy on through collisions (and, in metals, through free electrons). For a slab of area \(A\), the heat crossing a section per unit time is \[ \frac{Q}{t} = -KA\frac{dT}{dx} \] where \(K\) is the thermal conductivity of the material (W/m K) and \(\frac{dT}{dx}\) the temperature gradient. The minus sign shows that heat flows towards lower temperature.

1. Steady state means the temperature at every section no longer changes with time (the temperature still varies from place to place). This is different from thermal equilibrium, where the temperature is the same everywhere.
2. In steady state with all other faces insulated, the same heat current crosses every section, so the gradient is constant and \(\frac{Q}{t} = \frac{KA(T_H - T_C)}{L}\).
3. Thermal resistance: \(R = \frac{L}{KA}\), so \(\frac{Q}{t} = \frac{\Delta T}{R}\). This is analogous to Ohm's law with temperature difference as voltage and heat current as current.
4. In a non-steady state, a section absorbs heat equal to the difference between the heat entering and leaving it, and its temperature rises at the rate \(\frac{kA(g_1 - g_2)}{C}\), where \(g_1\) and \(g_2\) are the gradients at the two ends and \(C\) is its heat capacity.

Slabs in Series and Parallel

1. The temperature drop across a layer in series is larger for the poorer conductor.
2. At the junction of two rods of equal size, \(T_j = \frac{K_1T_1 + K_2T_2}{K_1 + K_2}\); for lengths \(L_1, L_2\): \(T_j = \frac{K_1L_2T_1 + K_2L_1T_2}{K_1L_2 + K_2L_1}\).
3. For a Y-shaped joint, reduce the network step by step: parallel branches first, then add series resistances.

Worked example: a wall made of 2.5 cm of wood (K = 0.125 W/m K) and 25 cm of brick (K = 1.0 W/m K), with 30°C between its faces. Resistance per unit area is \(\frac{0.025}{0.125} + \frac{0.25}{1.0} = 0.45\ \text{m}^2\text{K/W}\), so the heat flux is \(\frac{30}{0.45} = 66.7\ \text{W/m}^2\). The wood layer takes \(\frac{0.2}{0.45} \times 30 = 13.3^\circ\)C of the drop.

Radial Conduction

1. Cylindrical shell (inner radius \(r_1\), outer radius \(r_2\), length \(\ell\)), heat flowing radially: \(\frac{Q}{t} = \frac{2\pi K\ell(T_1 - T_2)}{\ln(r_2/r_1)}\), so \(R = \frac{\ln(r_2/r_1)}{2\pi K\ell}\).
2. Spherical shell: \(\frac{Q}{t} = \frac{4\pi K\,r_1r_2(T_1 - T_2)}{r_2 - r_1}\), so \(R = \frac{r_2 - r_1}{4\pi K\,r_1r_2}\). The temperature varies as \(\frac{1}{r}\) across the shell, so the mid-temperature surface is at \(\frac{2r_1r_2}{r_1 + r_2}\).
3. The method is to take a thin shell of thickness \(dr\), find its resistance \(\frac{dr}{K\cdot\text{area}(r)}\) and integrate, because the area depends on \(r\).
4. Flow along the axis of a hollow tube uses the annulus cross-section \(\pi(R_2^2 - R_1^2)\) and the length, as in a slab.

Conduction that is not Steady

A container of water of mass \(m\), specific heat \(s\) connected to a heat source at \(T_0\) by a rod of length \(l\), area \(A\) and conductivity \(K\) obeys \(ms\frac{dT}{dt} = \frac{KA(T_0 - T)}{l}\). The solution is \(T = T_0 - (T_0 - T_i)e^{-t/\tau}\) with \(\tau = \frac{msl}{KA}\). In the same way, two bodies of heat capacities \(C_1\), \(C_2\) joined by a rod approach each other's temperature with \(\Delta T = (\Delta T)_0e^{-\lambda t}\), \(\lambda = \left(\frac{1}{C_1} + \frac{1}{C_2}\right)\frac{KA}{l}\). A layer of ice growing on a pond thickens as \(x^2 = \frac{2K\Delta T\,t}{\rho L}\).

Convection

Convection transfers heat by actual movement of a heated fluid. It can be natural (a heated fluid expands, becomes lighter and rises, as in sea breezes and a heated room) or forced (a fan or pump). Land and sea breezes and ocean currents are examples.

Radiation

Every body at a temperature above absolute zero emits electromagnetic radiation. Thermal radiation travels at the speed of light, needs no medium and obeys the laws of reflection and refraction. A body that is in thermal equilibrium with its surroundings still radiates, but absorbs the same amount (Prevost's theory of exchange).

1. Absorptive power \(a\): the fraction of the incident radiation that is absorbed. For a perfectly black body \(a = 1\). Absorptivity, reflectivity and transmissivity add to 1.
2. Emissive power \(E\): energy radiated per unit area per unit time. Spectral emissive power \(E_\lambda\) is the emissive power per unit wavelength. Emissivity \(e\) is the ratio of the emissive power of a body to that of a black body at the same temperature.
3. Kirchhoff's law: for a body in thermal equilibrium, the ratio \(\frac{E}{a}\) is the same for all bodies at a given temperature and equals the emissive power of a black body. Hence \(e = a\), and good absorbers are good emitters, bad absorbers are bad emitters.
4. A black body absorbs all radiation falling on it and emits the maximum possible at its temperature. A small hole in a hollow enclosure behaves like a black body.

Wien's Displacement Law and Stefan's Law

1. Wien's law: the wavelength of maximum intensity satisfies \(\lambda_mT = b\), where \(b = 2.9\times10^{-3}\ \text{m K}\). As temperature rises, the peak moves to shorter wavelength, and the peak frequency is proportional to \(T\). The total area under the curve increases rapidly with \(T\), but the curves for different temperatures never cross.
2. Stefan-Boltzmann law: the power radiated by a black body is \(P = \sigma AT^4\), with \(\sigma = 5.67\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}\). For a real body \(P = e\sigma AT^4\).
3. Net power lost by a body at \(T\) in surroundings at \(T_0\): \(P = e\sigma A(T^4 - T_0^4)\), with temperatures in kelvin.
4. Power radiated by a body of radius \(r\) varies as \(r^2\), but its heat capacity as \(r^3\), so the rate of cooling varies as \(\frac{1}{r}\).
5. Equilibrium temperature of a planet or sphere in sunlight: absorbed power \(\pi r^2I\) equals radiated \(4\pi r^2\sigma T^4\), so \(T^4 = \frac{I}{4\sigma}\) for a black body.

Worked examples: a body of emissivity 0.75, area 300 cm\(^2\) at 227°C in surroundings at 27°C loses \(0.75 \times 5.67\times10^{-8} \times 0.03 \times (500^4 - 300^4) = 69.4\) W. If the sun's radiation peaks near 470 nm, its surface temperature is \(\frac{2.9\times10^{-3}}{4.7\times10^{-7}} \approx 6200\) K.

Newton's Law of Cooling

1. When the temperature excess over the surroundings is small, the rate of loss of heat is proportional to the excess: \(\frac{dT}{dt} = -k(T - T_0)\), where \(k = \frac{4e\sigma AT_0^3}{ms}\).
2. The solution is \(T - T_0 = (T_i - T_0)e^{-kt}\), an exponential decay that never quite reaches \(T_0\). The graph of \(\ln(T - T_0)\) against \(t\) is a straight line with slope \(-k\).
3. Approximate method for a small interval (the average temperature is used): \(\frac{T_1 - T_2}{t} = k\left(\frac{T_1 + T_2}{2} - T_0\right)\). This is accurate when the fall in temperature is small compared with the excess.
4. The law is valid only for small temperature differences, with forced convection or in a steady wind, and when the surroundings stay at a constant temperature. It is also approximately followed with natural convection if the excess is small.

Worked example: a liquid cools from 80°C to 64°C in 5 minutes in surroundings at 30°C. Exact method: \(k = \frac{1}{5}\ln\frac{50}{34} = 0.077\ \text{min}^{-1}\), so it takes \(\frac{1}{k}\ln\frac{34}{22} \approx 5.6\) min to reach 52°C. With the approximate method \(k = \frac{16/5}{72 - 30} = 0.076\ \text{min}^{-1}\), and in the next 5 minutes the liquid cools from 64°C to about 53°C.

Common Mistakes

1. Using Celsius temperatures in Stefan's law. Always convert to kelvin.
2. Forgetting that in series every layer carries the same heat current, and in parallel every branch has the same temperature difference.
3. Using the slab formula for radial flow through a cylinder or a sphere.
4. Assuming radiation stops when a body reaches the temperature of its surroundings.