JEE Physics Thermal Properties Of Matter Notes Set 06

Temperature Scales

Temperature is read on a scale with two fixed points. On Celsius the ice point is 0 and the steam point 100, on Fahrenheit they are 32 and 212, and on Kelvin 273.15 and 373.15. Any two linear scales are related by the fact that the same fraction of the interval is covered:

\( \dfrac{C}{5}=\dfrac{F-32}{9}=\dfrac{K-273.15}{5} \)

A faulty thermometer with fixed points at readings \( L \) and \( U \) gives the true temperature through \( \dfrac{\text{reading}-L}{U-L}=\dfrac{\text{true}}{100} \).

Thermal Expansion of Solids

On heating, atoms vibrate with larger amplitude and the mean separation grows, so every dimension of a solid increases. For small temperature changes:

The three coefficients of an isotropic solid are in the ratio \( \alpha:\beta:\gamma=1:2:3 \). A hole, a cavity or the gap in a ring expands exactly as the surrounding material would, as if the whole body were photographically enlarged. If \( \alpha \) varies with position or temperature, the expansion is found by integrating: \( \Delta L=\int\alpha L_0\,dT \).

Heating changes the density: \( \rho=\rho_0(1-\gamma\Delta T) \) for small changes. A body with moment of inertia \( I \) has \( \Delta I=2\alpha I\Delta T \) because \( I\propto l^{2} \).

Composite Rods, Bimetallic Strips and Scales

1. Two rods of lengths \( L_1,L_2 \) joined end to end have an average coefficient \( \alpha=\dfrac{\alpha_1L_1+\alpha_2L_2}{L_1+L_2} \).
2. Two rods whose difference in length stays the same at all temperatures need \( \alpha_1L_1=\alpha_2L_2 \).
3. A bimetallic strip bends on heating, with the metal of larger \( \alpha \) on the outside of the curve. The radius of curvature for thickness \( d \) is \( R=\dfrac{d}{(\alpha_2-\alpha_1)\Delta T} \).
4. A metal scale calibrated at a lower temperature reads too small a value when hot (its divisions have grown), and too large a value when colder than its calibration temperature.
5. A pendulum clock runs slow when the temperature rises. The fractional change in time period is \( \dfrac{\Delta T_p}{T_p}=\tfrac12\alpha\Delta\theta \), so the time lost in a time \( t \) is \( \tfrac12\alpha\Delta\theta\,t \).

Expansion of Liquids

Liquids have only a volume expansion coefficient \( \gamma_r \) (real). When a liquid is heated in a container, the container also expands, so the observed rise is the apparent expansion:

\( \gamma_a=\gamma_r-\gamma_g \)

where \( \gamma_g=3\alpha_g \) is the volume coefficient of the vessel. If \( \gamma_r>\gamma_g \) the level rises, and if \( \gamma_r<\gamma_g \) it falls. Water behaves anomalously between 0 °C and 4 °C: it contracts on heating, and has maximum density at 4 °C. A solid floating in a liquid sinks deeper when the liquid expands more than the solid, because the liquid density falls faster.

Heat, Specific Heat and Calorimetry

Heat is energy in transit due to a temperature difference. The heat needed to change the temperature of mass \( m \) by \( \Delta T \) is \( Q=ms\Delta T \); for a temperature-dependent specific heat, \( Q=m\int s\,dT \). The heat capacity \( C=ms \) and the water equivalent \( W=ms/s_w \) (the mass of water that absorbs the same heat) simplify problems with calorimeters. Specific heat is infinite during a change of state because the temperature does not change, and zero for a rapid adiabatic change in which no heat flows.

During a change of state \( Q=mL \). For water: \( L_f=80\ \text{cal/g}=3.36\times10^{5}\ \text{J/kg} \), \( L_v=540\ \text{cal/g}=2.26\times10^{6}\ \text{J/kg} \), \( s_{water}=4200\ \text{J/kg K} \) and \( s_{ice}\approx2100\ \text{J/kg K} \). The mechanical equivalent of heat is \( J=4.2\ \text{J/cal} \).

Principle of calorimetry: in an isolated system, heat lost by hot bodies equals heat gained by cold bodies. For a mixture of ice, water and steam always check first whether the ice can melt completely and whether the steam can condense completely; if neither happens fully the final temperature is the melting or boiling point. On a heating curve of temperature against heat supplied at a steady rate, flat portions mark changes of state, the slope of a sloping portion is inversely proportional to the specific heat, and the length of a flat portion is proportional to the latent heat.

For a flow heater of power \( P \) and efficiency \( \eta \): \( \eta P=\dot m\,s\,\Delta T \). When a falling or sliding body loses energy as heat, equate the lost mechanical energy to \( ms\Delta T \) or \( mL \).

Conduction

In conduction, energy passes from particle to particle with no bulk movement of the material. In steady state the heat current through a slab of area \( A \) and length \( l \) is

\( H=\dfrac{dQ}{dt}=\dfrac{KA(T_1-T_2)}{l}=\dfrac{T_1-T_2}{R_{th}},\qquad R_{th}=\dfrac{l}{KA} \)

Thermal resistances combine like electrical resistances and temperature difference plays the role of potential difference.

In series the same heat current flows in each part, so the temperature drop is larger across the poorer conductor, and the interface temperature is \( \dfrac{K_1l_2T_1+K_2l_1T_2}{K_1l_2+K_2l_1} \). In steady state the temperature of a uniform insulated bar falls linearly along its length. For radial flow through a spherical shell between radii \( r_1,r_2 \), \( H=4\pi K\dfrac{r_1r_2}{r_2-r_1}(T_1-T_2) \). While a layer of ice grows on a lake, the thickness \( x \) obeys \( t=\dfrac{\rho L}{2K\Delta T}x^{2} \).

Convection and Radiation

Convection transfers heat by bulk motion of a fluid, and so needs a medium. Radiation is carried by electromagnetic waves and needs no medium. For a surface, absorptive power \( a \), reflective power \( r \) and transmissive power \( t \) satisfy \( a+r+t=1 \). A perfect black body has \( a=1 \).

1. Kirchhoff's law: at a given temperature the ratio of emissive power to absorptive power is the same for all bodies and equals the emissive power of a black body. Good absorbers are good emitters; bad absorbers are good reflectors.
2. Stefan's law: a body of emissivity \( e \) and area \( A \) radiates \( P=e\sigma AT^{4} \). In surroundings at \( T_0 \) the net power is \( e\sigma A(T^{4}-T_0^{4}) \), with \( \sigma=5.67\times10^{-8}\ \text{W/m}^2\text{K}^4 \).
3. Wien's displacement law: \( \lambda_mT=b \) with \( b=2.9\times10^{-3}\ \text{m K} \). The spectral curve of a hotter body peaks at a shorter wavelength, and the area under the curve grows as \( T^{4} \).
4. Newton's law of cooling: for a small temperature difference the rate of cooling is proportional to the excess over the surroundings, \( \dfrac{dT}{dt}=-k(T-T_0) \), giving \( T-T_0=(T_i-T_0)e^{-kt} \). For a short interval use \( \dfrac{T_1-T_2}{t}=k\left(\dfrac{T_1+T_2}{2}-T_0\right) \).
5. Rate of fall of temperature by radiation is \( \dfrac{e\sigma A(T^{4}-T_0^{4})}{ms} \), so bodies with larger area-to-mass ratio cool faster. Small spheres of the same material cool at a rate proportional to \( 1/r \).
6. Intensity from a small source of power \( P \) at distance \( d \) is \( \dfrac{P}{4\pi d^{2}} \).