Thermal Properties of Matter

In this chapter: temperature and heat, temperature scales, the ideal gas equation and absolute temperature, linear, area and volume expansion, anomalous expansion of water, specific heat capacity, calorimetry, change of state and latent heat, conduction, convection, radiation, and Newton's law of cooling.

Temperature and heat

Temperature is a measure of how hot or cold a body is. Heat is energy transferred from one body to another because of a temperature difference. A body does not "contain" heat; it has internal energy, and heat is energy in transit. The SI unit of heat is the joule; the SI unit of temperature is the kelvin.

Temperature scales

The two fixed points of the modern Kelvin scale are absolute zero and the triple point of water, 273.16 K, where ice, water and water vapour coexist.

Ideal gas equation and absolute temperature

For a low-density gas, Boyle's law (PV constant at fixed T) and Charles' law (V/T constant at fixed P) combine into the ideal gas equation PV = μRT, where μ is the number of moles and R = 8.31 J mol⁻¹ K⁻¹. Plotting pressure against temperature for a gas at constant volume and extending the straight line down, the pressure becomes zero at −273.15 °C. This temperature is absolute zero, the zero of the Kelvin scale.

Thermal expansion

Metals expand more than glass, which is why a tight metal lid on a glass jar loosens under hot water. Gases have much larger expansion coefficients than solids or liquids; for an ideal gas at constant pressure, γ = 1/T. A rod clamped at both ends and then heated cannot expand, so it develops a thermal stress equal to YαΔT. This is why gaps are left between railway rails.

Anomalous expansion of water

Water contracts when heated from 0 °C to 4 °C and expands above 4 °C, so its density is maximum at 4 °C. In winter, when the surface of a lake cools, the denser water at 4 °C sinks and the top layer freezes first. The ice floats and insulates the water below, so fish and other aquatic life survive at the bottom.

Specific heat and calorimetry

The heat needed to change the temperature of a body is ΔQ = msΔT, where s is the specific heat capacity of the substance (J kg⁻¹ K⁻¹). The molar specific heat C is the heat needed per mole per kelvin. For gases, C_p (at constant pressure) is greater than C_v (at constant volume), because at constant pressure some of the heat does work in expanding the gas.

Water has the highest specific heat among common substances, about 4186 J kg⁻¹ K⁻¹. For this reason it is used in car radiators and hot-water bags, and coastal places have milder climates than inland ones.

Calorimetry rests on energy conservation: in an isolated system, heat lost by the hot bodies equals heat gained by the cold ones.

Worked example: A 0.047 kg aluminium sphere is heated to 100 °C and dropped into a 0.14 kg copper calorimeter containing 0.25 kg of water at 20 °C. The final temperature is 23 °C. Find the specific heat of aluminium. (s_water = 4.18 × 10³, s_copper = 0.387 × 10³ J kg⁻¹ K⁻¹)
Solution: Heat gained by water and calorimeter = (0.25 × 4180 + 0.14 × 387) × (23 − 20) = (1045 + 54.2) × 3 ≈ 3298 J. Heat lost by aluminium = 0.047 × s × (100 − 23) = 3.619 s. Equating, s ≈ 911 J kg⁻¹ K⁻¹.

Change of state and latent heat

During melting or boiling, the temperature stays constant even though heat is being supplied; the heat goes into breaking the bonds between molecules. The heat needed per unit mass for a change of state at constant temperature is the latent heat L, so Q = mL.

• For water, the latent heat of fusion L_f = 3.33 × 10⁵ J kg⁻¹ and the latent heat of vaporisation L_v = 22.6 × 10⁵ J kg⁻¹.
• The boiling point rises with pressure. A pressure cooker cooks faster because water boils above 100 °C inside it. At high altitudes, where pressure is low, water boils below 100 °C.
• The melting point of ice falls with increasing pressure. A wire loaded at both ends passes through a slab of ice without cutting it into two, because the ice melts under the wire and refreezes above it. This is regelation.
• Some substances pass directly from solid to vapour, which is sublimation: dry ice (solid CO₂) and iodine are examples.

Worked example: How much heat is needed to turn 0.5 kg of water at 20 °C completely into steam at 100 °C? (s = 4186 J kg⁻¹ K⁻¹, L_v = 22.6 × 10⁵ J kg⁻¹)
Solution: Heating to 100 °C: 0.5 × 4186 × 80 = 167 440 J. Boiling: 0.5 × 22.6 × 10⁵ = 1 130 000 J. Total ≈ 1.30 × 10⁶ J. Most of the heat goes into the change of state, not the temperature rise.

Heat transfer

Conduction

Conduction transfers heat between neighbouring parts of a body through molecular collisions, without bulk movement of matter. For a bar of length L and cross-section A with its ends at T_C and T_D (in steady state),

Cooking pots have copper bottoms because copper conducts well. Houses with thick walls or double-glazed windows stay comfortable because trapped air is a poor conductor.

Convection

Convection is heat transfer by the actual movement of the fluid. In natural convection, heated fluid becomes less dense and rises while cooler fluid sinks, as in sea breezes during the day, land breezes at night, and the trade winds. In forced convection the fluid is pushed by a pump or fan, as in a car's cooling system or the human circulatory system.

Radiation

Radiation transfers energy as electromagnetic waves and needs no medium, which is how the Sun's energy reaches the Earth. Every body radiates, and a perfect absorber, called a black body, is also the best emitter.

As a body gets hotter, the wavelength of maximum emission moves to shorter values: a heated iron rod glows dull red, then orange, then yellow-white. Taking λ_m ≈ 480 nm for the Sun gives a surface temperature of about 2.9 × 10⁻³/(4.8 × 10⁻⁷) ≈ 6000 K.

Newton's law of cooling

The rate of loss of heat of a body is proportional to the difference between its temperature T and that of its surroundings T_s, provided the difference is small:

For example, if a body cools from 80 °C to 60 °C in 10 minutes in a room at 20 °C, then 20/10 = K′(70 − 20), so K′ = 0.04 per minute. In the next 10 minutes it cools to T where (60 − T)/10 = 0.04[(60 + T)/2 − 20], which gives T ≈ 46.7 °C. It falls by only about 13 °C this time, because the body is now closer to room temperature.

Common mistakes: (1) Using kelvin in Stefan's law but Celsius in the calculation; T⁴ needs absolute temperature. (2) Forgetting the latent heat step in calorimetry problems with ice or steam, or not checking whether all the ice melts. (3) Writing γ = α or β = α instead of 3α and 2α. (4) Applying Newton's law of cooling to large temperature differences. (5) Thinking the density of water is highest at 0 °C; it is highest at 4 °C.

JEE and NEET focus

• Scale conversions and the −40° point where Celsius and Fahrenheit agree.
• Expansion: change in length, area, volume; thermal stress; the relation between α, β and γ.
• Calorimetry with change of state, especially mixing ice with water or steam with water.
• Conduction through slabs in series and parallel (thermal resistance L/KA), and the temperature at the junction.
• Stefan's law and Wien's law ratio questions, and Newton's law of cooling with the average-temperature formula.

Practice questions