Thermal Properties of Matter

Heat is energy that flows between two bodies because they are at different temperatures. Temperature is the physical quantity that tells us which way heat will flow — heat always flows from a body at higher temperature to one at lower temperature, until they reach a common temperature called thermal equilibrium. This chapter studies how temperature is measured, how objects respond to heating (they expand, they store heat, they change state), and how heat actually travels from one place to another.

Keep the distinction clear for the exam: heat is energy in transit (measured in joule), while temperature is the degree of hotness (measured in kelvin or degree Celsius). A hot body does not 'contain' heat; it contains internal energy, and heat is what flows when there is a temperature difference.

Temperature measures the average kinetic energy of the molecules of a substance. The SI unit of temperature is the kelvin (K); Celsius (°C) is the commonly used practical scale. Heat is the energy transferred due to temperature difference and is measured in joule (older texts also use calorie; 1 cal = 4.186 J).

A thermometer works because some measurable property of a substance changes uniformly with temperature — this is called a thermometric property. Examples: the length of a mercury column, the pressure of a gas at constant volume, the resistance of a metal wire. The constant-volume gas thermometer is the most accurate and is used as the standard against which other thermometers are calibrated, because all gases (at low pressure) expand by nearly the same fraction for the same rise in temperature, independent of the gas used.

A thermometer is calibrated by choosing two fixed, reproducible reference temperatures and dividing the interval between them into a fixed number of equal parts. On the Celsius scale the ice point (melting point of pure ice at 1 atm) is 0 °C and the steam point (boiling point of water at 1 atm) is 100 °C, with 100 equal divisions between them. On the Fahrenheit scale these are 32 °F and 212 °F, with 180 divisions.

For a linear thermometric property X, temperature is found from t_X = [(X_T − X₀)/(X₁₀₀ − X₀)] × 100 °C, where X₀ and X₁₀₀ are the property's values at the ice point and steam point. Since the same interval is divided differently on different scales, a general relation connects any two scales: the fraction of the way up from the lower fixed point to the upper fixed point is the same number on every scale.

At low density, all gases obey the ideal gas equation PV = μRT, where P is pressure, V is volume, μ is the number of moles, R is the universal gas constant (8.31 J mol⁻¹ K⁻¹) and T is absolute temperature. A constant-volume gas thermometer plots pressure against temperature; extrapolating this straight-line graph backwards, all gases (in the low-pressure, ideal limit) reach zero pressure at the same temperature, −273.15 °C. This temperature is called absolute zero, and it defines the zero of the Kelvin scale. The Kelvin scale (SI unit of temperature) is therefore related to Celsius by T(K) = t(°C) + 273.15, and its size of one degree is identical to that of the Celsius scale.

The triple point of water — the unique temperature and pressure at which ice, water and water vapour coexist in equilibrium — is used as the single fixed reference point for the modern Kelvin scale and is assigned the value 273.16 K exactly.

Most substances expand when heated because the average distance between their molecules increases as thermal vibrations grow with temperature. Three types of expansion are defined for solids: linear expansion (change in length), area (superficial) expansion (change in area), and volume (cubical) expansion (change in volume).

For a small temperature change ΔT, the change is proportional to the original dimension and to ΔT:

Here α, β and γ are the coefficients of linear, area and volume expansion respectively, each with SI unit K⁻¹ (or °C⁻¹). For an isotropic solid, it can be shown that β = 2α and γ = 3α, because area scales as length² and volume as length³, and the cross terms in α·ΔT are negligible for small ΔT.

Liquids only have a volume expansion coefficient (they have no fixed shape). Because the containing vessel also expands, we distinguish apparent expansion of the liquid (relative to the vessel) from its real (absolute) expansion: γ_real = γ_apparent + γ_vessel.

Anomalous expansion of water: water contracts on heating from 0 °C to 4 °C and expands beyond 4 °C — its density is maximum at 4 °C. This is why ice forms first at the top of a lake in winter and floats, insulating the water below and letting aquatic life survive; the water at the bottom stays near 4 °C, its densest state.

Gases expand far more than solids or liquids for the same rise in temperature, and at constant pressure obey V = V₀(1 + γΔT) with γ nearly the same (1/273 per °C) for all gases at low pressure.

The heat capacity of a body is the amount of heat needed to raise its temperature by 1 K: S = ΔQ/ΔT. Specific heat capacity is heat capacity per unit mass, s = S/m = ΔQ/(mΔT), and molar specific heat capacity is per mole, C = ΔQ/(μΔT). Specific heat capacity is a property of the material, not of a particular body, and it can depend on the process (constant pressure vs constant volume for gases — hence C_P and C_V).

Water has an unusually high specific heat capacity (4186 J kg⁻¹ K⁻¹, i.e. 1 cal g⁻¹ °C⁻¹). This is why coastal regions have a milder climate than inland regions at the same latitude: large water bodies absorb or release a great deal of heat with only a small temperature change, moderating the surrounding air temperature; it is also why water is used as a coolant in car radiators.

Calorimetry is the measurement of heat exchanged. It rests on the principle of calorimetry (principle of mixtures): when two bodies at different temperatures are mixed in an isolated system (no heat lost to surroundings), heat lost by the hotter body equals heat gained by the colder body, because energy is conserved.

Heat lost by hot body = Heat gained by cold body.

A calorimeter is an insulated vessel (usually with a stirrer and a low specific heat, e.g. copper) used to measure specific heat or latent heat experimentally by mixing a known mass at a known temperature with the substance under test and recording the common final temperature.

Worked method: a hot substance of mass m₁, specific heat s₁, at temperature T₁ is mixed with a colder substance of mass m₂, specific heat s₂, at temperature T₂ (T₁ > T₂). If T_f is the common final temperature, m₁s₁(T₁ − T_f) = m₂s₂(T_f − T₂), which can be solved for any one unknown, typically s₁ or T_f.

Matter exists as solid, liquid or gas, and can change from one state to another at fixed temperature and pressure conditions. Melting is solid → liquid at the melting point; freezing is liquid → solid at the freezing point (same temperature as melting point for a given pressure). Boiling/vaporisation is liquid → gas at the boiling point; condensation is the reverse. Sublimation is the direct change of solid to vapour without passing through the liquid state (e.g. camphor, dry ice).

During a change of state, temperature stays constant even though heat is continuously supplied — this heat goes into breaking intermolecular bonds (changing the state), not into raising kinetic energy/temperature. The heat required per unit mass for this, at constant temperature, is called latent heat: L = Q/m. Latent heat of fusion (L_f) is for melting/freezing (for ice, 3.33 × 10⁵ J kg⁻¹); latent heat of vaporisation (L_v) is for boiling/condensation (for water, 22.6 × 10⁵ J kg⁻¹).

A temperature-vs-heat-supplied graph for ice heated steadily from below 0 °C to steam above 100 °C shows: a rising slope (ice warming), a flat plateau at 0 °C (melting), a rising slope (water warming), a flat plateau at 100 °C (boiling), then a rising slope (steam warming). The plateaus are wider than the slopes because latent heat of vaporisation is much larger than that of fusion.

Melting point changes with pressure: it decreases with increasing pressure for substances that expand on melting is wrong for ice — ice contracts on melting, so increasing pressure lowers its melting point (this is why pressure from a skate blade can melt ice slightly). Boiling point always increases with increasing pressure (this is the principle of the pressure cooker).

Heat travels from a hotter region to a colder one by three distinct mechanisms.

Conduction is the transfer of heat through a medium (usually solids) without bulk motion of the material — energy passes from molecule to molecule through collisions and, in metals, through free electrons. Consider a rod of length L, uniform cross-section A, with its ends kept at steady temperatures T_C and T_D (T_D > T_C), the sides insulated. In the steady state, the rate of heat flow H = ΔQ/Δt is the same at every cross-section and is given by H = KA(T_D − T_C)/L, where K is the thermal conductivity of the material — a measure of how readily it conducts heat. SI unit of K is J s⁻¹ m⁻¹ K⁻¹ (or W m⁻¹ K⁻¹). Metals (especially silver and copper) have high K; most non-metals, and gases in particular, are poor conductors and are used as insulators (thermos flasks trap air/vacuum for exactly this reason).

Convection is heat transfer by actual movement of the heated material itself — it occurs in fluids (liquids and gases), where the heated portion becomes less dense, rises, and is replaced by cooler fluid, setting up convection currents. Natural convection (density-driven, e.g. land and sea breezes, boiling water) is different from forced convection (fluid driven by a pump/fan, e.g. blood circulation, forced-air heating). Land and sea breezes occur because land heats up and cools down faster than the sea; during the day, warm air over land rises and the cooler sea breeze blows in to replace it; at night the reverse (land breeze) occurs.

Radiation is heat transfer by electromagnetic waves and needs no medium — this is how heat from the Sun reaches Earth through empty space. Every body above absolute zero radiates thermal (infrared) energy; a body that is a good absorber of radiation is also a good emitter (this is the basis of a perfectly black body, which absorbs all radiation falling on it and, at a given temperature, is the best possible emitter).

Newton's law of cooling states that the rate of loss of heat of a body is directly proportional to the temperature difference between the body and its surroundings, provided this difference is small. If T is the body's temperature and T_s the (constant) surrounding temperature, −dQ/dt ∝ (T − T_s).

Derivation: Let a body of mass m and specific heat s cool from T to T_s. Rate of loss of heat: −dQ/dt = ms(dT/dt) in magnitude. By Newton's law, −dQ/dt = k(T − T_s), where k is a positive constant depending on the area and nature of the surface. So −ms(dT/dt) = k(T − T_s), i.e. dT/(T − T_s) = −(k/ms) dt = −K dt, where K = k/ms. Integrating between t = 0 (temperature T₁) and time t (temperature T₂): ln[(T₂ − T_s)/(T₁ − T_s)] = −Kt, giving T₂ − T_s = (T₁ − T_s)e^−Kt. This shows temperature excess over the surroundings falls off exponentially with time.

For exam numericals, a simpler averaged form is commonly used: (T₁ − T₂)/t = K[(T₁ + T₂)/2 − T_s], which avoids needing calculus — it approximates the instantaneous rate by the average temperature over the interval. A graph of temperature versus time for a cooling body is a curve that falls steeply at first (large T − T_s) and flattens as T approaches T_s, asymptotically approaching the surrounding temperature; a graph of log_e(T − T_s) versus t is a straight line with negative slope −K.

The spectrum of radiation emitted by a hot body depends on its temperature. As temperature increases, the wavelength at which the emission is maximum shifts towards shorter wavelengths — this is Wien's displacement law: λ_mT = b, where λ_m is the wavelength of maximum intensity and b is Wien's constant (2.9 × 10⁻³ m K). This is why an iron rod glows dull red, then orange, then white as it is heated further — the peak wavelength shifts from infrared towards the visible and then towards blue.

The total energy radiated per unit area per unit time by a perfect black body is given by the Stefan–Boltzmann law: E = σT⁴, where σ is Stefan's constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴). For a body that is not a perfect black body, E = εσT⁴, where ε (emissivity, 0 < ε ≤ 1) measures how close the body is to an ideal black body. If such a body radiates into surroundings at temperature T₀, the net rate of loss of energy per unit area is E_net = εσ(T⁴ − T₀⁴) — this is the more complete law that Newton's law of cooling approximates for small (T − T₀).