understand that a physical property that varies with temperature may be used for the measurement of temperature and state examples of such properties, including the density of a liquid, volume of a gas at constant pressure, resistance of a metal, e.m

14 Temperature

14.1 Thermal equilibrium

When two bodies are in thermal contact heat flows from the hotter to the colder body until both reach the same temperature. At that point there is no net flow of heat and the system is said to be in thermal equilibrium.

A thermometer must be allowed to reach thermal equilibrium with the system before a reading is taken.

Two objects at different temperatures exchanging heat until they reach the same temperature

Heat flows from the hotter body to the colder body until both reach a common temperature.

14.2 Temperature scales

Temperature is an intensive property that can be expressed on different scales. The Cambridge 9702 syllabus requires students to recognise the three common scales and to convert between kelvin (K) and Celsius (°C).

ScaleSymbolZero pointDegree sizeConversion to/from Kelvin
Celsius°C0 °C = 273.15 K (freezing point of water)1 °C = 1 K\(K = ^\circ\!C + 273.15\)  \(^{\circ}\!C = K - 273.15\)
Fahrenheit°F32 °F = 273.15 K (freezing point of water)1 °F = 5/9 K\(K = \frac{5}{9}(^\circ\!F - 32) + 273.15\)
KelvinK0 K = absolute zero (the lowest possible temperature)1 K = 1 °CDefinition

Worked example – Celsius to Kelvin

Convert 25 °C to kelvin.

  1. Use the conversion \(K = ^\circ\!C + 273.15\).

  2. \(K = 25 + 273.15 = 298.15\; \text{K}\).

Thus 25 °C = 298.15 K.

14.3 Specific heat capacity and latent heat

  • Specific heat capacity (c) – the amount of energy required to raise the temperature of 1 kg of a substance by 1 K.

    \[Q = mc\Delta T\]

  • Latent heat (L) – the amount of energy required for a phase change at constant temperature (fusion or vaporisation).

    \[Q = mL\]

Both \(c\) and \(L\) are properties of the material; they do not depend on the mass of the sample.

Worked example – Heat required to melt ice

How much heat is needed to melt 0.5 kg of ice at 0 °C? (Latent heat of fusion of water \(L_f = 334\,\text{kJ kg}^{-1}\)).

\[

Q = mL_f = 0.5 \times 334\,\text{kJ kg}^{-1}=167\;\text{kJ}

\]

The temperature does not change during the melting process.

14.4 Using Physical Properties to Measure Temperature

What makes a good thermometric property?

  • Monotonic (always increases or always decreases) with temperature over the range of interest.

  • Reproducible and easy to measure accurately.

  • Preferably linear, or at least a well‑characterised mathematical relationship.

  • Only weakly affected by other variables such as pressure or composition.

Temperature‑dependent properties used in thermometry (Cambridge 9702)

  1. Density of a liquid – Most liquids expand on heating, so density \(\rho\) falls.

    \[

    \rho(T) \approx \rho0\bigl[1-\beta\,(T-T0)\bigr]

    \]

    \(\beta\) = volumetric expansion coefficient (≈ 2.1 × 10⁻⁴ K⁻¹ for water). A liquid‑in‑glass thermometer is calibrated using this relationship.

  2. Volume of a gas at constant pressure (Charles’s law) – For a fixed amount of gas,

    \[

    \frac{V}{T}= \text{constant (at fixed }p\text{)}\qquad\Longrightarrow\qquad V = V0\frac{T}{T0}

    \]

    Measuring the volume of a known mass of gas gives its temperature. This is the principle of the constant‑pressure gas thermometer.

  3. Electrical resistance of a metal – For many pure metals the resistance varies approximately linearly with temperature:

    \[

    R = R0\,[1+\alpha\,(T-T0)]

    \]

    \(\alpha\) = temperature coefficient of resistance (≈ 0.0039 K⁻¹ for copper). Resistance thermometers (RTDs) exploit this relationship.

  4. Electromotive force (e.m.f.) of a thermocouple – Two dissimilar metals joined at a hot junction develop a voltage proportional to the temperature difference:

    \[

    E = S\,\Delta T

    \]

    \(S\) = Seebeck coefficient (varies with the metal pair). Thermocouples are widely used for rapid temperature measurement.

Worked example – Resistance thermometer

A platinum resistance thermometer has \(R_0 = 100.0\;\Omega\) at 0 °C and \(\alpha = 3.85\times10^{-3}\;\text{K}^{-1}\). If the measured resistance is 138.5 Ω, find the temperature.

\[

T = T0 + \frac{R-R0}{\alpha R_0}

= 0 + \frac{138.5-100.0}{3.85\times10^{-3}\times100.0}

= \frac{38.5}{0.385}=100\;\text{°C}

\]

General calibration procedure

  1. Choose a recognised reference scale (usually kelvin or Celsius).

  2. Measure the chosen property at several fixed reference temperatures (e.g. 0 °C, 25 °C, 100 °C).

  3. Plot the data and fit an appropriate function (linear, polynomial, or tabulated).

  4. Use the fitted equation or calibration table to convert future measurements of the property into temperature values.

Practical example – Thermocouple thermometer

A copper‑constantan thermocouple produces a small e.m.f. that is measured with a digital voltmeter. The voltage reading is converted to temperature using the calibrated \(E\)–\(T\) table supplied by the manufacturer.

Schematic of a copper‑constantan thermocouple showing hot and reference junctions and the voltmeter connection

Schematic of a copper‑constantan thermocouple.

Glossary (AO1 terminology)

TemperatureIntensive property that indicates the average kinetic energy of particles.
Thermal equilibriumState in which no net heat flows between bodies in contact.
Absolute zero0 K, the lowest possible temperature; molecular motion ceases in the classical sense.
Intensive propertyProperty independent of the amount of material (e.g., temperature, density).
Extensive propertyProperty that scales with the amount of material (e.g., mass, volume, internal energy).
Specific heat capacity (c)Energy required to raise 1 kg of a substance by 1 K.
Latent heat (L)Energy required for a phase change at constant temperature.
Thermometric propertyA physical property that varies in a known way with temperature and can therefore be used to indicate temperature.
Coefficient of linear expansion (\(\alpha\))Fractional change in length (or resistance) per kelvin.
Seebeck coefficient (S)Proportionality constant between the e.m.f. of a thermocouple and the temperature difference of its junctions.

Summary

  • Thermal equilibrium is a prerequisite for any temperature measurement.

  • Kelvin is the SI unit of thermodynamic temperature; 0 K is absolute zero.

  • Only the conversion between kelvin and Celsius is required by the syllabus; the relationship is \(K = ^\circ\!C + 273.15\).

  • Specific heat capacity and latent heat are material properties that relate heat transfer to temperature change or phase change.

  • Four properties commonly used for thermometry are liquid density, gas volume at constant pressure, metal resistance, and thermocouple e.m.f.; each has a simple, calibrated relationship with temperature.

  • Accurate temperature measurement always involves calibration against a recognised standard scale.