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

Learning Objective

Understand that any physical property which varies in a known, monotonic way with temperature can be used as a thermometer, and be able to state and explain the four textbook examples required by the Cambridge IGCSE/A‑Level syllabus.

1. Thermal Equilibrium

• Definition: Two or more bodies are in thermal equilibrium when, after being placed in thermal contact, they reach the same temperature and no net heat flows between them.
• Experimental test: Place a hot metal block and a cold water bath in an insulated container; after a short time the metal and the water have the same temperature (as shown by a single thermometer).
• This concept underpins all temperature measurements – a thermometer only gives a reliable reading when it is in thermal equilibrium with the system being measured.

2. Temperature Scales

2.1 Main Scales

• Celsius (°C) – 0 °C = freezing point of pure water, 100 °C = boiling point of pure water at 1 atm.
• Kelvin (K) – the true thermodynamic scale. Its zero is defined by the triple‑point of water (273.16 K) and the numerical value of the Boltzmann constant is fixed. One kelvin has the same magnitude as one degree Celsius.
• Fahrenheit (°F) – 32 °F = freezing point of water, 212 °F = boiling point of water at 1 atm.
• Rankine (°R) – absolute scale based on the Fahrenheit degree; 0 °R corresponds to absolute zero.

2.2 Conversion Formulas

3. Quantitative Temperature Concepts

3.1 Specific Heat Capacity

The amount of heat \(Q\) required to raise the temperature of a mass \(m\) of a substance by \(\Delta T\) is

\[

Q = mc\Delta T

\]

• \(c\) – specific heat capacity (J kg\(^{-1}\) K\(^{-1}\)).
• Example: To raise 0.5 kg of water ( \(c = 4180\) J kg\(^{-1}\) K\(^{-1}\) ) from 20 °C to 80 °C, \(Q = 0.5 \times 4180 \times 60 = 1.254\times10^{5}\) J.

3.2 Latent Heat

When a substance changes phase at constant temperature, the heat supplied (or removed) is

\[

Q = mL

\]

• \(L\) – latent heat of fusion (solid–liquid) or vaporisation (liquid–gas) (J kg\(^{-1}\)).
• Example: To melt 0.2 kg of ice at 0 °C, using \(L_{\text{fusion}} = 3.34\times10^{5}\) J kg\(^{-1}\), the required heat is \(Q = 0.2 \times 3.34\times10^{5} = 6.68\times10^{4}\) J.

4. Principle of Using Physical Properties as Thermometers

A thermometer measures a property \(P\) that changes in a known, monotonic way with temperature \(T\). If the functional relationship \(P(T)\) is established (by calibration), measuring \(P\) gives the temperature directly.

4.1 Requirements for a Good Thermometric Property

1. Strong, reproducible dependence on temperature (large change of \(P\) per degree).
2. Linear (or easily linearised) behaviour over the range of interest.
3. Minimal influence of other variables (pressure, composition, magnetic fields, etc.).
4. Stability and ease of measurement – the property should not drift with time and should be measurable with simple, reliable instrumentation.

5. Syllabus‑Required Examples

5.1 Density of a Liquid (e.g., mercury or coloured alcohol)

For most liquids the density \(\rho\) falls as temperature rises because the volume expands. Over a limited range

\[

\rho(T) \approx \rho{0}\bigl[1-\beta\,(T-T{0})\bigr]

\]

• \(\beta\) – volumetric expansion coefficient (≈ \(10^{-4}\,\text{K}^{-1}\) for water, larger for mercury).
• \(\rho{0}\) – density at a reference temperature \(T{0}\) (usually 0 °C).

Principle of the thermometer: The liquid is confined in a narrow glass capillary. As temperature increases the liquid expands, the height \(h\) of the column rises. Since the mass of liquid in the column is constant, \(h\) is inversely proportional to \(\rho\); the scale on the glass is calibrated in temperature units.

Instrument: Mercury or coloured‑alcohol glass thermometer.

5.2 Volume of a Gas at Constant Pressure (Charles’s Law)

At constant pressure an (ideal) gas obeys

\[

\frac{V}{T}= \text{constant}\qquad\Longrightarrow\qquad V = V{0}\frac{T}{T{0}}

\]

• \(V{0}\) – volume at a reference absolute temperature \(T{0}\) (usually 273.15 K).
• Valid for ideal gases; real gases follow the same trend over moderate temperature ranges.

Principle of the thermometer: A sealed bulb of known gas is connected to a movable piston or a calibrated volume chamber. As the gas temperature changes, the piston moves proportionally to the change in volume, giving a direct read‑out of \(T\) on an absolute scale.

Instrument: Constant‑pressure gas thermometer (used as a primary standard for calibrating other thermometers).

5.3 Electrical Resistance of a Metal

For many metals the resistance increases approximately linearly with temperature:

\[

R = R{0}\bigl[1+\alpha\,(T-T{0})\bigr]

\]

• \(\alpha\) – temperature coefficient of resistance (≈ \(3.9\times10^{-3}\,\text{K}^{-1}\) for platinum).
• \(R{0}\) – resistance at the reference temperature \(T{0}\) (commonly 0 °C).

Principle of the thermometer: The metal element (often a fine platinum wire) is placed in the medium whose temperature is to be measured. The resistance is measured with a Wheatstone bridge or a digital ohmmeter; the calibrated \(R\)–\(T\) relation yields the temperature.

Instrument: Platinum Resistance Thermometer (PRT).

5.4 e.m.f. of a Thermocouple

A thermocouple consists of two dissimilar metals joined at a hot junction. A temperature difference \(\Delta T = T{\text{hot}}-T{\text{cold}}\) generates an e.m.f. \(E\) (the Seebeck effect):

\[

E = a\,\Delta T

\]

• \(a\) – Seebeck coefficient for the metal pair (e.g., ≈ \(40\;\mu\text{V K}^{-1}\) for a Type K thermocouple).
• For large temperature spans the \(E\)‑vs‑\(\Delta T\) curve is slightly non‑linear; tables or polynomial fits are used for accurate work.

Principle of the thermometer: The hot junction is placed in the medium; the cold (reference) junction is kept at a known temperature (often 0 °C ice‑water bath) or compensated electronically. The measured e.m.f. is converted to temperature using the calibrated \(E\)–\(\Delta T\) relationship.

Instrument: Thermocouple (various types: J, K, T, etc.).

6. Comparative Summary of Thermometric Properties

7. Concluding Remarks

Any physical quantity that varies in a known, monotonic way with temperature can serve as a thermometer. The choice of property depends on:

• Temperature range required (e.g., liquid density for 0–100 °C, thermocouples for > 1000 °C).
• Desired accuracy and precision.
• Response time (electrical methods are fastest).
• Practical considerations such as durability, cost, and the need for regular calibration.

Understanding these principles enables you to select the most appropriate temperature‑measuring device for laboratory experiments, industrial processes, and everyday applications.