Thermal Measurement and Scales

Branch note: This page deepens one part of Thermal Physics A.

Overview

This page focuses on how temperature is defined and measured. Since temperature cannot be measured directly, we use physical properties that vary with temperature and calibrate them against agreed reference points.

Core ideas:

• temperature and thermal equilibrium
• thermometric properties
• thermometer calibration
• Celsius scale
• constant-volume gas thermometer
• absolute zero
• Kelvin scale
• zeroth law of thermodynamics

This topic supports the wider chapter:

• Thermal Physics A

Definition

Temperature measurement is the process of using a measurable thermometric property and a calibrated scale to assign a numerical value to the thermal state of a body.

Why It Matters

This topic matters because thermal calculations depend on clear distinction between temperature, heat, equilibrium, and absolute temperature. It also explains why kelvin is the preferred physics scale and why thermometers can work at all.

Key Representations

Core Ideas

• Temperature determines the direction of thermal energy transfer.
• A thermometer uses a thermometric property that varies reproducibly with temperature.
• Calibration assigns numerical temperature values using fixed points or a defined scale.
• Kelvin is the thermodynamic temperature scale; absolute zero is $0\mathrm{K}$.
• The zeroth law explains why a thermometer can compare temperatures.

Exam Relevance

The 2026 core is thermal equilibrium, the empirical route to an absolute scale, absolute zero and Celsius–kelvin conversion. Fixed-point calibration, particular thermometer designs and the zeroth law are retained as supporting/enrichment context.

Temperature and Thermal Equilibrium

What is Temperature?

Temperature is a measure of the degree of hotness of a body.

It determines the direction of thermal energy transfer.

If two bodies are placed in contact:

• thermal energy is transferred from higher temperature to lower temperature
• transfer continues until both reach the same temperature

Thermal Equilibrium

Two objects are in thermal equilibrium when:

• they have the same temperature
• there is no net thermal-energy transfer between them

This idea allows thermometers to work.

Figure: If $T_A>T_B$, the arrow shows the direction of net thermal-energy transfer. At thermal equilibrium, $T_A=T_B$ and microscopic exchanges may still occur, but there is no net transfer.

Heat vs Temperature

Students often confuse these ideas.

Temperature

• indicates hotness
• measured in ${}^{\circ }\mathrm{C}$ or $\mathrm{K}$

Heat

• energy transferred due to temperature difference
• measured in joules

A body does not “contain heat”: heat is a transfer. A hot small body can have less internal energy than a cooler large body because internal energy also depends on amount and material.

Thermometric Properties

A thermometric property is a measurable physical property that changes with temperature.

Examples:

• length of metal strip
• volume of liquid
• pressure of gas
• electrical resistance
• e.m.f. of thermocouple

Desirable Thermometric Properties

A good thermometric property should be:

• easy to measure
• sensitive to temperature changes
• reproducible
• approximately linear with temperature
• stable over operating range
• unaffected by unrelated factors

Supporting context: common thermometers and calibration

Scope

These examples explain how temperature scales are implemented, but the current syllabus does not prescribe this full catalogue of thermometer construction.

Liquid-in-Glass Thermometer

Uses expansion of liquid such as mercury or alcohol.

Advantages:

• simple
• inexpensive

Limitations:

• slow response
• limited range
• may be non-linear

Resistance Thermometer / Thermistor

Uses resistance change with temperature.

Advantages:

• sensitive
• suitable for electronic systems

Gas Thermometer

Uses gas pressure or volume.

More fundamental and useful in defining absolute temperature.

Fixed Points and Calibration

To create a scale, two reference temperatures are chosen.

Traditional Celsius scale uses:

• ice point = $0^{\circ }\mathrm{C}$
• steam point = $100^{\circ }\mathrm{C}$

These correspond to melting ice and boiling water under standard atmospheric pressure.

Celsius Calibration

Suppose a thermometer property $x$ has:

• $x_0$ at ice point
• $x_{100}$ at steam point
• $x$ at unknown temperature

Then:

$$T=\frac{x-x_0}{x_{100}-x_0}\times 100$$

This assumes linear variation of the property.

Figure: If the chosen thermometric property $x$ varies linearly with the numerical Celsius reading, the points $(0,x_0)$ and $(100,x_{100})$ define the straight calibration line. An intermediate measurement $x$ is converted using $\theta /{}^{\circ }\mathrm{C}=100(x-x_0)/(x_{100}-x_0)$. The interpolation is valid only over a range where the linearity assumption is adequate.

The key assumption is that the thermometric property varies linearly between the two fixed points.

Worked Example 1

A mercury column has:

• $x_0=2.0\mathrm{cm}$
• $x_{100}=24.0\mathrm{cm}$

Unknown reading:

• $x=18.0\mathrm{cm}$

Find temperature.

$$T=\frac{18.0-2.0}{24.0-2.0}\times 100$$ $$T=\frac{16}{22}\times 100=72.7^{\circ }\mathrm{C}$$

General Calibration Formula

If lower and upper fixed points are not 0 and 100:

$$T=\frac{x-x_{\text{lower}}}{x_{\text{upper}}-x_{\text{lower}}}(T_{\text{upper}}-T_{\text{lower}})+T_{\text{lower}}$$

Useful for resistance thermometers or custom scales.

Worked Example 2

A resistance thermometer gives:

• $20\Omega$ at $-10^{\circ }\mathrm{C}$
• $500\Omega$ at $110^{\circ }\mathrm{C}$

Find temperature when resistance is $360\Omega$.

$$T=\frac{360-20}{500-20}(110-(-10))+(-10)$$ $$T=\frac{340}{480}(120)-10=75^{\circ }\mathrm{C}$$

Constant-Volume Gas Thermometer

Principle

For a fixed mass of gas at constant volume:

• pressure increases with temperature

So gas pressure can be used as a thermometric property.

Figure: At each temperature, adjust the reservoir until the gas-side mercury surface returns to the fiducial mark. This restores the same gas volume. Then read $h$ and infer the absolute gas pressure from the actual manometer geometry.

In practice, the mercury levels are adjusted to keep the gas volume fixed, and the resulting pressure is inferred from the height difference.

The pressure relation depends on the manometer geometry. The pressure of the gas may be $P_{\text{atm}}+\rho gh$ or $P_{\text{atm}}-\rho gh$ depending on which mercury surface is higher.

Why Important?

Gas behaviour becomes nearly universal at low pressure, making it more reliable than material expansion thermometers.

Pressure-Temperature Graph

At constant volume:

$$P\propto T$$

When pressure is plotted against Celsius temperature, extrapolation gives zero pressure near:

$$-273.15^{\circ }\mathrm{C}$$

This leads to the idea of absolute zero.

Figure: For fixed amounts of low-density gas at constant volume, pressure varies approximately linearly with Celsius temperature. The dashed continuation is an extrapolation because real gases condense before zero pressure is reached. Different gases approach the common intercept near $-273.15^{\circ }\mathrm{C}$ in the low-density limit.

The important exam idea is that different gas samples give different lines, but the extrapolated zero-pressure intercept is the same.

Absolute Zero

Absolute zero is:

• the lowest possible thermodynamic temperature
• temperature where substances have minimum internal energy

Minimum internal energy does not mean that every microscopic energy contribution must be zero. The H2-level point is that thermodynamic temperature cannot fall below $0\mathrm{K}$.

Value:

$$0\mathrm{K}=-273.15^{\circ }\mathrm{C}$$

Kelvin Scale

The Kelvin scale is an absolute scale based on thermodynamic principles.

Conversion:

$$T(\mathrm{K})=T{(}^{\circ }\mathrm{C})+273.15$$

Examples:

• $25^{\circ }\mathrm{C}=298.15\mathrm{K}$
• $0^{\circ }\mathrm{C}=273.15\mathrm{K}$

Figure: The aligned scales show that $0\mathrm{K}$ corresponds to $-273.15^{\circ }\mathrm{C}$ and that the kelvin has no degree sign. The two scales have equal interval sizes, so a change of $1^{\circ }\mathrm{C}$ has the same magnitude as a change of $1\mathrm{K}$ even though the readings differ by 273.15.

Notes on Kelvin

• no degree sign
• is not negative in the H2 thermodynamic-temperature scale
• same interval size as Celsius

So:

• rise of $5^{\circ }\mathrm{C}$ = rise of $5\mathrm{K}$

Zeroth Law of Thermodynamics

If:

• A is in thermal equilibrium with C
• B is in thermal equilibrium with C

Then:

• A is in thermal equilibrium with B

This allows a thermometer (object C) to compare temperatures of different bodies.

Why Celsius is Less Fundamental

Celsius scale depends on properties of water:

• melting point
• boiling point

These can vary slightly with pressure and impurities.

Kelvin scale is preferred in physics because it is absolute.

Worked Example 3

Convert:

(a) $37^{\circ }\mathrm{C}$

$$T=37+273.15=310.15\mathrm{K}$$

(b) $400\mathrm{K}$

$$T=400-273.15=126.85^{\circ }\mathrm{C}$$

Experimental Notes

Ice Point

Use:

• pure melting ice
• mixture of ice and water

Steam Point

Use:

• pure steam above boiling water
• standard atmospheric pressure

If conditions are poor, calibration errors occur.

Links

• Thermal Physics A
• Heat Capacity and Latent Heat
• Thermal Practicals
• Thermal Physics A Common Exam Traps

Summary

You should be able to explain:

• what temperature means
• thermal equilibrium
• thermometric properties
• thermometer calibration
• constant-volume gas thermometer
• absolute zero
• Kelvin conversion
• zeroth law of thermodynamics