Measurement of Temperature

Temperature is one of the most fundamental quantities in physics, yet measuring it precisely is far from trivial. Unlike length or mass, you cannot directly compare one object's temperature to another with a ruler or balance — you need a thermometer and an agreed-upon scale. Thermometry is the science of temperature measurement, and it relies on thermometric properties: physical quantities that change in a predictable, reproducible way as temperature changes. Common examples include the volume of a liquid, the electrical resistance of a metal, and the pressure of a gas at constant volume. Three temperature scales are used in science and everyday life: Celsius (°C), Fahrenheit (°F), and Kelvin (K). The Celsius scale sets 0 °C at the ice point of water and 100 °C at the steam point, giving 100 equally spaced divisions between them. Fahrenheit places the ice point at 32 °F and the steam point at 212 °F. The Kelvin scale — the SI unit — is an absolute scale anchored to absolute zero (−273.15 °C), the theoretical minimum temperature where particles have minimum thermal energy. Converting between scales is a routine but important skill for every physics student. A thermometer works by exploiting thermal equilibrium: when two objects are in thermal contact long enough, they reach the same temperature. This is formalised in the Zeroth Law of Thermodynamics, which states that if object A is in thermal equilibrium with object C, and object B is also in thermal equilibrium with object C, then A and B are in thermal equilibrium with each other. This seemingly obvious statement is the logical foundation that makes thermometry meaningful — the thermometer (object C) can act as a universal comparator between any two systems.

Key takeaways
  • Choose a Thermometric Property
  • Define Two Fixed Points
  • Divide and Interpolate
  • Convert Between Scales
  • Why the Kelvin Scale is Preferred in Physics

Choose a Thermometric Property

Select a physical property X that varies continuously and reproducibly with temperature. In a liquid-in-glass thermometer, X is the length of the liquid column. In a constant-volume gas thermometer (the most accurate laboratory instrument), X is the gas pressure P. The property must be single-valued — each temperature must correspond to exactly one value of X.

Define Two Fixed Points

Fixed points are precisely reproducible temperatures. For the Celsius scale: the lower fixed point is 0 °C (pure ice in equilibrium with air-saturated water at 1 atm) and the upper fixed point is 100 °C (pure water boiling at 1 atm). These anchor the scale. The modern SI definition uses the triple point of water (0.01 °C = 273.16 K) as the single defining fixed point for the Kelvin scale.

Divide and Interpolate

Between the two fixed points, assume the thermometric property changes linearly with temperature. If the property has value X₀ at 0 °C and X₁₀₀ at 100 °C, then any intermediate temperature θ (in °C) is found by linear interpolation: θ = [(X − X₀) / (X₁₀₀ − X₀)] × 100. This formula is the operational definition of temperature on that particular scale.

Convert Between Scales

Once a temperature is known in one scale, conversion is algebraic. Celsius to Kelvin: T(K) = T(°C) + 273.15. Celsius to Fahrenheit: T(°F) = (9/5)·T(°C) + 32. Note that a temperature difference of 1 °C equals a difference of 1 K, but equals a difference of 1.8 °F. This distinction between a temperature and a temperature interval is crucial in thermodynamics calculations.

Why the Kelvin Scale is Preferred in Physics

Many physical laws — such as the ideal gas law PV = nRT and Wien's displacement law — require absolute temperature in Kelvin. At absolute zero (0 K = −273.15 °C), the classical kinetic energy of gas molecules approaches its minimum. Using Celsius in these formulas would give nonsensical negative values. The Kelvin scale removes this problem entirely, making calculations consistent and physically meaningful.

The Key Equation

θ=Xθ−X0X100−X0×100\theta = \frac{X_{\theta} - X_0}{X_{100} - X_0} \times 100
  • \theta — Temperature (°C): The unknown temperature being determined on the Celsius scale.
  • X_{\theta} — Property at unknown temperature (varies (e.g. mm, Pa)): The measured value of the thermometric property at the unknown temperature θ.
  • X_0 — Property at ice point (varies): The value of the thermometric property at exactly 0 °C (lower fixed point).
  • X_{100} — Property at steam point (varies): The value of the thermometric property at exactly 100 °C (upper fixed point).

Worked Example

Problem. A resistance thermometer has a resistance of 25.0 Ω at the ice point (0 °C) and 35.0 Ω at the steam point (100 °C). When placed in a liquid, the resistance reads 31.5 Ω. (a) Calculate the temperature of the liquid in °C. (b) Convert this temperature to Kelvin and to Fahrenheit.

θ=65.0 ∘C,T=338 K,T=149 ∘F\theta = 65.0\,^{\circ}\text{C},\quad T = 338\,\text{K},\quad T = 149\,^{\circ}\text{F}

Answer. The liquid is at 65.0 °C, which equals 338 K and 149 °F.

Figure
Diagram
Comparison of Temperature ScalesCelsiusFahrenheitKelvinEvent100212373Water boils3798.6310Body temp032273Water freezes−273−4600Absolute zeroDashed lines show equivalent temperatures across all three scales.
Origins · 1 minAnders Celsius and how this was discovered
AC
Anders Celsius
1701–1744 · Swedish

Anders Celsius was a Swedish astronomer and physicist who, in 1742, proposed a centigrade temperature scale with 100 divisions between the freezing and boiling points of water. Ironically, his original scale was inverted — he set 0 at the boiling point and 100 at the freezing point. The scale was reversed after his death by fellow scientist Carl Linnaeus, and it eventually became the Celsius scale universally used in science today. Celsius was primarily an astronomer; he led an expedition to northern Sweden in 1736 to measure the shape of the Earth, helping confirm Newton's prediction that our planet is slightly flattened at the poles.

  • Thermal Equilibrium & the Zeroth Law — The Zeroth Law is the logical foundation of thermometry — it guarantees that a thermometer gives a meaningful, consistent reading when placed in contact with any system.
  • Internal Energy and Heat Transfer — Temperature measures the tendency to transfer heat; understanding internal energy explains why two objects reach the same temperature at thermal equilibrium.
  • The Ideal Gas Law — PV = nRT requires absolute temperature in Kelvin, directly illustrating why the Kelvin scale is indispensable in quantitative thermodynamics.
  • Specific Heat Capacity — Calculations of heat transferred (Q = mcΔT) use temperature differences, making accurate temperature measurement and scale conversions essential.