Measurement of temperature

Put a mercury thermometer under your tongue and a doctor reads " $98.6^\circ\text{F}$ "; hand the same body to a lab technician anywhere outside the United States and she writes " $37.0^\circ\text{C}$ ." Both are correct. Neither number is the temperature in any absolute sense — each is a reading on a scale that someone built by picking two fixed points, marking them with arbitrary numbers, and assuming everything in between varies smoothly. Temperature itself is a statement about the average kinetic energy of jostling molecules; a thermometer is simply a device with some other, easily read property — the length of a liquid column, the pressure of a trapped gas, the resistance of a wire — that changes reliably as that molecular agitation changes. This article is about the machinery that turns molecular chaos into a number on a dial: how the Celsius, Fahrenheit and Kelvin scales are constructed from fixed points, how the algebra of converting between them works, and why physicists trust a column of gas at constant volume more than a column of mercury. Along the way we meet the triple point of water, a uniquely reproducible reference condition, and see why the constant-volume gas thermometer became the practical backbone of the absolute Kelvin scale used throughout physics and chemistry. None of this needs anything beyond straight-line algebra — the entire subject rests on one idea, that a thermometric property is assumed to vary linearly with temperature between two calibration points. Get that one idea solid and every scale-conversion question, and every "which thermometer is more accurate" question, falls out of it.

Key takeaways

A thermometer measures temperature indirectly, through a thermometric property (length, pressure, resistance...) that changes measurably and reproducibly with temperature.
Two fixed points calibrate any scale: for Celsius and Fahrenheit these are the ice point and the steam point of water at standard pressure.
Celsius: $0^\circ\text{C}$ to $100^\circ\text{C}$ over $100$ divisions. Fahrenheit: $32^\circ\text{F}$ to $212^\circ\text{F}$ over $180$ divisions. Kelvin: absolute scale, $0\ \text{K}$ at absolute zero, same degree size as Celsius.
The general two-point linear interpolation formula converts between any two scales; applied to Celsius and Fahrenheit it gives $F = \frac{9}{5}C + 32$ , and $T(\text{K}) = C + 273.15$ links Celsius to Kelvin.
A liquid-in-glass thermometer exploits differential expansion of liquid over glass but is only approximately linear; a constant-volume gas thermometer measures pressure at fixed volume and is far closer to ideal, ultra-reproducible behaviour.
The triple point of water, $273.16\ \text{K}$ , is the single fixed point (not two) used to anchor the modern Kelvin scale, because it occurs at one exact combination of temperature and pressure.
Key idea: every temperature scale is a linear map built from two calibration points on some thermometric property, and the constant-volume gas thermometer is preferred because gas pressure tracks absolute temperature more faithfully and universally than any liquid column.

The core insight

A thermometer reads temperature through a thermometric property assumed linear between two fixed points, so any two scales convert by the ratio $\frac{X-X_1}{X_2-X_1}=\frac{Y-Y_1}{Y_2-Y_1}$ , and the constant-volume gas thermometer — whose pressure at fixed volume tracks absolute temperature almost perfectly, anchored at the triple point of water, $273.16\ \text{K}$ — is the practical route to the absolute Kelvin scale.

Where it fits in the syllabus

This sits inside thermometry, the opening topic of the CBSE Class 11 chapter Thermal Properties of Matter, immediately after the heat-versus-temperature distinction and just before thermal expansion. The prescribed scope asks a student to know the working principle of a liquid-in-glass thermometer and a constant-volume gas thermometer, the fixed points used to calibrate a scale, and the linear relations connecting Celsius, Fahrenheit and Kelvin — all squarely examinable, with scale-conversion arithmetic a recurring numerical-question type. The chapter does not ask for a derivation of the ideal-gas law itself (that belongs to the later Kinetic Theory chapter); here the gas thermometer is used only as a device whose pressure reading is taken to be proportional to absolute temperature.

The physical picture

A thermometer cannot measure "hotness" directly — no instrument reads molecular kinetic energy off a dial. Instead every thermometer relies on some other physical quantity, a thermometric property, that happens to change smoothly and repeatably as temperature changes: the length of a mercury column, the pressure of a gas held at fixed volume, or the electrical resistance of a platinum wire. To turn that property into a usable scale, two fixed points — reproducible physical states everyone can recreate, such as pure melting ice or boiling water at standard atmospheric pressure — are chosen and assigned agreed numbers, and the space between them is divided into equal parts on the assumption that the property varies uniformly with temperature. The Celsius scale sets the ice point at $0^\circ\text{C}$ and the steam point at $100^\circ\text{C}$ , split into $100$ equal divisions. The Fahrenheit scale, still in everyday use in the United States, sets the same two physical states at $32^\circ\text{F}$ and $212^\circ\text{F}$ , split into $180$ divisions — a Fahrenheit degree is therefore a smaller step than a Celsius degree. The Kelvin scale is the SI absolute scale: its zero, $0\ \text{K}$ , is absolute zero, the theoretical temperature at which molecular translational kinetic energy would fall to its minimum, and its degree (the kelvin) is deliberately made the same size as the Celsius degree, so that only the position of the zero differs between the two. Because every scale is just a straight-line relabelling of the same underlying physical temperature, any scale reading $X$ (with fixed points $X_1$ and $X_2$ ) converts to any other scale reading $Y$ (with fixed points $Y_1$ and $Y_2$ ) through one universal ratio.

\frac{X - X_1}{X_2 - X_1} = \frac{Y - Y_1}{Y_2 - Y_1}

where: $X$ and $Y$ are the temperature readings on the two scales being compared, $X_1, Y_1$ are their respective lower fixed points (the ice point), and $X_2, Y_2$ are their respective upper fixed points (the steam point). The physical reading is that the fractional distance travelled from the lower fixed point to the upper fixed point must be identical on both scales, because both scales are describing the same physical warming, just with different rulers laid over it. Applying this with $X = C,\ X_1=0,\ X_2=100$ and $Y=F,\ Y_1=32,\ Y_2=212$ gives the everyday Celsius–Fahrenheit relation, and because the Kelvin degree is the same size as the Celsius degree with only the zero shifted, the Celsius–Kelvin relation is a pure offset:

F = \frac{9}{5}C + 32 \qquad\qquad T(\text{K}) = C + 273.15

Figure

Three vertical thermometer scales side by side labelled Celsius Fahrenheit and Kelvin, each marked with its ice point and steam point, showing zero at the ice point for Celsius, thirty two degrees at the ice point for Fahrenheit, and two hundred seventy three point one five kelvin at the ice point for Kelvin, with a horizontal dashed line connecting the same physical hotness across all three scales

Two families of practical thermometer put this idea to work. A liquid-in-glass thermometer — the familiar mercury or alcohol thermometer — works because the liquid expands more than the surrounding glass as temperature rises, so it visibly rises up a narrow, uniform capillary tube. The tube is marked at the two fixed points and divided into equal intervals on the assumption that the liquid's expansion is exactly linear with temperature. That assumption is only approximately true: real thermometric liquids deviate slightly from perfect linearity, and different liquids deviate by different, non-matching amounts, so two mercury and alcohol thermometers calibrated at the same two fixed points can disagree slightly at points in between. A constant-volume gas thermometer takes a different route: it keeps a fixed mass of gas at fixed volume in a bulb and measures how its pressure changes with temperature, because the pressure of a low-density gas held at constant volume varies almost exactly linearly with absolute temperature. This closeness to ideal behaviour makes the gas thermometer both more accurate and more universal than any liquid-in-glass design — remarkably, different gases (hydrogen, nitrogen, helium) at low density all give nearly identical pressure–temperature readings when cross-checked against each other, unlike mercury thermometers, whose readings vary slightly with the specific liquid and the glass used. This near-universality is exactly why the gas thermometer, not the mercury thermometer, is the practical instrument behind the definition of the absolute Kelvin scale.

Derivation ADeriving the Celsius-to-Fahrenheit conversion from two fixed points

The goal here is to derive the single most-used formula in this topic, $F = \frac{9}{5}C + 32$ , starting from nothing more than the definition of a fixed point and the assumption that a thermometric scale divides the interval between its two fixed points into equal steps. The derivation is pure ratio-and-proportion algebra — no calculus and no physics beyond "equal fractional progress on any linear scale represents the same physical warming" — and the same method, run with different fixed-point numbers, produces every other scale-conversion formula a student will ever need, including the Celsius-to-Kelvin relation.

Step 1. Write down the two fixed points on each scale. On the Celsius scale the ice point is $C_1 = 0$ and the steam point is $C_2 = 100$ . On the Fahrenheit scale the same two physical states are $F_1 = 32$ and $F_2 = 212$ . Both scales describe the same interval of physical hotness — from melting ice to boiling water at standard pressure — just labelled with different numbers and split into a different number of divisions.

Step 2. State the linearity assumption as a ratio. Because both scales are assumed to vary linearly with the underlying physical temperature over this range, a reading that is a certain fraction of the way from the lower fixed point to the upper fixed point on the Celsius scale must be that exact same fraction of the way from the lower to the upper fixed point on the Fahrenheit scale. Writing $C$ for an arbitrary Celsius reading and $F$ for the corresponding Fahrenheit reading, this fraction equality is

\frac{C - C_1}{C_2 - C_1} = \frac{F - F_1}{F_2 - F_1}.

Step 3. Substitute the fixed-point numbers. $C_2 - C_1 = 100 - 0 = 100$ and $F_2 - F_1 = 212 - 32 = 180$ , so

\frac{C - 0}{100} = \frac{F - 32}{180}.

Step 4. Solve for $F$ in terms of $C$ . Cross-multiplying, $180\,C = 100\,(F-32)$ , so $F - 32 = \frac{180}{100}C = \frac{9}{5}C$ , which rearranges to the familiar result

F = \frac{9}{5}C + 32.

Notice where the $\frac{9}{5}$ comes from: it is simply the ratio of the number of Fahrenheit divisions to Celsius divisions, $180/100$ , over the same physical span — Fahrenheit degrees are smaller, so more of them are needed to cover the same warming, hence the factor greater than one.

Step 5. Read off the Kelvin case as a special, simpler instance of the same rule. Because the kelvin is deliberately defined to be the same size as the Celsius degree ( $100$ Kelvin divisions also span ice point to steam point), the ratio in Step 4 collapses to $1$ , and the two scales differ only by an additive offset equal to where Celsius's zero sits on the Kelvin scale: $T(\text{K}) = C + 273.15$ . The general two-fixed-point method used above always reduces to a pure offset whenever the two scales share the same degree size, and to a scaling-plus-offset (like Celsius–Fahrenheit) whenever they do not.

Worked examples

Board level — body temperature in Celsius and Kelvin

Normal human body temperature is often quoted as $98.6^\circ\text{F}$ . Convert this to Celsius and to Kelvin.

Step 1. Start from $F = \frac{9}{5}C + 32$ and solve for $C$ : $C = \frac{5}{9}(F-32)$ .

Step 2. Substitute $F = 98.6$ : $C = \frac{5}{9}(98.6 - 32) = \frac{5}{9}(66.6) = 37.0^\circ\text{C}$ .

Step 3. Convert to Kelvin using the pure offset: $T = C + 273.15 = 37.0 + 273.15 = 310.15\ \text{K}$ .

So normal body temperature is $37.0^\circ\text{C}$ , or $310.15\ \text{K}$ . As a sanity check, this sits comfortably between the ice point ( $273.15\ \text{K}$ ) and the steam point ( $373.15\ \text{K}$ ), exactly where a warm human body should fall.

Competitive level — a constant-volume gas thermometer reading

A constant-volume gas thermometer reads a pressure of $80\ \text{kPa}$ at the triple point of water ( $273.16\ \text{K}$ ). When placed in contact with an unknown liquid, the pressure reading becomes $95\ \text{kPa}$ . Assuming pressure is directly proportional to absolute temperature at constant volume, find the temperature of the liquid in kelvin and in Celsius.

Step 1. Write down the constant-volume gas-thermometer relation. Since the gas is held at fixed volume and low density, $P \propto T$ , so the ratio of any two pressure readings equals the ratio of the corresponding absolute temperatures: $\frac{T}{273.16} = \frac{P}{P_{\text{triple}}}$ .

Step 2. Substitute the known pressures: $T = 273.16 \times \frac{95}{80} = 273.16 \times 1.1875$ .

Step 3. Carry out the multiplication: $T = 324.5\ \text{K}$ (to four significant figures).

Step 4. Convert to Celsius for a physical sense of the number: $T = 324.5 - 273.15 \approx 51.3^\circ\text{C}$ — noticeably warmer than body temperature but well below boiling, a physically sensible reading for, say, a warm bath.

The liquid is at approximately $324.5\ \text{K}$ , or $51.3^\circ\text{C}$ . Timing note: under a minute. The whole calculation is one proportion, $T = 273.16 \times (P/P_{\text{triple}})$ — the only trap is reaching for $273.15$ (the ordinary ice point) instead of $273.16$ (the triple point, the actual calibration point of a real gas thermometer); the two are numerically close enough that a careless swap rarely gets caught unless you deliberately check which fixed point the question specifies.

Common traps

Students lose marks on this topic almost entirely through arithmetic and definition slips, not conceptual confusion, so it is worth naming the recurring ones explicitly.

Writing $F = \frac{5}{9}C + 32$ instead of $F = \frac{9}{5}C + 32$ — the fraction is inverted. Remember Fahrenheit degrees are smaller than Celsius degrees, so more of them are needed to span the same physical change, which means the multiplier on $C$ must be greater than one, not less.
Forgetting that $-40^\circ$ is the one temperature where the Celsius and Fahrenheit readings coincide numerically ( $-40^\circ\text{C} = -40^\circ\text{F}$ ) — a fast way to sanity-check any conversion formula you have derived under exam pressure, rather than a trap in itself.
Assuming every liquid-in-glass thermometer is equally accurate. Real thermometric liquids like mercury and alcohol deviate slightly from perfect linearity between the fixed points, and different liquids deviate differently — which is exactly why the gas thermometer, whose behaviour is much closer to ideal, is treated as the more fundamental practical standard.
Confusing the triple point of water ( $273.16\ \text{K}$ , a single unique state where ice, liquid water and water vapour coexist in equilibrium at one specific pressure) with the ordinary ice point ( $273.15\ \text{K}$ at standard atmospheric pressure, one of two fixed points used on the everyday Celsius scale) — they are numerically close but conceptually distinct, and the modern Kelvin scale is anchored on the triple point precisely because it needs no external pressure specification to reproduce.
Plugging a Celsius or Fahrenheit reading directly into $P \propto T$ for the gas thermometer — that relation only holds for the absolute (Kelvin) temperature; forgetting to convert first gives a nonsensical negative or wildly wrong pressure ratio.

Common trapThe gas thermometer's P proportional to T rule needs absolute temperature, not Celsius

The single biggest scale-conversion error in this topic is applying $P \propto T$ with a Celsius or Fahrenheit number sitting in for $T$ . Proportionality statements like this only hold for a scale whose zero is the true physical zero of the quantity involved — here, absolute zero, where (ideally) the pressure of a fixed-volume gas would fall to zero. Celsius and Fahrenheit have arbitrary zeros fixed by the freezing point of water, not by any physical "zero hotness," so doubling a Celsius reading does not double the underlying physical temperature, and $P \propto C$ is simply false. Always convert to Kelvin first, do the proportion, and only convert back to Celsius or Fahrenheit at the very end if the question asks for it.

You should be able to…

Explain what a thermometric property is and name at least two examples used in real thermometers.
Derive the Celsius–Fahrenheit conversion formula from the two-fixed-point linear interpolation rule, and quote the Celsius–Kelvin offset.
Apply the general two-point interpolation formula to convert a temperature between any two linear scales, including the constant-volume gas thermometer's pressure-to-temperature relation.
Describe how a liquid-in-glass thermometer and a constant-volume gas thermometer each work, and state why the gas thermometer is considered more accurate and more universal.
Distinguish the triple point of water ( $273.16\ \text{K}$ ) from the ordinary ice point ( $273.15\ \text{K}$ ), and explain why the triple point is preferred as a calibration standard.
Recognise when a proportionality like $P \propto T$ requires absolute (Kelvin) temperature rather than a Celsius or Fahrenheit reading.

Ask the tutor: want more scale-conversion drills, or a walk-through of how the constant-volume gas thermometer's pressure–temperature graph is used to extrapolate toward absolute zero?

Why does the Fahrenheit scale even exist if Celsius and Kelvin are so much simpler

It is mostly historical accident and inertia. Daniel Fahrenheit proposed his scale in the early eighteenth century, before the ice-and-steam-point convention was standardised, using a brine mixture and body temperature as early reference points; it later got redefined onto the same ice-and-steam fixed points but kept its original $180$ -division spacing. Celsius came later with a cleaner $0$ -to- $100$ design, and Kelvin came later still as the scientific absolute scale. The United States and a few other countries simply never switched their everyday usage away from Fahrenheit, so it survives in weather reports and thermostats even though virtually all of science, and most of the world, uses Celsius and Kelvin.

Why do we need three scales instead of just one

Because they serve different purposes. Celsius is convenient for everyday and most laboratory use because its zero sits at a familiar, easily reproduced physical event, water freezing. Kelvin is essential for physics calculations — anything involving proportionality to temperature, like the gas thermometer relation $P \propto T$ or the ideal gas law you will meet next, only works cleanly if the zero is the true physical zero of temperature, absolute zero. Fahrenheit persists purely by regional convention. In exams, always check which scale a formula assumes before plugging numbers in.

Is absolute zero actually reachable in a lab

No, and this is a genuinely deep result rather than just an engineering limitation — the third law of thermodynamics (a topic beyond this chapter) says absolute zero can be approached arbitrarily closely but never exactly reached in a finite number of steps. Laboratories have cooled small samples to within a billionth of a kelvin of $0\ \text{K}$ using techniques like laser cooling, but the true zero itself stays permanently out of reach. For this chapter, treat absolute zero as the theoretical reference point that fixes where the Kelvin scale starts, not as a temperature you would expect any real experiment to hit.

Why is the triple point of water used to define the Kelvin scale instead of the ordinary ice and steam points

Because the ordinary ice point and steam point both depend on atmospheric pressure — melting ice sits at exactly $0^\circ\text{C}$ only at standard pressure, and boiling water's temperature shifts noticeably with altitude. The triple point, where ice, liquid water and water vapour all coexist in equilibrium, occurs at one single, sharply defined combination of temperature and pressure with no external pressure choice needed, which makes it a far more precise and universally reproducible calibration point. That is why the Kelvin scale is anchored on the single number $273.16\ \text{K}$ at the triple point, rather than on two separate ice-and-steam fixed points the way Celsius and Fahrenheit are.