Thermometers and thermometric properties
A measure of temperature is obtained using a thermometer. Its principle is simple: many physical properties of matter change appreciably with temperature, and any such property that varies smoothly and reproducibly can be pressed into service as a thermometric property. NCERT highlights the most common one — the volume of a liquid. In liquid-in-glass thermometers, mercury or alcohol expands almost linearly with temperature over a wide range, so the height of the column becomes a stand-in for the temperature.
The volume of a liquid is not the only choice. Different thermometers exploit different thermometric properties, and the property chosen sets the instrument's range, sensitivity and accuracy.
| Thermometer type | Thermometric property | Typical use |
|---|---|---|
| Liquid-in-glass (mercury, alcohol) | Volume / length of the liquid column | Everyday and laboratory measurements over a wide range |
| Constant-volume gas thermometer | Pressure of a fixed mass of gas at constant volume | Standard reference; defines the absolute scale |
| Resistance thermometer | Electrical resistance of a metal (e.g. platinum) | Precise wide-range measurements |
| Thermocouple | Thermo-emf across a junction of two metals | Rapid and high-temperature measurements |
Whatever the property \(X\), a thermometer is useful only after it is calibrated — a numerical value must be assigned to each value of \(X\) in some agreed scale. Calibration needs reference temperatures, and that is where fixed points enter.
Fixed points and calibration
To define a standard scale, two fixed reference points are required. There is a subtlety NCERT is careful about: because all substances change their dimensions with temperature, there is no absolute, material-independent reference for expansion. The way out is to anchor the scale to physical phenomena that always occur at the same temperature. The two conventional anchors are the ice point and the steam point of water.
| Fixed point | Physical definition | Celsius | Fahrenheit |
|---|---|---|---|
| Ice point (lower) | Pure water freezes under standard pressure | 0 °C | 32 °F |
| Steam point (upper) | Pure water boils under standard pressure | 100 °C | 212 °F |
| Interval between them | Number of equal divisions assigned | 100 divisions | 180 divisions |
Once the two fixed points are pinned, the rest of the scale follows by interpolation. The thermometric property is assumed to vary linearly between the lower and upper points, and the interval is split into equal divisions — 100 on Celsius, 180 on Fahrenheit. The choice of how many divisions to use, and where to put the zero, is exactly what distinguishes one scale from another.
The three scales at a glance
Three temperature scales matter for NEET: the Celsius scale, the Fahrenheit scale and the Kelvin (absolute) scale. They differ only in two decisions — where the zero sits, and how big one division is. The figure below sets the three side by side against the same two physical anchors.
Converting between scales
The Celsius–Fahrenheit relation comes straight from the fixed points. Both scales measure the same physical interval between ice and steam, but Celsius cuts it into 100 parts starting at 0, while Fahrenheit cuts it into 180 parts starting at 32. Equating the fraction of the interval covered on each scale gives NCERT Eq. 10.1:
$$\frac{t_F - 32}{180} = \frac{t_C - 0}{100} \quad\Longrightarrow\quad \frac{t_F - 32}{9} = \frac{t_C}{5}.$$
Rearranging into the working form most useful in the exam,
$$\boxed{\,t_F = \frac{9}{5}\,t_C + 32\,} \qquad t_C = \frac{5}{9}\,(t_F - 32).$$
The link to the Kelvin scale is even simpler, because Kelvin keeps the Celsius division size and only shifts the zero:
$$\boxed{\,T(\text{K}) = t_C + 273.15\,}.$$
| Conversion | Relation | One-line check |
|---|---|---|
| Celsius → Fahrenheit | t_F = (9/5) t_C + 32 | 0 °C → 32 °F; 100 °C → 212 °F |
| Fahrenheit → Celsius | t_C = (5/9)(t_F − 32) | 32 °F → 0 °C; 212 °F → 100 °C |
| Celsius → Kelvin | T = t_C + 273.15 | 0 °C → 273.15 K; 100 °C → 373.15 K |
| Kelvin → Celsius | t_C = T − 273.15 | 0 K → −273.15 °C (absolute zero) |
The general linear scale rule
Every one of these conversions is a special case of one idea: between two fixed points, a thermometric property varies linearly, so the fraction of the way from the lower fixed point to the upper one is the same on every scale. If a thermometric quantity \(X\) (a column length, a pressure, a resistance) reads \(X\) at the unknown temperature, with \(X_{\text{lower}}\) and \(X_{\text{upper}}\) at the two fixed points, then
$$\frac{X - X_{\text{lower}}}{X_{\text{upper}} - X_{\text{lower}}} = \text{invariant across all scales}.$$
Set the right-hand side equal for two scales and the conversion drops out. For Celsius and Fahrenheit the lower and upper points are \((0, 100)\) and \((32, 212)\), which reproduces \(\dfrac{t_C - 0}{100 - 0} = \dfrac{t_F - 32}{212 - 32}\) — the very relation above. This invariant is also how an uncalibrated thermometer is read: measure the property at the two fixed points, measure it at the unknown temperature, and substitute.
Why Kelvin is the absolute scale
Celsius and Fahrenheit both place their zero at a convenient but arbitrary spot — the ice point, or 32 divisions below it. The Kelvin scale, named after Lord Kelvin, instead puts its zero at the lowest temperature that can exist, absolute zero. NCERT locates this point experimentally: cool a low-density gas at constant volume and its pressure falls; extrapolate the pressure–temperature line to zero pressure and, for every gas, the line meets the temperature axis at the same value, \(-273.15\,^\circ\text{C}\). That common intercept is absolute zero, defined as \(0\) K.
Because its zero is physical rather than chosen, the Kelvin scale is the SI base scale for thermodynamic temperature and the only scale on which ratios are meaningful. The size of one kelvin is the same as the size of one degree Celsius, so the scale is just Celsius slid down by 273.15:
$$T(\text{K}) = t_C + 273.15.$$
| Feature | Celsius | Fahrenheit | Kelvin (absolute) |
|---|---|---|---|
| Zero point | Ice point (arbitrary) | 32° below ice point (arbitrary) | Absolute zero (physical) |
| Symbol | °C | °F | K (no degree) |
| Ice point reads | 0 °C | 32 °F | 273.15 K |
| Steam point reads | 100 °C | 212 °F | 373.15 K |
| Divisions ice→steam | 100 | 180 | 100 |
| SI status | Common unit | Non-SI | SI base unit |
The constant-volume gas thermometer
The reason a gas underlies the absolute scale is that gases behave alike where liquids do not. Liquid-in-glass thermometers disagree away from the fixed points because mercury and alcohol expand by different amounts; a gas thermometer gives the same reading whichever gas is used, because all gases at low density expand in the same way. NCERT states the working relation: at constant volume the pressure of a fixed mass of gas is proportional to its absolute temperature, \(P \propto T\), so temperature can be read directly off pressure.
A plot of pressure against temperature is therefore a straight line, and extrapolating that line to \(P = 0\) gives the absolute-zero intercept used to fix the Kelvin scale. Real gases deviate from the ideal prediction at low temperature, but the relation is linear over a wide range, which is enough to define the scale.
The full \(PV = \mu R T\) treatment, Boyle's and Charles's laws, and the formal definition of absolute temperature are developed in ideal-gas equation & absolute temperature.
Worked conversions
Normal human body temperature is 37 °C. Express it in (a) Fahrenheit and (b) Kelvin.
(a) \(t_F = \tfrac{9}{5}t_C + 32 = \tfrac{9}{5}(37) + 32 = 66.6 + 32 = 98.6~^\circ\text{F}\) — the familiar "98.6".
(b) \(T = t_C + 273.15 = 37 + 273.15 = 310.15~\text{K}\).
At what temperature is the Fahrenheit reading numerically equal to the Celsius reading?
Set \(t_F = t_C = x\) in \(t_F = \tfrac{9}{5}t_C + 32\): \(x = \tfrac{9}{5}x + 32\), so \(x - \tfrac{9}{5}x = 32\), i.e. \(-\tfrac{4}{5}x = 32\), giving \(x = -40\).
Answer: \(-40~^\circ\text{C} = -40~^\circ\text{F}\). This is the single coincidence point of the two scales.
A faulty thermometer reads 0 °C at the ice point and 99 °C at the steam point. What true Celsius temperature does it show when it reads 33 °C?
Use the invariant fraction. The instrument's span is 0 to 99 (99 divisions), the true span is 0 to 100. The fraction of the way up is the same: \(\dfrac{t_{\text{true}} - 0}{100 - 0} = \dfrac{33 - 0}{99 - 0} = \dfrac{1}{3}\).
So \(t_{\text{true}} = \tfrac{1}{3}\times 100 = 33.3~^\circ\text{C}\). The same linear-scale rule that derives the C–F conversion also corrects a mis-graduated thermometer.
Temperature scales in one breath
- A thermometer reads temperature from a thermometric property (liquid volume, gas pressure, resistance) calibrated against two fixed points.
- Fixed points: ice point (0 °C, 32 °F, 273.15 K) and steam point (100 °C, 212 °F, 373.15 K).
- Celsius–Fahrenheit: \(t_F = \tfrac{9}{5}t_C + 32\); equivalently \(\dfrac{t_F-32}{180} = \dfrac{t_C}{100}\).
- Celsius–Kelvin: \(T = t_C + 273.15\); 1 K interval = 1 °C interval.
- General rule: \(\dfrac{X - X_{\text{lower}}}{X_{\text{upper}} - X_{\text{lower}}}\) is the same on every scale.
- Kelvin is the SI absolute scale: zero is absolute zero (\(-273.15\) °C), no degree symbol, ratios meaningful.
- The constant-volume gas thermometer reads \(P \propto T\) and fixes absolute zero by extrapolating to \(P = 0\).
- Memorise: \(-40~^\circ\text{C} = -40~^\circ\text{F}\); 37 °C = 98.6 °F = 310.15 K.