Temperature, Scales, and Thermal Expansion

Temperature is a measure of the average kinetic energy of the particles of a substance. The three common scales:

• Celsius (°C) — ice point 0°C, steam point 100°C (100 divisions).
• Fahrenheit (°F) — ice point 32°F, steam point 212°F (180 divisions).
• Kelvin (K) — the SI absolute scale; 0 K is absolute zero. A change of 1 K equals a change of 1°C (same size step); they differ only by the offset 273.15.

Kelvin is never written with a degree sign — it is '300 K', not '300°K'.

Absolute zero is the lowest possible temperature: 0 K = −273.15°C (often quoted as −273°C). At this point the thermal kinetic energy of particles is at its minimum. Thermometer choice by range:

• Liquid-in-glass (mercury / alcohol) — ordinary lab and clinical use.
• Thermocouple thermometer — wide range including very low (around −250°C) and very high temperatures; based on the voltage produced at a junction of two metals.
• Constant-volume gas thermometer — the most accurate standard, used to define the Kelvin scale.

To find where two scales read the same numerical value, set their variables equal and solve:

• C = F: put $F = C$ into $F = \frac{9}{5}C + 32$ and solve. Answer: −40° (the one temperature where Celsius and Fahrenheit coincide).
• K = F: put $K = F$ into $K = C + 273$ and $F = \frac{9}{5}C + 32$, then eliminate C.

The trick is purely algebraic: two linear relations, one unknown.

For a solid heated through $\Delta\theta$:

• Linear: $\Delta L = L\alpha\,\Delta\theta$ — $\alpha$ is the coefficient of linear expansion.
• Areal (superficial): $\Delta A = A\beta\,\Delta\theta$ — $\beta$ is the coefficient of areal expansion.
• Volume (cubical): $\Delta V = V\gamma\,\Delta\theta$ — $\gamma$ is the coefficient of volume expansion.

They are related by the fixed ratio $\alpha : \beta : \gamma = 1 : 2 : 3$, so $\beta = 2\alpha$ and $\gamma = 3\alpha = \tfrac{3}{2}\beta$. Anomalous expansion of water: water is the famous exception — between 0°C and 4°C it CONTRACTS on heating, reaching maximum density at 4°C. Above 4°C it expands normally. This is why ice floats and ponds freeze from the top down (the diagram below shows the density peak).

• Pendulum clock: the period is $T = 2\pi\sqrt{L/g}$. When the rod is heated, $L$ grows, so $T$ increases — the clock loses time (runs slow) in hot weather. The fractional change is small: $\frac{\Delta T}{T} = \tfrac{1}{2}\alpha\,\Delta\theta$.
• Liquid expansion: a liquid is held in a container that ALSO expands. The observed rise gives only the apparent expansion; the real (absolute) expansion = apparent expansion + expansion of the container. This is why a liquid's coefficient is harder to measure than a solid's.