Physics 2nd Paper · Classes 11-12 · Chapter 1 · 1.1

Principle of Measurement of Temperature.

A wooden ball and an iron ball lie in the same room at the same temperature. Touch them and the iron one feels colder. So your hand cannot be trusted. Measuring temperature needs an instrument that watches some property of a substance instead of how it feels. This note covers the principle behind that instrument, its scales and a few calculations.

1. Why touch is not enough

A wooden cricket ball and an iron ball at the same temperature feel different to the touch: the iron ball feels colder. So touch alone cannot give a correct idea or a measure of temperature.

Temperature is a thermal state of a body. That state controls whether heat flows from the body to another one, and in which direction. How hot or cold a body feels is its apparent temperature: apparently, temperature means the degree of heat of the body. An instrument that measures temperature accurately is a thermometer.

2. What a thermometer actually watches

When temperature changes, some properties of a substance change regularly with it. A property whose change lets you find temperature easily and accurately is a thermometric property, and the substance whose property goes into the thermometer is the thermometric substance. In a mercury thermometer the expansion of mercury is the thermometric property and mercury is the thermometric substance.

Three thermometers sit in one bath: mercury, platinum resistance and a thermocouple. Heat the water and see which property each one reads the temperature by.

Not every thermometer works by expansion. A platinum resistance thermometer uses the electrical resistance of platinum, a thermocouple uses the thermal e.m.f. of a pair of metals, and a radiation pyrometer uses the radiation of a hot body. The pyrometer measures temperatures above 500°C. Match all four to their property.

3. Thermal equilibrium and the zeroth law

Bodies that are thermally connected and exchange no heat are in thermal equilibrium. Heat flows from the hotter body to the colder one until both reach the same temperature. That is the moment they are in equilibrium.

Zeroth law: if two bodies are each in thermal equilibrium with a third body, the first two are in thermal equilibrium with each other. A conclusion follows: every body has a property that, when equal for two bodies, puts them in thermal equilibrium. That property is temperature. Thermometers are built on this law. Step through it and see why the wall can be removed.

4. Three scales, one relation

The Celsius (C), Fahrenheit (F) and Kelvin (K) scales are related by C/5 = (F − 32)/9 = (K − 273)/5. In 1927 the international committee on weights and measures approved a practical scale of temperature, the international temperature scale.

On the Celsius scale the ice point is 0°, the steam point is 100°, and this fundamental interval is divided into 100 parts.

On the Fahrenheit scale the ice point (lower fixed point) is 32°F, the steam point (upper fixed point) is 212°F, and the fundamental interval is divided into 180 parts, each division being 1°F. Fever is measured on a Fahrenheit thermometer: the clinical (doctor) thermometer reads from 95°F to 110°F.

Which of the three equal ratios to use, and the signs at −40 and absolute zero, are where slips happen. Check both by hand here. Choose the scale you know and type a number, or pick one of the common values. None of this goes anywhere; the notebook is yours.

When are the Celsius and Fahrenheit readings equal? Put C = F = x in x/5 = (x − 32)/9, so 9x = 5x − 160 and x = −40. At −40° the two scales read the same. For readings 40° apart, take the Celsius reading x and the Fahrenheit reading x ± 40. Solving gives x = 10°C (F = 50°) or x = −90°C (F = −130°).

Absolute zero, −273°C, reads F = (9/5)(−273) + 32 = −459.4°F. And the Fahrenheit and Kelvin readings are equal (F = K = x) at x = 574.25, from the book's relation. Two quick conversions: 5°C = (9/5)(5) + 32 = 41°F, and body heat of 98.6°F = 37°C ≈ 310.15 K.

5. Faulty thermometers and the resistance thermometer

For a faulty thermometer with ice point M, steam point B and reading S, C/100 = (K − 273.15)/100 = (F − 32)/180 = (S − M)/(B − M). The 273.15 appears only in this relation; in ordinary conversions the book uses K = C + 273.

A resistance thermometer follows the same idea: θ°C comes from θ = (R_θ − R₀)/(R₁₀₀ − R₀) × 100. Here R₀ is the resistance at the ice point, R₁₀₀ at the steam point, and R_θ at the unknown temperature.

A book example: a platinum resistance thermometer reads 2.57 Ω at 0°C and 3.53 Ω at 100°C. From 33.3 = (R_t − 2.57)/(3.53 − 2.57) × 100, the resistance at 33.3°C is R_t = 2.889 Ω.

In another, the resistance is 9 Ω at 0°C, 22 Ω at 100°C and 36 Ω at a liquid's boiling point. So θ = (36 − 9)/(22 − 9) × 100 = 207.7°C.

A faulty thermometer has lower fixed point 4°C and upper fixed point 98°C, and it reads 51°C. The true reading is C = 100 × (51 − 4)/(98 − 4) = 50°C.

6. The triple point and the kelvin

Triple point of water: the temperature at which pure ice, water and water vapour coexist in thermal equilibrium at a pressure of 4.58 mm of mercury. Its value is T_tr = 273.16 K. The S.I. unit of temperature is the kelvin: 1/273.16 of the temperature of the triple point of water is one kelvin (1 K).

The thermodynamic or absolute scale takes the triple point as 273.16 K and 1/273.16 of it as 1 K. It does not depend on the nature of any substance, only on temperature, which is why it is also called the absolute scale.

The international scale of temperature is the practical scale approved by the international weights and measures body, using the triple point as 273.16 K plus further easily reproduced fixed points. Examples: triple point of neon 24.5561 K, oxygen 54.3584 K, argon 83.8058 K, mercury 234.3156 K; freezing point of copper 1357.77 K.

If a thermometric property X varies uniformly with temperature T, then T = aX. With a fixed point of temperature T_p where the property is X_p, a = T_p/X_p, so T = T_p X/X_p. Taking the triple point as the fixed point, T = 273.16 × X/X_tr. In other words, the property is proportional to the temperature. For a resistance thermometer reading 30 Ω at the triple point and 45 Ω against some body, T = 273.16 × 45/30 = 409.74 K.

If the resistance is 32.316 Ω at the triple point and 27.316 Ω at some temperature, T = 273.16 × 27.316/32.316 = 230.9 K.

Fixed points and the number of divisions (100, 180, 273.16) get swapped between scales in exams. Match each fact to its scale.

7. Check yourself

Three questions in the exam's style. Start with the first.

Think a little on the second.

In the last one, watch the signs and numbers of the scales.

The whole note on eight cards.

Source: Physics 2nd Paper, Classes 11-12 (NCTB), Chapter 1 “Thermodynamics”, section 1.1.