Temperature Scales & Thermometers
Pick a property that changes with temperature, two fixed points, and a straight line between.
Lesson at a glanceShow the key points
- Choose a property that changes steadily with temperature (mercury column length, resistance) and assume a linear relation t = al + b.
- Fixed points: ice point and steam point at 1 atm. Centigrade: 0 and 100; t = (l − l₀)/(l₁₀₀ − l₀) × 100.
- Fahrenheit: 32°F and 212°F, so F = 32 + 9C/5.
- Platinum scale: t = (R_t − R₀)/(R₁₀₀ − R₀) × 100, measured with a Wheatstone bridge (P/Q = R/X at balance) and a compensating wire to cancel the leads: t = (x_t − x₀)/(x₁₀₀ − x₀) × 100.
- Two scales agree at the fixed points but not in between: each depends on its substance.
The zeroth law tells us when two things are equally hot. To say how hot, we need a scale: a recipe for turning some measurable property into a number.
Building a scale
Pick a substance and a property of it that changes steadily as it gets hotter, such as the length l of a mercury column in a glass capillary. Then:
- Choose two easily reproduced fixed points: melting ice at 1 atm (the ice point) and boiling water at 1 atm (the steam point).
- Assign them temperatures, and assume temperature is a linear function of the property: t = al + b.
The centigrade scale puts the ice point at 0° and the steam point at 100°. With l₀ and l₁₀₀ the column lengths there:
Fahrenheit
The Fahrenheit scale puts the ice point at 32°F and the steam point at 212°F, so 180 Fahrenheit degrees span the same interval as 100 Celsius degrees:
A patient’s temperature is 39.5°C. What is it in °F? And at what temperature do the two scales read the same?
The platinum resistance thermometer
The electrical resistance of a metal wire also rises steadily with temperature. Define the platinum scale by
A fine platinum coil wound on mica sits in a glass tube dipped into the bath. Its resistance is measured with a Wheatstone bridge: four resistances P, Q, R and X in a loop, with a galvanometer across the middle. There is no deflection (the bridge is balanced) when .
The leads to the platinum coil also have resistance, which changes with temperature too. A compensating wire of the same kind runs alongside them into the tube and is placed in the opposite arm, so the leads cancel. With P = Q and a sliding contact at distance x along a uniform bridge wire, the balance point moves linearly with R_t, and
A platinum coil has resistance 5.00 Ω at the ice point and 5.39 Ω at the steam point. In a furnace it reads 5.80 Ω. Find the temperature on the platinum scale.
Different scales disagree
By definition the platinum scale makes resistance rise uniformly with its own temperature, and the mercury scale makes mercury expand uniformly with its own. Compare them and they agree at 0 and 100 (they were forced to) but not in between: measured on the mercury scale, platinum’s resistance rises slowly at low temperatures and faster at high ones. Each scale depends on its substance.
- 1A scale: a thermometric property, two fixed points (ice point and steam point), and an assumed linear relation.
- 2Centigrade: t = (l − l₀)/(l₁₀₀ − l₀) × 100. Fahrenheit: F = 32 + 9C/5.
- 3Platinum scale from resistance; measured with a Wheatstone bridge and compensating leads.
- 4Scales based on different substances agree only at the fixed points.
Check yourself
Keep exploring
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