Gas Thermometers & the Kelvin Scale
Gases agree with each other at low pressure, and give a scale independent of any one substance.
Lesson at a glanceShow the key points
- Constant-volume gas thermometer: keep the gas volume fixed with a mercury manometer and reservoir; p = p₀ + hρg.
- Define T ∝ p with the triple point of water at 273.16 K: T = (p/p_tr) × 273.16 K. A centigrade scale: t = (p − p₀)/(p₁₀₀ − p₀) × 100.
- Errors: gas in the connecting capillary is outside the bath; the bulb expands.
- Different gases agree in the limit p_tr → 0: the ideal-gas scale, identical to the absolute (kelvin) scale. Steam point 373.15 K.
- Celsius: θ = T − 273.15; same size degree as the kelvin.
- Ideal gas equation pV = nRT, R = 8.314 J K⁻¹ mol⁻¹.
- Callendar's thermometer compensates the capillary error: T = V T₀/(V − v′).
Mercury and platinum disagree about temperature. Gases, especially thin ones, almost all agree, and their agreement gives us the kelvin scale.
The constant-volume gas thermometer
A gas is sealed in a bulb A connected by a capillary to a mercury manometer. The manometer’s other arm is open to the air and joined by a rubber tube to a movable mercury reservoir. Raising or lowering the reservoir keeps the mercury in the near arm at a fixed mark C, so the gas volume stays constant. The gas pressure is then
where p₀ is atmospheric pressure and h the difference in mercury levels. At constant volume, p rises steadily as the gas warms.
Defining T
Take the temperature to be proportional to the pressure, T = cp, and fix c with one reference: the triple point of water (ice, water and vapour together in equilibrium) is assigned 273.16 K. If the gas pressure there is p_tr:
A centigrade scale can be built the same way, from the pressures at the ice and steam points:
A constant-volume gas thermometer reads 74.0 cm of mercury at 0°C, 101.5 cm at 100°C and 79.0 cm in a room. Find the room temperature.
A gas thermometer reads 75 cm of mercury in melting ice (273.15 K) and 180 cm in hot oil. Find the oil’s temperature.
The ideal-gas scale
Gas thermometers using different gases, or different amounts of the same gas, give slightly different readings for the steam point. But as the amount of gas (measured by p_tr) is made smaller, they converge to the same value, 373.15 K. So we define
This scale doesn’t depend on which gas is used. It turns out to be identical to the absolute temperature scale, which can be defined without reference to any substance at all.
The Celsius scale
On the ideal-gas scale the ice point is 273.15 K and the steam point 373.15 K. The Celsius scale uses degrees of exactly the same size, shifted so the ice point is 0°C:
The old mercury centigrade scale is very close to it.
The ideal gas equation
At low pressures (and far above their condensing temperatures) gases behave alike and obey
Callendar’s compensated thermometer
Callendar removed the capillary error with a twin design. Bulb A (in the bath) and its capillary connect to a mercury-filled, graduated bulb B. An identical bulb C and capillary, holding an equal amount of the same gas, connect to the other side of a manometer. B and C sit in melting ice. When A is heated, mercury is drawn off from B until the manometer levels are equal again. Equating the amounts of gas on the two sides (the capillary terms cancel):
The bulbs have volume 100 cm³. To balance the manometer, 30 cm³ of mercury is drawn from B. What is the bath temperature?
- 1Constant-volume gas thermometer: p = p₀ + hρg; T = (p/p_tr) × 273.16 K, triple point of water = 273.16 K.
- 2Different gases agree as p_tr → 0: the ideal-gas scale, identical to the absolute (kelvin) scale.
- 3Celsius: θ = T − 273.15, same size degree.
- 4pV = nRT with R = 8.314 J K⁻¹ mol⁻¹.
- 5Callendar's thermometer cancels the capillary error: T = V T₀/(V − v′).
Check yourself
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