During an epidemic, a guard at a railway station points a small gun at each passenger's forehead. He reads a number in one second. Nothing touches the skin. A doctor's mercury thermometer needs a minute under the tongue. A furnace at a steel plant is too hot for either. All three are thermometers.
What makes a property useful
Lesson 1 used the length of a mercury column to measure temperature. Many other properties change with temperature. The pressure of a gas held in a closed bulb rises as it warms. The electrical resistance of a metal wire rises too. A property used to measure temperature is a thermometric property.
Three conditions make a property useful. It must change steadily as the temperature changes, so that small changes can be read. One value of the property must mean one temperature only. And the same temperature must always give the same value, today and next year. A property that fails the second test cannot make a thermometer, however large its change.
Water fails the second test near its freezing point. Between \(0\,°\text{C}\) and \(8\,°\text{C}\) a mass of water first shrinks and then grows as it warms. So \(2\,°\text{C}\) and \(6\,°\text{C}\) give it the same volume. Lesson 5 shows why.
Each liquid also has a range. Mercury freezes at \(-39\,°\text{C}\) and boils at \(357\,°\text{C}\), so a mercury thermometer can read neither \(-50\,°\text{C}\) nor \(400\,°\text{C}\). Alcohol stays liquid down to about \(-115\,°\text{C}\) and serves in cold places, but it boils at \(78\,°\text{C}\).
Which of these could serve as the thermometric property of a thermometer between \(10\,°\text{C}\) and \(90\,°\text{C}\)?
The gas thermometer
A constant-volume gas thermometer is a glass bulb of gas joined by a narrow tube to a mercury manometer. The bulb is put in the body whose temperature is wanted. As the gas warms, it pushes the mercury in the near arm down. The open far arm is then raised until the near level is back at a fixed mark A. So the gas always fills the same volume, and only its pressure changes.
The mercury in the open arm then stands a height \(h\) above the mark. The air presses on the open arm with the atmospheric pressure \(p_0\). So the pressure of the gas is \(p = p_0 + h\rho g\), where \(\rho\) is the density of mercury. It is often quoted as a length of mercury: \(p_0 + h\), in centimetres.
In Lesson 2 the pressure of a gas at constant volume was a straight line through absolute zero. So the pressure is proportional to the absolute temperature, \(p \propto T\). A thermometer needs only one fixed point to set the constant. The modern choice is the triple point of water, the one state where ice, liquid water and water vapour exist together. It occurs at a single pressure, \(611\,\text{Pa}\), and its temperature is defined to be exactly \(273.16\,\text{K}\).
In the sim at \(25\,°\text{C}\), the mercury stands \(6.95\,\text{cm}\) higher, so the gas is at \(82.95\,\text{cm}\) of mercury. Then \(T = 273.16 \times 82.95/76.0 = 298.1\,\text{K}\), which is \(25\,°\text{C}\). At \(100\,°\text{C}\) the head grows to \(27.8\,\text{cm}\).
Why the gas thermometer is the standard
Different gases in different bulbs give slightly different readings at ordinary pressures. As the amount of gas in the bulb is reduced, the differences shrink. In the limit of low pressure every gas gives the same temperature. So the scale does not depend on the gas chosen. It also works over a very wide range, from a few kelvin to well over \(1000\,°\text{C}\). For these reasons it is the standard against which other thermometers are checked.
The triple point is a better fixed point than the ice point. The melting point of ice shifts slightly with the air pressure and with air dissolved in the water. The triple point has only one possible pressure and temperature, so any laboratory can reproduce it exactly in a sealed glass cell.
(a) The pressure in a constant-volume gas thermometer is \(1.2\) times its triple-point value. Find the temperature. (b) Another gas thermometer reads \(72\,\text{cm}\) of mercury at the ice point and \(96\,\text{cm}\) in a hot bath. Find the temperature of the bath, taking the ice point as \(273\,\text{K}\).
(a) \(T = 273.16 \times 1.2 = 327.8\,\text{K}\), which is \(327.8 - 273.15 = 54.6\,°\text{C}\).
(b) Here the fixed point is the ice point: \(T = 273 \times \dfrac{96}{72} = 364\,\text{K}\), which is \(91\,°\text{C}\).
The pressure of the gas in a constant-volume gas thermometer is \(1.5\) times its value at the triple point of water. What is the temperature, in kelvin?
Why is the constant-volume gas thermometer chosen as the standard thermometer?
The platinum resistance thermometer
The resistance of a pure platinum wire rises almost exactly in a straight line with temperature. Write it as \(R_t = R_0(1 + \alpha t)\). Here \(R_0\) is the resistance at \(0\,°\text{C}\) and \(\alpha\) is the temperature coefficient of resistance. The resistance is the property \(X\) of Lesson 1, so the temperature follows from the two fixed points. A platinum resistance thermometer is accurate to a hundredth of a degree and works from about \(-200\,°\text{C}\) to over \(600\,°\text{C}\).
For the sensor of Fig 3.1, \(\alpha = 38.5/(100 \times 100) = 0.00385\) per °C. A reading of \(119.25\,\Omega\) gives \(t = 100 \times 19.25/38.5 = 50\,°\text{C}\).
A Pt-100 sensor reads \(100.0\,\Omega\) at \(0\,°\text{C}\) and \(138.5\,\Omega\) at \(100\,°\text{C}\). In a tank of milk it reads \(127.0\,\Omega\). What is the temperature of the milk, in °C?
Thermocouples, clinical thermometers and infrared guns
A thermocouple is two wires of different metals joined at both ends. When the two junctions are at different temperatures, a small emf appears in the loop. The emf grows with the difference in temperature. Thermocouples are cheap, tiny and robust. They read furnaces at over \(1500\,°\text{C}\), and one guards the pilot flame of many gas geysers.
A clinical mercury thermometer reads only from about \(35\,°\text{C}\) to \(42\,°\text{C}\), the range of a human body, so its degrees can be widely spaced. Just above the bulb the bore has a narrow kink. When the thermometer is taken out of the mouth, the mercury in the bulb contracts. The thread breaks at the kink and stays up. So the thermometer holds its highest reading until it is shaken down.
Every warm body sends out infrared radiation, and a hotter body sends out more. An infrared thermometer collects this radiation through a lens and turns it into a temperature. It needs no contact and answers in a second, which is why it screens passengers at a railway station. The chapter on radiation explains the rule it uses.
Why does a clinical mercury thermometer keep showing the patient's temperature after it is taken out of the mouth?
Around 1593 Galileo, then a professor at Padua, made a glass bulb with a long open neck. He dipped the neck in a bowl of water. When the air in the bulb warmed, it expanded and pushed the water down the neck. When it cooled, the water rose. His thermoscope showed that one day was warmer than another, but it had no fixed points and no numbers. It also changed with the air pressure outside. Scales with fixed points came more than a century later.
In 2020 the entrances of Indian railway stations and airports were fitted with infrared thermometers. A guard aims the gun at a forehead from a few centimetres away. The gun reads the infrared radiation of the skin and shows a temperature at once. A mercury thermometer would have needed a minute per passenger and a fresh one for each mouth.
- Using a thermometer outside the range of its liquid.Mercury freezes at \(-39\,°\text{C}\) and boils at \(357\,°\text{C}\). For colder places use alcohol; for hotter use platinum, a gas or a thermocouple.
- Thinking a gas thermometer measures the volume of the gas.The volume is held fixed by returning the mercury to the mark. The pressure is what is measured.
- Using Celsius temperatures in \(T = 273.16 \times p/p_\text{tr}\).The pressure is proportional to the absolute temperature, so the result is in kelvin.