Concepts: temperature, thermometry and the ideal gas
Thermal Properties of Matter24 min readLesson 2 of 6
2.1 Temperature and thermal equilibrium
Temperature is the property that decides the direction of heat flow. When two bodies are placed in contact, heat flows from the hotter one to the colder one until no net flow remains; at that point the bodies are in thermal equilibrium and share the same temperature. The zeroth law of thermodynamics makes this statement precise: if two systems are each in equilibrium with a third, they are in equilibrium with each other. A thermometer is exactly such a third system. Any property that changes steadily with temperature, such as the length of a mercury column, the resistance of a wire or the pressure of a fixed volume of gas, can be used to build one.
2.2 The Celsius, Fahrenheit and Kelvin scales
The Celsius scale sets the ice point at 0 degrees C and the steam point at 100 degrees C, with 100 equal divisions between them. The Fahrenheit scale sets the same two points at 32 and 212 degrees F, with 180 divisions, which is why 1 Fahrenheit degree is 5/9 of a Celsius degree. The kelvin scale is the absolute scale used in science: its zero is absolute zero, the temperature at which a gas would have zero pressure if it could remain ideal, and its unit interval equals one Celsius degree. Absolute zero is about -273.15 degrees C. A temperature on the kelvin scale is written T and on the Celsius scale t, with T = t + 273.15.
2.3 Fixed points and thermometer calibration
A thermometer's fixed point has to give the same value in every laboratory. Ice and steam come close to that ideal, but their temperatures shift with atmospheric pressure, so modern thermometry drops both and keeps only the water triple point, where ice, liquid water and water vapour coexist at 273.16 K and a vapour pressure of 0.611 kPa. A constant-volume gas thermometer measures the pressure of a trapped gas at the unknown temperature and compares it with the pressure at the triple point. Since the pressure of a gas varies nearly linearly with absolute temperature, the unknown temperature is T = 273.16 (P/P_tr) K. Real gases differ slightly, so the readings are taken at lower and lower pressure and extrapolated to zero pressure to remove the gas-specific error.
2.4 Practical thermometers
A laboratory thermometer commonly uses the expansion of mercury or alcohol along a narrow tube, and its scale is calibrated at two fixed points. A clinical thermometer is a mercury thermometer with a constriction that holds the reading until it is shaken down, and it covers only a short range around body temperature. A resistance thermometer measures the change of electrical resistance of a platinum wire with temperature; platinum is chosen because its resistance grows smoothly and reproducibly over a wide range. A thermocouple uses the small voltage produced at the junction of two metals, and it is useful for measuring the temperature of a small or fast-changing object. All of these need the same kind of calibration, and all of them rely on a property that changes steadily with temperature. A thermometer must be allowed to reach thermal equilibrium with the body being measured before the reading is taken, and the choice of instrument depends on the range needed, the precision wanted and the size of the object.
2.5 The ideal-gas equation
For a dilute gas, the pressure, volume and absolute temperature satisfy P V = n R T, where n is the number of moles and R = 8.31 J mol^-1 K^-1. Equivalently, P V = N k_B T with N the number of molecules and k_B = 1.38 x 10^-23 J K^-1. For a fixed mass of gas, P V/T = constant, which contains Boyle's law (constant T), Charles's law (constant P) and Gay-Lussac's law (constant V) as special cases. The absolute zero follows from extrapolating the pressure against temperature graph of a gas at constant volume to zero pressure; the intercept is -273.15 degrees C for every dilute gas. Real gases liquefy before reaching that point, so the absolute zero is an extrapolated limit.
2.6 Worked example: reading a gas thermometer
A constant-volume gas thermometer reads a pressure of 0.200 x 10^5 Pa at the triple point of water. At the normal melting point of sulphur it reads 0.287 x 10^5 Pa. The temperature is T = 273.16 x (0.287/0.200) = 392 K, or about 119 degrees C. A second thermometer using a different gas might give 391 K, and the small difference comes from the non-ideal behaviour of the gas. Repeating the measurement at lower pressures and extrapolating to zero pressure would bring both thermometers to the same value.
Common mistake
Do not add 273 to a temperature difference. The conversion T = t + 273.15 applies to a temperature, not to an interval: a rise of 10 degrees C is a rise of 10 K, not 283 K.