# Temperature

The concept of temperature has evolved from man’s experience of hot and cold conditions with temperature scales devised on the basis of changes in the physical properties of substances that depend on temperature. Practical examples of thermometers for temperature measurement include the following:

• Constant volume gas thermometers, which make use of the pressure of a fixed quantity of gas maintained at a constant volume as an indicator

• • Liquid in glass thermometers, which use the volume of a liquid, such as mercury or alcohol, contained in a reservoir attached to a capillary tube with a calibrated scale
• • Electrical resistance thermometers, which use the variation of the resistance of a metal, such as platinum, or of a doped semiconductor, such as GaAs, to obtain temperature
• • Vapor pressure and paramagnet thermometers for special purposes particularly at low temperatures

Most of the thermometers listed above are secondary thermometers that are calibrated against agreed standards. The constant volume gas thermometer has a more fundamental significance as explained in Section 1.3. For everyday purposes, various empirical scales have been established. Commonly used scales are the Fahrenheit and Celsius scales. For reasons that become clear below, we consider the Celsius scale, which chooses two fixed reference temperatures. The lower reference point is at 0°C, which corresponds to the triple point of water, the point at which water, ice, and water vapor coexist, and the higher reference point at 100°C, which corresponds to the steam point, where water and steam coexist at a pressure of 1 atm. Degrees Celsius are obtained by dividing the range between the triple point and the steam point into 100°. Figure 1.1 shows a schematic drawing of a triple-point cell.

Thermodynamics shows that it is possible to establish an absolute temperature scale called the Kelvin scale in honor of Lord Kelvin, who introduced it and first appreciated its significance. The absolute scale does not depend on the properties of a particular substance. Absolute zero on the Kelvin scale, designated as 0 K, corresponds to -273.16°C. For convenience, 1 К is chosen to correspond to 1°C. This gives T(K) = t(°C) + 273.16. We can gain insight into why the absolute zero of temperature is of fundamental importance by considering the equation of state for an ideal gas.

# Ideal Gas Equation of State

An equation of state establishes a relationship among thermodynamic variables. For an ideal gas, the variables chosen are the pressure P, the volume К and the absolute temperature T. Experiments carried out on real gases, such as helium, under conditions of low density have shown that the following equation describes the behavior of many gases:

where n is the number of moles of gas and R is a constant called the gas constant with a value of 8.314J mol-1 К-1. As mentioned above, the constant volume gas thermometer involves the measurement of the pressure of a constant volume of gas as a function of temperature. Figure 1.2 gives a sketch of a constant volume gas thermometer with a representative P versus T plot shown in Figure 1.3.

FIGURE 1.1 Schematic depiction of a triple-point cell in which water, ice, and water vapor coexist. The cell is used to fix 0°C on the Celsius scale.

FIGURE 1.2 Sketch of a constant volume gas thermometer in which the pressure of a fixed volume of gas, held at various fixed temperatures by use of heat baths, is measured on a pressure gauge.

FIGURE 1.3 Calculated pressure variation with temperature for a constant mass of an ideal gas (lCP'mol) kept at constant volume (10",m3). The low-temperature portion of the graph, where gases in general liquefy as a result of intermolecular forces, is in practice obtained by extrapolation from higher temperatures.

From Equation l.l, it follows that the temperature T= OK corresponds to a pressure P = 0 Pa. Zero pressure implies that the particles of the gas have zero kinetic energy at О К and do not exchange momentum with the walls of the container. We can therefore view the absolute zero of temperature as the temperature at which the energy of particles in the system is effectively zero. This is of fundamental significance.

As the temperature is lowered, gases normally liquefy, and most solidify at temperatures much higher than 0 K. This is because real gases have interactions between particles, which lead to departures from ideal gas behavior. Extrapolation from the high-temperature, low-density region, where gases obey the ideal gas equation of state, shows what would happen at much lower temperatures if the gas were to remain ideal. The ideal gas equation of state, expressed in Equation 1.1, is extremely useful in considering processes in which gases are involved. P, V, and Tare called state variables, and because they are related by the equation of state, the specification of any two of the variables immediately fixes the value of the third variable. Examples of applications of the ideal gas law, as it is also called, are given in later chapters. The ideal gas equation provides a fairly good description of the behavior of many gases over a range of conditions. Under conditions of high density, however, the description may not be adequate, and empirical equations of state that work better under these conditions have been developed. Two of these equations are briefly discussed in the next section.