8.2: Empirical and Thermodynamic Temperature

The Second Law of Thermodynamics defines the transport of entropy in terms of the heat transfer and the thermodynamic temperature on the boundary where the heat transfer occurs. What precisely is a thermodynamic temperature? Because of its importance in the definition of entropy and the technological importance of temperature measurement, we will address the issue of temperature measurement and temperature scales before we proceed further.

8.2.1 Temperature and Thermal Equilibrium

The first question one might ask is "what is temperature?" Most of us have a very common-sense understanding of this. We understand temperature to be a characteristic of a system that is related to how "hot" or "cold" it feels. As these sensations are relative, it is highly possible that different individuals might have different perceptions of the temperature of any given object.

Temperature is intimately related with the concept of thermal equilibrium. When two objects are brought into contact for a sufficiently long period of time, their properties will eventually stop changing and we say that the two objects are in thermal equilibrium. Temperature is the property of the objects that indicates whether they are in thermal equilibrium or not. When we use a thermometer to compare the temperature of two objects, we are assuming that if both objects are independently in thermal equilibrium with the thermometer (have the same temperature as the thermometer), the two objects would also be in thermal equilibrium with each other (have the same temperature). This empirical result is known as the Zeroth Law of Thermodynamics. If this were not true, it would be impossible to use a thermometer to measure temperatures.

8.2.2 Empirical Temperature

To measure the temperature of a system, we need a thermometer with a property that changes with temperature (a thermometric property), we need a set of fixed points for reference, and we need a temperature scale to interpolate between the fixed points.

One of the most common thermometers is a liquid-in-glass thermometer where the liquid is mercury or alcohol. Imagine that you are given a mercury-in-glass thermometer without a scale engraved on the glass and asked to use it to measure the temperature of various objects. From experience, you know that the mercury in the thermometer (and the glass) will expand with increasing temperature. To quantify the temperature, you must establish a scale and some fixed reference points. To do this, you get a thermos bottle and fill it with a mixture of ice and water. Then you set a pan of water on the stove and get it boiling. You dip the bulb of the thermometer into the ice-water mixture, wait until the mercury stops moving, and scratch a line on the stem of the thermometer. You then repeat the process with the boiling water. You now have two fixed points - the boiling point of water and the ice-point of water. If you arbitrarily assign a temperature of \(0^{\circ}\) to the ice-point mark and a temperature of \(180^{\circ}\) to the boiling point of water you have two fixed points. If we divide the distance between the two marks on the thermometer stem into 90 equally spaced divisions, we now have a scale to interpolate the temperature anywhere between these two fixed points. This will work fine for comparing the temperature of various objects as long as their temperatures fall within the given range.

We are all familiar with numerous devices for measuring temperatures and each one depends on the thermometric behavior of a substance, e.g. the expansion of mercury and glass (mercury-in-glass thermometer), the change in electrical resistance (electric resistance thermometer), or the pressure-temperature behavior of a gas (Constant-volume ideal-gas thermometer). With each of these devices it is possible to establish an empirical temperature scale by assigning fixed temperatures to repeatable physical phenomena and then using a thermometer to interpolate between these points. Typical reference points include the boiling point of water at one atmosphere; the ice-point of water (an equilibrium mixture of ice, liquid water, and saturated air) at one atmosphere; and the melting points of various metals at one atmosphere.

8.2.3 Thermodynamic Temperature

One of the major achievements of thermodynamics has been the development of an absolute or thermodynamic temperature scale that is independent of any substance. Following a suggestion made by Lord Kelvin in 1848, we can establish such a temperature scale using the Second Law of Thermodynamics.

Figure \(\PageIndex{1}\): Power cycle with heat transfer of entropy at two different surfaces.

To see what this means, consider a power cycle as shown in Figure \(\PageIndex{1}\). The power cycle receives energy by heat transfer across a boundary at temperature \(T_{H}\) and rejects energy by heat transfer across a boundary at temperature \(T_{L}\). If we write the rate form of the entropy accounting equation for the power cycle, we have the following result: \[\underbrace{ \cancel{ \frac{d S_{sys}}{dt} }^{=0} }_{\text{Steady-state}} = \frac{\dot{Q}_{H, \text{ in}}}{T_{H}} - \frac{\dot{Q}_{L, \text{ out}}}{T_{L}} + \dot{S}_{gen} \quad \rightarrow \quad \frac{\dot{Q}_{L, \text{ out}}}{T_{L}} = \frac{\dot{Q}_{H, \text{ in}}}{T_{H}} + \dot{S}_{gen} \nonumber \]

Recall that we are looking for a way to define a temperature scale that is independent of any specific physical thermometer or thermometric property.

If we rearrange Eq. \(\PageIndex{1}\) to find the ratio of temperatures, we have the following relationship between the ratio of heat transfer rates and the ratio of temperatures: \[\frac{\dot{Q}_{L, \text { out}}}{\dot{Q}_{H, \text { in}}} = \frac{T_{L}}{T_{H}} + \dot{S}_{gen} \left(\frac{T_{L}}{\dot{Q}_{H, \text { in}}}\right) \nonumber \] If we restrict ourselves to an internally reversible power cycle, \(\dot{S}_{gen}=0\) and this equation simplifies to: \[\left(\frac{\dot{Q}_{L, \text { out}}}{\dot{Q}_{H, \text { in}}}\right)_{\begin{array}{l} \text {internally} \\ \text{reversible} \end{array}} = \frac{T_{L}}{T_{H}} \nonumber \] This result is independent of the physical properties of the working fluid of the power cycle and only requires that the power cycle operate in an internally reversible fashion. This satisfies our criterion for establishing a thermodynamic temperature scale. Equation \(\PageIndex{3}\) is the defining equation for thermodynamic temperature.

The ratio of any two temperatures on a thermodynamic temperature scale is equal to the ratio of the heat transfer rates for an internally reversible power cycle (heat engine) that operates between the same two temperatures. The minimum temperature on any absolute temperature scale is zero degrees, often called "absolute zero."

The Kelvin temperature scale is the thermodynamic temperature scale used with the SI system of units. (The Rankine temperature scale is the thermodynamic temperature scale used with the USCS system of units.) On the Kelvin scale, the triple point (tp) of water is assigned a temperature of exactly \(T_{tp}=273.16 \mathrm{~K}\). Thus, the defining equation for all other temperatures on this scale is \[T = (273.16 \mathrm{~K}) \left(\frac{\dot{Q}_{T}}{Q_{tp}}\right)_{\begin{array}{c} \text {internally} \\ \text {reversible} \end{array}} \nonumber \] where \(\dot{Q}_{T}\) and \(\dot{Q}_{tp}\) are the heat transfer rates on the boundary of an internally reversible power cycle that occur at temperature \(T\) and at \(T_{tp}\), respectively.

There are four temperature scales that are commonly used in engineering work. The Celsius scale and the Fahrenheit scale are empirical scales and originally were established by interpolating between two fixed points. The Kelvin scale and the Rankine scale are both thermodynamic (absolute) temperature scales. As demonstrated above, an absolute scale is set when a numerical value is assigned to one fixed reference point, e.g. the triple point.

The relationship between temperatures and temperature differences on the four scales are described by the following equations:

\[ \begin{array}{ll} \text{Temperatures } & \quad\quad\quad \dfrac{T_{\mathrm{R}}}{\text{}^{\circ} \mathrm{R}} = \mathbf{1.8} \left(\dfrac{T_{\mathrm{K}}}{\mathrm{K}}\right) & \\ &\dfrac{T_{\mathrm{K}}}{\mathrm{K}} = \dfrac{t_{\mathrm{C}}}{\text{}^{\circ} \mathrm{C}} + \mathbf{273.15} & \dfrac{T_{\mathrm{R}}}{\text{}^{\circ} \mathrm{R}} = \dfrac{t_{\mathrm{F}}}{\text{}^{\circ} \mathrm{F}} + \mathbf{459.67} \\ & \dfrac{t_{\mathrm{C}}}{\text{}^{\circ} \mathrm{C}} = \dfrac{1}{\mathbf{1.8}} \left(\dfrac{t_{\mathrm{F}}}{\text{}^{\circ} \mathrm{F}} - \mathbf{32} \right) & \dfrac{t_{\mathrm{F}}}{\text{}^{\circ} \mathrm{F}} = \mathbf{1.8} \left(\dfrac{t_{\mathrm{C}}}{\text{}^{\circ} \mathrm{C}}\right) + \mathbf{32} \\ { } \\ \text { Temperature differences } & \quad\quad\quad \dfrac{\Delta T_{\mathrm{R}}}{\Delta T_{\mathrm{K}}} = \dfrac{\Delta t_{\mathrm{F}}}{\Delta t_{\mathrm{C}}} = \mathbf{1.8} \\ & \Delta T_{\mathrm{K}} = \Delta t_{\mathrm{C}} & \Delta T_{\mathrm{R}} = \Delta t_{\mathrm{F}} \end{array} \nonumber \]

where \(T_{\mathrm{K}}\), \(T_{\mathrm{R}}\), \(t_{\mathrm{C}}\), and \(t_{\mathrm{F}}\) are temperature values measured on the Kelvin, Rankine, Celsius, and Fahrenheit scales, respectively. The bold numbers in Eq. \(\PageIndex{5}\) are exact.

In practice, the International Practical Temperature Scale (IPTS) is used to establish a practical scale for temperature measurement. The IPTS is based on of a number of easily reproducible and fixed points with assigned temperature values and prescribed instruments and formulas for interpolating between the points. The assigned temperature values are equal to the best experimental values of the thermodynamic temperatures of the fixed points.

Assigned temperatures of some fixed points used in defining the International Practical Temperature Scale of 1968 (IPTS-68)

Fixed Point	\(T_{68} / \mathrm{K}\)
Triple point of hydrogen	\(13.80\)
Boiling point of neon	\(27.102\)
Triple point of oxygen	\(54.361\)
Triple point of water	\(273.16\)
Boiling point of water	\(373.15\)
Freezing point of zinc	\(692.73\)
Freezing point of silver	\(1235.08\)
Freezing point of gold	\(1337.58\)