Physics Bootcamp

Subsection 19.4.1 Thermometric Property

The temperature of a system in a thermodynamic equilibrium can be measured by bringing the given system in equilibrium with a reference system, called the thermometric substance, whose temperature has been calibrated somehow. The reference system used for measuring a temperature is called a thermometer. But the question is how to find systems with known temperature readings. This is akin to the chicken-and-egg problem: you need temperature to establish a state of thermal equilibrium, and a state of thermal equilibrium is necessary to define a temperature.

To break this cycle, we introduce another measurable property that varies with temperature - a property used for this purpose is called thermometric property. Table 19.4.1 lists some common thermometric properties used and the materials used in the instruments. Assuming that temperature \(t\) varies linearly with property \(X \text{,}\) we can write \(t\) as a linear function of \(X\text{.}\)

where \(a \) is a constant of proportionality to be fixed by choosing a reference condition of matter to which we arbitrarily assign a values of \(t \) corresponding to the measured value of the property \(X\) in that condition. This is also called calibrating a thermometer. I will now describe a method of calibration of thermometers based on the triple point of water that has been agreed upon internationally.

Subsection 19.4.2 Calibration Using The Triple Point of Water

Since 1954 the triple point of water, a state in which a mixture of ice, liquid water and vapor coexist in equilibrium, has been chosen as the standard reference which is arbitrarily assigned a value of \(273.16\text{K}\text{.}\) This fixes the value of constant \(a \) in Eq. (19.4.1) in terms of the assigned value of the temperature of triple point and the value of \(X\) at \(X=X_{\text{TP}}\) of the thermometric property when the thermometer is in equilibrium at the triple point of water.

Eliminating \(a \) from Eqs. (19.4.1) and (19.4.2), we are able to immediately determine temperature \(t \) when the value of the thermometric property \(X \) is known.

A Big Problem: Suppose you follow this procedure for different thermometer, e.g. a gas thermometer, a resistance thermometer, or a thermocouple, will you get the same readings in them? The answer is a resounding no, if the unknown temperature is not near the triple point of water. The temperature of an unknown will be different even for two gas thermometer that are somehow different, either having different size bulbs or different amount of gas.

When you use formula given in Eq. (19.4.3) for different thermometers you get different values for the temperature of the same thermal bath.

The Solution: To solve this problem, we need a standard thermometer to calibrate other thermometers so that we do not assign multiple values for the temperature of the same system. The constant volume gas thermometer provides a procedure for obtaining a unique temperature independent of the thermometric material as we will see below, and therefore it is used as a standard thermometer.

Subsection 19.4.3 Constant Volume Gas Thermometer

A constant volume gas Thermometer is a thermometer that uses a gas as the thermometric medium and its pressure as the property \(X\text{.}\) It consists of a bulb B of a gas that connects to a reservoir R of mercury and a manometer M as shown in Figure 19.4.2.

When the gas bulb is immersed in a thermal bath, the pressure in the volume of the gas changes changing the mercury level from the indicial point C in arm M. By moving the reservoir R up or down the mercury level in arm M can be brought to the reference point C. Movement of reservoir also changes the level C' in arm M'. The difference \(h\) in the level C' and C is measured when the bulb is immersed in the unknown temperature \(t\text{.}\) From the difference in heights we can determine the absolute pressure \(p\) in the bulb.

where \(p_0\) is the atmospheric pressure since the tube on the M' side is open to the atmosphere. The temperature of the unknown is determined from the absolute pressure reading by equating the ratio of the absolute pressures to the ratio of temperatures expressed in kelvin scale.

where \(p_{\text{TP}}\) is the absolute pressure when the bulb is immersed in water at triple point \((T_{\text{TP}}= 273.16\text{K})\text{.}\)

Experiments with gas thermometer have shown that there are minor differences in temperature readings for different gases placed in the bulb as shown in Figure 19.4.3. Furthermore, even these minor differences can disappear when the pressure in the bulb is reduced by using less gas.

The graphs of different gases show two important results:

1. Real thermometers are inconsistent
2. The extrapolation of gas thermometers to zero pressure give consistent and unique temperature value.

A disturbing fact is that a unique temperature cannot be assigned to an unknown thermal state for a non-zero pressure since the temperature value assigned depends on the thermometric gas. For instance, if you draw a vertical line at \(p = 100\text{ kPa} \) in Figure 19.4.3, you will find that there are three different values of the temperature for the same sample depending on whether you have \(\text{He} \text{,}\) \(\text{Ar} \text{,}\) or \(\text{N}_2 \) in the bulb as you can see by drawing a vertical line for \(p = 100\text{ kPa} \text{.}\)

You can also see in Figure 19.4.3 that the extrapolations of the graphs for different gases meet at one point, giving a unique reading of the temperature independent of the nature of gas in the bulb. This procedure defines an Absolute scale of temperature also called the kelvin scale.

Subsection 19.4.4 Important Fixed Points for Standard Thermal Baths

The conditions of the triple point and boiling point of water are not convenient when you need thermometers for other ranges of temperatures. Over the years, other suitable fixed points have been utilized, some of which are shown in Figure 19.4.4.

Subsection 19.4.5 Calibration With Two Fixed Points

A Thermometer is sometimes calibrated to operate over a small range of temperatures by picking two reproducible thermal conditions of known temperatures. Readily available thermal conditions are the conditions that exist in which a material can coexist between two different material phases, such as solid and liquid, or three different phases simultaneously, such as the triple point. These conditions are also called fixed points. Some of the fixed points of common use are shown in Figure 19.4.4.

The procedure for extrapolating between two nearby fixed points are based on the assumption of the linear dependence of the change in the Thermometric property \(\Delta X \) on the change in temperature \(\Delta T \text{.}\) Let \(X_1 \) and \(X_2 \) be the value of the property \(X \) at reference reproducible baths with temperature \(t_1 \) and \(t_2 \text{.}\)

If the temperature range between the baths is not too great it is reasonable to assume that the property \(X \) will vary linearly with temperature \((t) \) between the two temperatures \(t_1 \) and \(t_2 \text{,}\) and can be extended on either side of \(t_1 \) and \(t_2 \text{.}\) Then the linear relation shown in Figure 19.4.5 will have the following analytic relation.

Unfortunately, most properties do not vary linearly with temperature over a wide range, and therefore, we need many reference baths to recalibrate, and use different thermometers in different ranges. We make sure that all thermometers give the same readings for the same sample in the range where they overlap.