One goal of thermodynamics is to minimize the amount of experiment that must be done, though measurements cannot be eliminated completely. Experiments necessarily reveal and quantify the rich behavior of materials. Moreover, experiment both motivates and guides theoretical efforts aimed at interpreting and correlating phenomena. Even when theoretical models are reliable, experimental data are needed to provide values for parameters in the models. Thus, thermodynamics is intimately and irrevocably tied to experiment.

This article provides a superficial introduction to common devices for measuring pressure. We do not delve into experimental protocols; for a more thorough introduction to pressure measuring devices, see Benedict [1].

Absolute pressure P is defined to be the force applied per unit area, P = F/A. Force, in turn, can be defined operationally as anything that is measured by a balancing operation [2]. Thus, we have two general approaches to measuring pressure, either directly, by determining the force applied to a known area, or indirectly, by determining some effect of an applied pressure.

Many indirect measuring devices are *transducers*: devices that take a force from one system (that providing the unknown pressure) and apply a portion of that force, in a different form, to another system (a measuring system). Mechanical transducers convert the unknown pressure into the mechanical displacement of a tube, diaphragm, or bellows. Electrical transducers further convert a mechanical displacement into an electrical signal: a current, voltage, or capacitance. Here we briefly describe three simple pressure-measuring devices: the free-piston gage, the liquid manometer, and the Bourdon tube.

These devices are also known as deadweight gages or pressure balances. They operate directly by balancing the unknown pressure with a known force applied to a known area. The gage consists of a finely machined piston riding in a close-fitting cylinder, as shown in Figure 1. The known pressure is applied to the bottom of the piston, usually via an oil. This applied pressure is balanced by weights added to the top of the piston. When the piston is balanced, the unknown pressure is determined by

P = F/A

(1)

where A is the cross-sectional area of the piston and F is the combined force of the piston, weights, and atmosphere.

For accurate results several corrections must be applied to the value provided by (1). These include a correction for distortion of the piston area caused by changes in temperature or pressure or both, corrections for fluid leaking between the piston and cylinder, and perhaps even a correction for buoyancy of the piston and weights caused by displaced air.

The free-piston gage has an operating range from just above atmospheric pressure to values as high as one kilobar. The gage is particularly useful at high pressures because it can be constructed to withstand extreme forces. The free-piston gage can also achieve high accuracy: typically ±0.01% of the measured value, that is, ±0.01 bar at 100 bar. Its principal disadvantages are that it is a large cumbersome device and it cannot measure pressures below local atmospheric pressure. Besides high-pressure work, the free-piston gage is routinely used to calibrate other pressure-measuring devices.

The typical manometer is a glass U-tube that connects an unknown pressure P to a known reference pressure P_{r}, as in Figure 2. The manometer is partially filled with a liquid that is immiscible in both the system fluid and the reference fluid; mercury is often used, though if P is close to P_{r}, then a low density fluid, such as water or an alcohol, may be used. Often, a closed tube is used as the reference system, so P_{r} = 0.

A displacement h_{m} of manometer fluid balances the forces F exerted in the two legs of the U-tube,

F_{left leg} = F_{right leg}

(2)

This force balance must apply at any vertical position along the U-tube; if we choose to apply it at the system fluid-manometer fluid interface, as in the figure, then we can write

P + ρ_{s} g h_{s} = P_{r} + ρ_{r} g h_{r} + ρ_{m} g h_{m}

(3)

Here, ρ_{s} is the density of the system fluid, h_{s} is the height of the system fluid in the left leg (see Figure 2), and g is the acceleration due to gravity. Similarly, ρ_{r} is the density of the reference fluid and ρ_{m} is the density of the manometer fluid. Likewise, h_{r} is the height of the reference fluid is the right leg, while h_{m} is the difference in heights for the manometer fluid in the two legs (see Figure 2). Equation (3) follows from (2) so long as the cross-sectional areas are the same for the two legs of the U-tube.

The principal corrections to manometer readings (a) account for effects of temperature on the densities of the fluids and (b) account for capillary effects that distort the shapes of fluid interfaces. The typical range of application is from about 0.1 inch of water to about 7 bar; the typical accuracy is about 0.1% of the scale reading.

Note that the manometer does not measure an absolute pressure; instead, as (3) shows, it measures a difference relative to a known reference pressure. In many applications only a difference is needed; for example, to obtain a flow rate through an orifice meter we need measure only the pressure drop across the meter. Further, because it measures differences, a manometer can be used to determine pressures in vacuum systems. The McLeod gage is an elaborate modification of the mercury manometer for measuring pressures in vacuum systems [13].

This device is a mechanical transducer, constructed of an oval cross-sectioned tube, bent into an arc. One end of the arc is fixed to a support and the unknown pressure is admitted to that end. The other end is sealed, free to move, and mechanically linked to a pointer. When pressure is applied, stresses around the oval cross-section and the circular arc resolve themselves by displacing the free end of the tube and hence moving the pointer.

Many variations of this basic design have been constructed, including gages that measure either absolute or differential pressures over extensive ranges. An accurate Bourdon tube gage can achieve uncertainties of ±0.1% of its scale reading.

[1] R. P. Benedict, *Fundamentals of Temperature, Pressure, and Flow Measurements*, 3rd ed., Wiley, New York, 1984.

[2] O. Redlich, *Thermodynamics: Fundamentals, Applications*, Elsevier, Amsterdam, 1976.

[3] J. H. Leck, *Pressure Measurement in Vacuum Systems*, 2nd ed., Chapman and Hall, London, 1964.