# Gass Law – Pressure Notes

A manometer is an device employed to measure pressure. There are a variety of manometer designs. A simple, common design is to seal a length of glass tubing and bend the glass tube into a U-shape. The glass tube is then filled with a liquid, typically mercury, so that all trapped air is removed from the sealed end of the tube. The glass tube is then positioned with the curved region at the bottom. The mercury settles to the bottom (see the picture at the left).
After the mercury settles to the bottom of the manometer, a vacuum is produced in the sealed tube (the left tube in the picture). The open tube is connected to the system whose pressure is being measured. In the sealed tube, there is no gas to exert a force on the mercury (except for some mercury vapor). In the tube connected to the system, the gas in the system exerts a force on the mercury. The net result is that the column of mercury in the left (sealed) tube is higher than that in the right (unsealed) tube. The difference in the heights of the columns of mercury is a measure of the pressure of gas in the system. In the example at the left, the top of the left column of mercury corresponds to 875 mm on the scale. The top of the right column of mercury corresponds to 115 mm. The difference in heights is 875 mm – 115 mm = 760. mm, which indicates that the pressure is 760. mm Hg or 760. torr. |

This method for measuring pressure led to the use of millimeters of mercury (mm Hg) as a unit of pressure. Today 1 mm Hg is called 1 torr. A pressure of 1 torr or 1 mm Hg is literally the pressure that produces a 1 mm difference in the heights of the two columns of mercury in a manometer.

To understand how the height of a column of mercury can be used as a unit of pressure and how the unit of torr is related to the SI unit of pascal (1 Pa = 1 N/m^{2}), consider the following mathematical analysis of the behavior of the manometer.

The force exerted by the column of mercury in a tube arises from the gravitational acceleration of the column of mercury. Newton’s Second Law provides an expression for this force:

*F = m g*

In this equation, *m* is the mass of mercury in the column and *g* = 9.80665 m/sec^{2} is the gravitational acceleration. This force is distributed over the cross-sectional area of the column ( *A* ). The pressure resulting from the column of mercury is thus

P = |
m gA |

The mass of mercury is given by the product of the density of mercury ( *d _{Hg}* ) and the volume of mercury (

*V*). For a cylindrical column of mercury, the volume of mercury is the product of the cross-sectional area and the height of the column (

*h*). These relationships product the following equation.

P = |
m gA |
= | d_{Hg} V gA |
= | d_{Hg} A h gA |
= d_{Hg} h g |

This equation clearly shows that the height of a column of mercury is directly proportional to the pressure exerted by that column of mercury. The difference in heights of the two columns of mercury in a manometer can thus be used to measure the difference in pressures between the two sides of the manometer.

The relation between torr and Pa is also clearly evident. Using *d _{Hg}* = 13.5951 g cm

^{-3}, one finds that 1 torr = 133 Pa or 1 atm = 760 torr = 101 kPa.