Gas pressure is sometimes measured with a device called a manometer. A manometer, shown in FIGURE 14.13, is a U-shaped tube connected to the gas at one end and open to the air at the other end. The tube is filled with a liquid—usually mercury—of density $\rho$. The liquid is in equilibrium. A scale allows the user to measure the height $h$ of the right side above the left side.

FIGURE 14.13 A manometer is used to measure gas pressure.

Steps 1–3 from Tactics Box 14.1 lead to the conclusion that the pressures $p_1$ and $p_2$ must be equal. Pressure $p_1$, at the surface on the left, is simply the gas pressure: $p_1=p_{\text{gas}}$. Pressure $p_2$ is the hydrostatic pressure at depth $d=h$ in the liquid on the right: $p_2=1\text{atm}+\rho gh$. Equating these two pressures gives

Figure 14.13 assumed $p_{\text{gas}}>1\text{atm,}$ so the right side of the liquid is higher than the left. Equation 14.7 is also valid for $p_{\text{gas}}<1\text{atm}$ if the distance of the right side below the left side is considered to be a negative value of h.

Another important pressure-measuring instrument is the barometer, which is used to measure the atmospheric pressure $p_{\text{atmos}}$. FIGURE 14.14a shows a glass tube, sealed at the bottom, that has been completely filled with a liquid. If we temporarily seal the top end, we can invert the tube and place it in a beaker of the same liquid. When the temporary seal is removed, some, but not all, of the liquid runs out, leaving a liquid column in the tube that is a height $h$ above the surface of the liquid in the beaker. This device, shown in FIGURE 14.14b, is a barometer. What does it measure? And why doesn’t all the liquid in the tube run out?

FIGURE 14.14 A barometer.

We can analyze the barometer much as we did the manometer. Points 1 and 2 in Figure 14.14b are on a horizontal line drawn even with the surface of the liquid. The liquid is in hydrostatic equilibrium, so the pressure at these two points must be equal. Liquid runs out of the tube only until a balance is reached between the pressure at the base of the tube and the pressure of the air.

You can think of a barometer as rather like a seesaw. If the pressure of the atmosphere increases, it presses down on the liquid in the beaker. This forces liquid up the tube until the pressures at points 1 and 2 are equal. If the atmospheric pressure falls, liquid has to flow out of the tube to keep the pressures equal at these two points.

The pressure at point 2 is the pressure due to the weight of the liquid in the tube plus the pressure of the gas above the liquid. But in this case there is no gas above the liquid! Because the tube had been completely full of liquid when it was inverted, the space left behind when the liquid ran out is a vacuum (ignoring a very slight vapor pressure of the liquid, negligible except in extremely precise measurements). Thus pressure $p_2$ is simply $p_2=\rho gh$.

Equating $p_1$ and $p_2$ gives

Thus we can measure the atmosphere’s pressure by measuring the height of the liquid column in a barometer.

The average air pressure at sea level causes a column of mercury in a mercury barometer to stand 760 mm above the surface. Knowing that the density of mercury is $13\text{,}600\text{k}\text{g}\text{/}{\text{m}}^3$ (at $0\text{°}\text{C}$ we can use Equation 14.8 to find that the average atmospheric pressure is

This is the value given earlier as “one standard atmosphere.”

The barometric pressure varies slightly from day to day as the weather changes. Weather systems are called high-pressure systems or low-pressure systems, depending on whether the local sea-level pressure is higher or lower than one standard atmosphere. Higher pressure is usually associated with fair weather, while lower pressure portends rain.