Pressure Gauge: Barometer: Videos & Practice Problems

Pressure gauges, like barometers, measure atmospheric pressure using the height of a liquid column. The fundamental equation is $P_b=P_t+\rho gh$, where P_b is the pressure at the bottom, P_t is the pressure at the top (zero in a vacuum), ρ is the liquid density, g is the acceleration due to gravity, and h is the height of the liquid column. Mercury is commonly used due to its high density, allowing for a more compact design. Atmospheric pressure varies with altitude, making barometers essential for measuring local conditions.

Pressure Gauges: Barometer

Pressure gauges are essential devices used to measure pressure, with the barometer being a specific type that measures atmospheric pressure. The fundamental principle behind pressure measurement involves height differences and the densities of liquids, which can be expressed using the equation:

$$ P_{\text{bottom}} = P_{\text{top}} + \rho g h $$

In the context of a barometer, the setup typically includes a container filled with a heavy liquid, such as mercury, and a glass tube. The bottom of the container is open to the atmosphere, allowing air pressure to exert force on the liquid, causing it to rise in the tube. The top of the tube is sealed, creating a vacuum. At equilibrium, the height of the liquid column reflects the atmospheric pressure, leading to a simplified equation for barometers:

$$ P_{\text{air}} = \rho g h $$

Here, \( P_{\text{air}} \) represents the atmospheric pressure, \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid column. This equation indicates that to calculate atmospheric pressure, one needs to know the density of the liquid and measure the height of the liquid column.

Standard atmospheric pressure at sea level is defined as 1 atmosphere (atm), which corresponds to a height of approximately 76 cm of mercury. However, this value can change with altitude; for instance, at higher elevations, the atmospheric pressure decreases, resulting in a lower height of the liquid column in the barometer.

Mercury is commonly used in barometers due to its high density (approximately 13.6 times that of water), which allows for a more compact design. For example, if water were used instead, the height of the column would need to be about 10.3 meters to equal 1 atm, making the instrument impractical.

To illustrate the application of these principles, consider a barometer filled with an unknown liquid that rises to 76 cm under standard atmospheric pressure. By rearranging the equation to solve for density, we find:

$$ \rho = \frac{P_{\text{air}}}{g h} $$

Substituting \( P_{\text{air}} = 101,000 \, \text{Pa} \), \( g = 9.8 \, \text{m/s}^2 \), and \( h = 0.76 \, \text{m} \), we calculate the density to be approximately 13,560 kg/m³, which corresponds to the density of mercury.

In a different scenario, if the same barometer is taken to a location where the liquid rises to 84 cm, we can again use the equation to find the new atmospheric pressure:

$$ P_{\text{air}} = \rho g h $$

Using the previously determined density of mercury, we substitute the values to find that the atmospheric pressure is approximately 11,625 Pa, or about 1.1 atm. This increase in height indicates that the external air pressure is greater, pushing the liquid higher in the tube.

Understanding how barometers function and the relationship between liquid height, density, and atmospheric pressure is crucial for various applications in meteorology and environmental science.

A classic barometer (shown below) is built with a 1.0-m tall glass tube and filled with mercury (13,600 kg/m³). Calculate the atmospheric pressure, in ATM, surrounding the barometer if the column of liquid is 76 cm high. (Use g=9.8 m/s².)

• A. 0 Pa
• B. 1.01 Pa
• C. 1.01×10⁵ Pa
• D. 1.01×10⁷ Pa

Answer: (C)

Here’s what students ask on this topic:

How does a barometer measure atmospheric pressure?

A barometer measures atmospheric pressure by using the height of a liquid column, typically mercury, in a glass tube. The fundamental equation used is:

$P{b}_{}=P_{t_{}}+\rho gh$

where $P{b}_{}$ is the pressure at the bottom, $P_{t_{}}$ is the pressure at the top (zero in a vacuum), $\rho$ is the liquid density, $g$ is the acceleration due to gravity, and $h$ is the height of the liquid column. The atmospheric pressure forces the liquid up the tube, and the height of the liquid column is used to calculate the pressure.

Why is mercury used in barometers instead of water?

Mercury is used in barometers instead of water because it has a much higher density. Mercury's density is approximately 13.6 times that of water. This high density allows the barometer to be more compact. For example, if water were used, the height of the column would need to be about 10.3 meters to measure standard atmospheric pressure. In contrast, with mercury, the column height is only about 0.76 meters. This makes the instrument more practical and easier to use.

What is the equation used to calculate atmospheric pressure with a barometer?

The equation used to calculate atmospheric pressure with a barometer is:

$P{b}_{}=P_{t_{}}+\rho gh$

In a barometer, the pressure at the top $P_{t_{}}$ is zero because it is a vacuum. Therefore, the equation simplifies to:

$P{a}_{}=\rho gh$

where $P{a}_{}$ is the atmospheric pressure, $\rho$ is the density of the liquid, $g$ is the acceleration due to gravity, and $h$ is the height of the liquid column.

How does altitude affect the reading of a barometer?

Altitude affects the reading of a barometer because atmospheric pressure decreases with increasing altitude. At higher altitudes, there is less air above the barometer, resulting in lower atmospheric pressure. This causes the height of the liquid column in the barometer to decrease. Conversely, at lower altitudes, the atmospheric pressure is higher, causing the liquid column to rise. Therefore, barometers are essential for measuring local atmospheric pressure, which can vary significantly with changes in altitude.

How do you calculate the density of an unknown liquid using a barometer?

To calculate the density of an unknown liquid using a barometer, you can rearrange the barometer equation:

$P{a}_{}=\rho gh$

to solve for density $\rho$:

$\rho =\frac{P}{a_{}}$

Here, $P{a}_{}$ is the atmospheric pressure, $g$ is the acceleration due to gravity, and $h$ is the height of the liquid column. By measuring the height of the liquid column and knowing the atmospheric pressure and gravitational constant, you can calculate the density of the unknown liquid.