Pressure Gauge: Manometer: Videos & Practice Problems

A manometer is a pressure gauge that measures pressure using height differences in a liquid column. The fundamental equation used is p_bottom = p_top + ρgh, where ρ is the liquid density, g is acceleration due to gravity, and h is the height difference. Understanding how different gas pressures affect liquid height is crucial for accurate pressure measurement, reinforcing the principle that equal pressures yield equal heights in a manometer.

Pressure Gauges: Manometer

Manometers are essential instruments used to measure pressure by utilizing height differences in a liquid column. The fundamental principle behind a manometer is that pressure can be calculated using the equation:

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

In this equation, \( P_{\text{bottom}} \) represents the pressure at the bottom of the liquid column, \( P_{\text{top}} \) is the pressure at the top, \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( h \) is the height difference between the two liquid levels.

A typical manometer consists of a U-shaped tube filled with a liquid, where one side is exposed to a gas or vacuum, and the other side may be open to the atmosphere or another gas. The pressure exerted by the gases on either side causes the liquid to rise or fall, creating a measurable height difference.

In a closed manometer, if one side is at a vacuum (0 pressure), and the other side is at 1 atmosphere, the liquid levels will remain equal, as both sides exert the same pressure. Conversely, if one side is at a higher pressure, the liquid will shift, creating a height difference that can be measured and used to calculate the pressure of the gas.

When dealing with different pressures, it is crucial to remember that the pressure of a gas remains relatively constant throughout the column, unlike liquids where pressure can vary significantly with depth. This uniformity allows for straightforward calculations of pressure differences based on height measurements.

For example, if a manometer has a height difference of 60 cm between a vacuum and 1 atmosphere, the density of the liquid can be calculated using the rearranged pressure equation. Converting 1 atmosphere to pascals (101,000 Pa) and substituting into the equation yields:

\( 101,000 = 0 + \rho \cdot 9.8 \cdot 0.6 \)

Solving for \( \rho \) gives a density of approximately 17,177 kg/m³.

In a scenario where the gas pressure changes, the height difference will also change proportionally. For instance, if the height difference increases to 80 cm while maintaining the same liquid density, the new pressure can be calculated similarly, demonstrating the direct relationship between gas pressure and liquid height in the manometer.

Understanding these principles allows for accurate pressure measurements in various applications, reinforcing the importance of manometers in scientific and industrial settings.

A classic manometer (as shown below) has one of its ends open, and a 2 atm gas on the other. When mercury (13,600 kg/m³ ) is added to the manometer, you measure the top of the mercury column on the left to be 40 cm higher than the mercury column on the right. Calculate the atmospheric pressure that the manometer is exposed to, in units of atm. (Use g=9.8 m/s².)

• A. 0.477 atm
• B. 1.0 atm
• C. 1.47 atm
• D. 2.53 atm

Answer: (C)

Here’s what students ask on this topic:

What is a manometer and how does it work?

A manometer is a type of pressure gauge used to measure pressure by utilizing height differences in a liquid column. It typically consists of a U-shaped tube filled with a liquid, with one end connected to the gas whose pressure is to be measured and the other end open to the atmosphere or sealed. The fundamental equation used is $p_{\mathrm{bottom}}=p_{\mathrm{top}}+\; \rho gh$, where ρ is the liquid density, g is the acceleration due to gravity, and h is the height difference between the liquid columns. The height difference indicates the pressure difference between the gas and the reference pressure (atmospheric or vacuum).

How do you calculate the pressure using a manometer?

To calculate the pressure using a manometer, you use the equation $p_{\mathrm{bottom}}=p_{\mathrm{top}}+\; \rho gh$. Here, $p_{\mathrm{bottom}}$ is the pressure at the bottom of the liquid column, $p_{\mathrm{top}}$ is the pressure at the top of the liquid column, ρ is the density of the liquid, g is the acceleration due to gravity (9.8 m/s²), and h is the height difference between the two liquid columns. By measuring the height difference and knowing the liquid density, you can determine the pressure difference between the gas and the reference pressure.

What are the different types of manometers?

There are several types of manometers, including:

• U-tube Manometer: A simple U-shaped tube filled with liquid, used to measure pressure differences.
• Inclined Manometer: A U-tube manometer with one leg inclined to increase measurement sensitivity for small pressure differences.
• Digital Manometer: Uses electronic sensors to measure pressure and display the results digitally.
• Well-type Manometer: Has a large reservoir (well) on one side and a narrow tube on the other, used for measuring small pressure differences with high accuracy.

Each type has its specific applications and advantages depending on the required accuracy and pressure range.

How do you determine the density of a liquid using a manometer?

To determine the density of a liquid using a manometer, you can use the equation $p_{\mathrm{bottom}}=p_{\mathrm{top}}+\; \rho gh$. If you know the pressures at the top and bottom of the liquid column and the height difference (h), you can rearrange the equation to solve for the liquid density (ρ): $\rho \; =\; (p_{\mathrm{bottom}}-p_{\mathrm{top}})\; /\; (gh)$. By measuring the height difference and knowing the pressures, you can calculate the density of the liquid.

What is the significance of height difference in a manometer?

The height difference (h) in a manometer is crucial because it directly correlates to the pressure difference between the gas being measured and the reference pressure (atmospheric or vacuum). According to the equation $p_{\mathrm{bottom}}=p_{\mathrm{top}}+\; \rho gh$, the height difference, along with the liquid density (ρ) and gravitational acceleration (g), determines the pressure difference. A greater height difference indicates a larger pressure difference, making it a key parameter in accurately measuring and calculating pressure using a manometer.