Question

A vacuum gage indicates that the pressure of air in a closed chamber is \(0.2\) bar (vacuum). The pressure of the surrounding atmosphere is equivalent to a \(750-\mathrm{mm}\) column of mercury. The density of mercury is \(13.59 \mathrm{~g} / \mathrm{cm}^{3}\), and the acceleration of gravity is \(9.81 \mathrm{~m} / \mathrm{s}^{2}\). Determine the absolute pressure within the chamber, in bar.

Short answer

0.8003 bar

Step-by-step solution

- Understand the Problem

The goal is to determine the absolute pressure within a closed chamber given the vacuum gage reading and atmospheric pressure. The vacuum gage reads 0.2 bar for the vacuum, and the atmospheric pressure is given by a mercury column of height 750 mm.

- Convert Atmospheric Pressure

To convert the atmospheric pressure from the height of a mercury column to bar, use the formula: \[ P_{\text{atm}} = \rho_{\text{Hg}} \times g \times h_{\text{Hg}} \ \text{where} \ \rho_{\text{Hg}} = 13.59 \text{ g/cm}^3 \ g = 9.81 \text{ m/s}^2 \ h_{\text{Hg}} = 750 \text{ mm} \ \text{Convert to consistent units,...} \ \rho_{\text{Hg}} = 13.59 \times 10^3 \text{ kg/m}^3 \ h_{\text{Hg}} = 0.750 \text{ m} \ P_{\text{atm}} = (13.59 \times 10^3) \times 9.81 \times 0.750 \text{ Pa} \ P_{\text{atm}} = 100030.575 \text{ Pa} \]

- Convert Atmospheric Pressure to Bar

Convert the atmospheric pressure obtained to bar: \[ P_{\text{atm}} \text{ in bar} = \frac{100030.575}{100000} = 1.0003 \text{ bar} \]

- Calculate Absolute Pressure

Since the gauge provides the vacuum pressure, determine absolute pressure using: \[ P_{\text{abs}} = P_{\text{atm}} - P_{\text{vacuum}} \ P_{\text{abs}} = 1.0003 - 0.2 \text{ bar} \ P_{\text{abs}} = 0.8003 \text{ bar} \]

Key concepts

vacuum pressure

Vacuum pressure is the pressure below the atmospheric pressure. It is often measured with a vacuum gage. For example, when the vacuum gage of a closed chamber reads 0.2 bar, it indicates that the pressure inside the chamber is 0.2 bar less than the atmospheric pressure. Understanding vacuum pressure is crucial because it helps in calculating the absolute pressure within a system.

To find the absolute pressure, you need to know the atmospheric pressure and subtract the vacuum pressure from it. This gives a clear indication of the true pressure inside the chamber.

atmospheric pressure

Atmospheric pressure is the pressure exerted by the weight of the atmosphere. At sea level, this pressure is standard and about 1.013 bar or 1013 millibars. However, atmospheric pressure can vary based on altitude and weather conditions.

In the exercise, the atmospheric pressure is given as 750 mm of a mercury column. To convert this into bar, you first need to calculate the pressure in pascals (Pa) using the formula: \[ P_{\text{atm}} = \rho_{\text{Hg}} \times g \times h_{\text{Hg}} \] where \( \rho_{\text{Hg}} = 13590 \text{ kg/m}^3 \) is the density of mercury, **g** is **9.81 m/s^2**, and **h** is **0.750 m**. This results in 100030.575 Pa.

Finally, to convert pascals to bar: \[ 1 \text{ bar} = 100000 \text{ Pa} \] \[ P_{\text{atm}} \text{ in bar} = \frac{100030.575}{100000} = 1.0003 \text{ bar} \] Understanding atmospheric pressure and its conversion is crucial in calculating absolute pressure.

pressure conversion

Pressure conversion is essential in many scientific and engineering calculations. The need to convert pressure units arises because measurements might be given in different units. For instance, in this exercise, the atmospheric pressure is initially in millimeters of mercury (mm Hg) and must be converted to bar.

Here's the detailed process:

• Convert the given height of mercury column to meters (750 mm equals 0.750 m).
• Use the formula to convert to pascals, knowing \( 1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3 \), \( P_{\text{atm}} = \rho_{\text{Hg}} g h_{\text{Hg}} = (13.59 \times 10^3) \times 9.81 \times 0.750 \text{ Pa} = 100030.575 \text{ Pa} \).

After obtaining the pressure in pascals, convert to bar: \[ P_{\text{atm}} \text{ in bar} = \frac{100030.575}{100000} = 1.0003 \text{ bar} \]

For absolute pressure, combine atmospheric pressure and vacuum pressure using: \[ P_{\text{abs}} = P_{\text{atm}} - P_{\text{vac}} = 1.0003 \text{ bar} - 0.2 \text{ bar} = 0.8003 \text{ bar} \]
Understanding pressure conversion can simplify and solve many problems in physics and engineering.