Question

If a piston in an automobile engine compresses a fuel-air mixture to a pressure of \(9.50 \mathrm{~atm},\) what is the pressure expressed in each of the following units? (a) in. \(\mathrm{Hg}\) (b) \(\mathrm{mm} \mathrm{Hg}\)

Short answer

284.24 in. Hg; 7220 mm Hg

Step-by-step solution

Understand the Conversion to Inches of Mercury

To convert from atmospheres (atm) to inches of mercury (in. Hg), we need to use the conversion factor: 1 atm = 29.92 in. Hg. This conversion factor relates the standard pressure in atmospheres to the unit of mercury column height.

Calculate Pressure in Inches of Mercury

Multiply the given pressure in atm by the conversion factor: \[ 9.50 \text{ atm} \times 29.92 \text{ in. Hg/atm} = 284.24 \text{ in. Hg} \]

Understand the Conversion to Millimeters of Mercury

To convert from atmospheres (atm) to millimeters of mercury (mm Hg), we use the conversion factor: 1 atm = 760 mm Hg, which is another standard pressure equivalence.

Calculate Pressure in Millimeters of Mercury

Multiply the given pressure in atm by the conversion factor: \[ 9.50 \text{ atm} \times 760 \text{ mm Hg/atm} = 7220 \text{ mm Hg} \]

Key concepts

Atmospheres to Inches of Mercury

Atmospheres (atm) and inches of mercury (in. Hg) are both units used to measure pressure, each bringing its unique context and usage. The conversion from atmospheres to inches of mercury is a representation of pressure in terms of the height of a mercury column. Mercury is used because it's dense, so smaller physical columns can represent significant pressures.

To convert atmospheres to inches of mercury, we use the conversion factor for these two units. It states that 1 atmosphere is equivalent to 29.92 inches of mercury. This means that for every atmosphere of pressure, it can push a column of mercury to a height of 29.92 inches.

Using this conversion factor, you can easily convert any pressure in atmospheres to inches of mercury by multiplying the number of atmospheres by 29.92. For example, a pressure of 9.50 atm would be converted as follows:

• 9.50 atm × 29.92 in. Hg/atm = 284.24 in. Hg

Thus, a pressure of 9.50 atm is equivalent to 284.24 inches of mercury.

Atmospheres to Millimeters of Mercury

Millimeters of mercury (mm Hg), often called "Torr" after the Italian physicist Evangelista Torricelli, is a traditional unit of pressure. It is commonly used in meteorology, aviation, and medicine. The idea is similar to using inches of mercury; however, it is more prevalent internationally.

A straightforward conversion factor allows conversion between atmospheres and millimeters of mercury: 1 atm equals 760 mm Hg. This indicates the height a mercury column would reach under the pressure of one atmosphere, in millimeters rather than inches.

To convert pressure from atmospheres to millimeters of mercury, multiply the pressure in atm by 760.

Here is an example with 9.50 atm:

• 9.50 atm × 760 mm Hg/atm = 7220 mm Hg

By this calculation, a pressure of 9.50 atm is equivalent to 7220 mm Hg. Converting between these units can assist in understanding pressure in terms different from standard atmospheric pressure.

Conversion Factors in Pressure Measurement

Understanding conversion factors is crucial for effectively measuring and comparing pressure in different units. A conversion factor is simply a mathematical tool that transforms numbers from one unit to another in a reliable way.

When it comes to pressure, conversion factors help us understand the same physical quantity under different measurement systems. Here are some common conversion factors used for pressure:

• 1 atm = 101325 Pascals (Pa)
• 1 atm = 14.696 pounds per square inch (psi)
• 1 atm = 29.92 inches of mercury (in. Hg)
• 1 atm = 760 millimeters of mercury (mm Hg)

These conversion factors are necessary not just for academic exercises, but also in fields that require precise pressure measurements, such as engineering and chemistry.

For example, converting pressure readings from weather reports (often given in mm Hg) to psi might be essential for designing a product that must operate under certain weather conditions.

By using conversion factors, you can approach problems with more flexibility, ensuring results are applicable across different industrial contexts.