Problem 5 - Applied Physics

Question

Find the pressure of (a) 3 atm (in \(\mathrm{kPa}\) ), (b) \(2 \mathrm{~atm}\) (in \(\mathrm{kPa}\) ), (c) \(6 \mathrm{~atm}\) (in \(\mathrm{lb} / \mathrm{in}^{2}\) ), (d) 5 atm (in \(\mathrm{kPa}\) ), (e) \(\frac{1}{3}\) atm (in \(\mathrm{kPa}\) ), and (f) \(\frac{1}{4}\) atm (in \(\mathrm{kPa}\) ).

Step-by-Step Solution

Answer

(a) 303.975 kPa, (b) 202.65 kPa, (c) 88.176 psi, (d) 506.625 kPa, (e) 33.775 kPa, (f) 25.33125 kPa.

Step 1: Understand the Conversion Factor

To convert pressures from atmospheres (atm) to kiloPascals (kPa), use the conversion factor: \(1 \, \text{atm} = 101.325 \, \text{kPa}\). To convert pressures from atmospheres to pounds per square inch (psi), use: \(1 \, \text{atm} = 14.696 \, \text{lb/in}^2\).

Step 2: Convert 3 atm to kPa

Using the conversion factor, calculate the kPa for 3 atm: \(3 \, \text{atm} \times 101.325 \, \text{kPa/atm} = 303.975 \, \text{kPa}\).

Step 3: Convert 2 atm to kPa

Using the conversion factor, calculate the kPa for 2 atm: \(2 \, \text{atm} \times 101.325 \, \text{kPa/atm} = 202.65 \, \text{kPa}\).

Step 4: Convert 6 atm to psi

Using the conversion factor, calculate the psi for 6 atm: \(6 \, \text{atm} \times 14.696 \, \text{lb/in}^2/\text{atm} = 88.176 \, \text{lb/in}^2\).

Step 5: Convert 5 atm to kPa

Using the conversion factor, calculate the kPa for 5 atm: \(5 \, \text{atm} \times 101.325 \, \text{kPa/atm} = 506.625 \, \text{kPa}\).

Step 6: Convert \(\frac{1}{3}\) atm to kPa

Calculate the kPa for \(\frac{1}{3}\) atm: \(\frac{1}{3} \, \text{atm} \times 101.325 \, \text{kPa/atm} = 33.775 \, \text{kPa}\).

Step 7: Convert \(\frac{1}{4}\) atm to kPa

Calculate the kPa for \(\frac{1}{4}\) atm: \(\frac{1}{4} \, \text{atm} \times 101.325 \, \text{kPa/atm} = 25.33125 \, \text{kPa}\).

Key Concepts

Atmospheres to Kilopascals Conversion

Atmospheres (atm) and kilopascals (kPa) are common units of pressure measurement. To convert pressure values from atmospheres to kilopascals, we use a simple conversion factor. Knowing this conversion is crucial for many scientific and engineering applications.

The conversion factor is straightforward:

• 1 atmosphere is equal to 101.325 kilopascals.

This means, to convert from atm to kPa, you simply multiply the number of atmospheres by 101.325.

For example, if you have pressure in atmospheric units, let's say 3 atm, you calculate the equivalent in kilopascals using the following equation: \[3 \, \text{atm} \times 101.325 \, \text{kPa/atm} = 303.975 \, \text{kPa}\]This conversion finds multiple uses, especially in atmospheric sciences, where standard atmospheric pressure is often measured in kilopascals.

Pressure in PSI

Whenever dealing with pressure in industrial applications, psi or pounds per square inch is a popular unit that frequently appears. Psi is handy because it correlates well with the imperial measurement system, making it familiar in places like the United States.

To convert from atmospheres to psi, another specific conversion factor is needed:

• 1 atmosphere is equivalent to 14.696 psi.

Let's say you need to convert 6 atm to psi to understand how the pressure compares to, say, a car tire pressure gauge; you apply the conversion as follows: \[6 \, \text{atm} \times 14.696 \, \text{lb/in}^2/\text{atm} = 88.176 \, \text{psi}\]Understanding psi is especially important in fields like automotive mechanics and plumbing, where psi is a common unit for pressure gauges.

Unit Conversion Factors

Unit conversion factors are powerful tools in mathematics and science that allow us to switch between different measurement units easily. They act like bridges connecting various systems of units, making our calculations consistent and accurate.

In pressure conversion, the atmospheric pressure can be converted to a variety of units such as kPa or psi using predefined factors. For example:

• 1 atm to kPa: Multiply by 101.325.
• 1 atm to psi: Multiply by 14.696.

These factors work like magic numbers that facilitate swift and error-free conversions when you know how to use them. Applying the right conversion factor directly influences the accuracy of your results.

This understanding is central in experiments and real-world applications, ensuring measurements are communicated in the most practical units for a specific context.

Pressure Measurement

Pressure measurement is an integral aspect of science and engineering, impacting how we understand and interact with various systems and environments. It represents the force exerted per unit area and can be measured in different units, each suited to specific applications.

Some common units include:

• Atmospheres (atm): Often used in physics and chemistry.
• Kilopascals (kPa): Typically employed in meteorology and engineering.
• Pounds per square inch (psi): Common in industrial settings and mechanical systems.

Each unit serves a unique purpose, providing the precision and relevance needed in its operational area. For instance, kilopascals may be preferred in weather reporting to ensure clarity and uniformity across global contexts.

Measuring pressure accurately is critical for a host of tasks, from predicting weather patterns to assessing the integrity of structures, and calls for understanding not just the unit being used, but the conversion processes that might be needed.