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Converting pressure units is crucial, especially when applying formulas that require the pressure in a specific unit like Pascals. In this exercise, the initial pressure is provided in atmospheres (atm), a common unit often used when discussing gases at a macroscopic level. However, for calculations involving the ideal gas law, the pressure needs to be in Pascals (Pa), which is more widely used in scientific formulas due to being a standard SI unit.
To convert from atmospheres to Pascals, use the conversion factor:

• 1 atm = 101,325 Pa.

This gives us the formula:

• \[ P = 0.300 \times 101,325 \, \text{Pa} = 30,397.5 \, \text{Pa}. \]

Understanding how to switch between these two units ensures accuracy across physical equations and helps when learning about physical principles expressed in different systems.

Pistons are vital components in many engineering and physics applications because they can change the volume of gas containers, consequently affecting pressure and work.In the given problem, the piston is supported by gas within a cylindrical tank. Key aspects to consider include:

• The area of the piston, which corresponds to the area of the tank's top circle. This is calculated through:\[ A = \pi r^2 = \pi (0.10 \, \text{m})^2 = 0.0314 \, \text{m}^2. \]
• The force exerted by the gas on the piston. This force can be calculated by multiplying the pressure of the gas by the area of the piston:\[ F = PA = 30,397.5 \, \text{Pa} \times 0.0314 \, \text{m}^2 = 954.5 \, \text{N}. \]
• Finally, the mass of the piston can be determined since the force exerted by the gas balances the gravitational force on the piston (assuming equilibrium):\[ m = \frac{F}{g} = \frac{954.5}{9.81} \approx 97.3 \, \text{kg}. \]

Understanding these mechanics helps explain how gases can be used to exert and balance forces in mechanical systems, highlighting the practical applications of physics principles.

Temperature conversion is essential in the application of the ideal gas law, as the law requires temperature to be in Kelvin (K), the standard SI unit for temperature.In Celsius, water freezes at 0°C, but in Kelvin, the same point is 273.15 K. To convert Celsius to Kelvin, use the simple conversion formula:

• \[ T(\text{K}) = T(\text{°C}) + 273.15. \]

For this problem, where the temperature is 20°C, the conversion to Kelvin gives:

• \[ T = 20 + 273.15 = 293.15 \, \text{K}. \]

Using Kelvin allows for straightforward and correct use of the ideal gas law:

• \[ PV = nRT, \]where \( R \) is the universal gas constant.

Grasping temperature conversion is pivotal for accurately working with thermal expansion, reactions, and various gas laws across scientific disciplines.