Problem 3 - University Physics with Modern Physics

Question

A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110 \(\mathrm{m}^{3}\) of air at a pressure of 0.355 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.390 \(\mathrm{m}^{3} .\) If the temperature remains constant, what is the final value of the pressure?

Step-by-Step Solution

Answer

The final pressure is approximately 0.100 atm.

Step 1: Identify the Known Values

The initial volume of the gas is \( V_1 = 0.110 \, \mathrm{m}^3 \). The initial pressure is \( P_1 = 0.355 \, \mathrm{atm} \). The final volume is \( V_2 = 0.390 \, \mathrm{m}^3 \). The temperature remains constant throughout the process.

Step 2: Apply Boyle's Law

Boyle's Law states that for a given mass of an ideal gas at constant temperature, the volume of the gas is inversely proportional to its pressure. Mathematically, it is expressed as \( P_1 V_1 = P_2 V_2 \).

Step 3: Substitute Known Values into Boyle's Law

Using the equation \( P_1 V_1 = P_2 V_2 \), substitute known values: \( (0.355 \, \mathrm{atm})(0.110 \, \mathrm{m}^3) = P_2 (0.390 \, \mathrm{m}^3) \).

Step 4: Solve for Final Pressure \( P_2 \)

Rearrange the equation to find \( P_2 \): \( P_2 = \frac{P_1 V_1}{V_2} = \frac{(0.355)(0.110)}{0.390} \). Calculate this to determine \( P_2 \).

Step 5: Calculate Final Pressure

Perform the calculation: \( P_2 = \frac{0.03905}{0.390} = 0.1001 \, \mathrm{atm} \). Thus, the final pressure of the gas is approximately 0.100 \( \mathrm{atm} \).

Key Concepts

Ideal Gas

The concept of an ideal gas is a theoretical framework used in physics and chemistry to simplify the study of gases. An ideal gas is composed of particles that are so small compared to the distances between them that no interactions occur besides elastic collisions.
These particles are considered to be in random motion. The ideal gas law describes the behavior of such gases, which relates pressure, volume, temperature, and the number of molecules present. It's worth noting that while no gas is truly "ideal," many gases behave like an ideal gas under certain conditions, especially at high temperatures and low pressures.
In problems involving gases like the one at hand, assumptions of ideal behavior often make calculations with laws such as Boyle's more straightforward. Boyle's law assumes ideal conditions where only pressure and volume are taken into account, given that temperature remains constant.

Pressure

Pressure is a crucial parameter when discussing gases. It is defined as the force applied perpendicular to the surface of an object per unit area over which that force is distributed. In the context of gases, pressure emerges from the collisions of gas molecules with the container walls.
As gas molecules move around, they impact the walls of their container, causing forceful interactions. Gas pressure is usually measured in units like Pascals (Pa) or atmospheres (atm).

• High pressure means frequent collisions within a tight space.
• Low pressure means fewer collisions, often associated with an increased volume.

The exercise demonstrates this as the piston is pulled out, reducing the pressure by increasing the volume, perfectly embodying Boyle's Law which relates the variables of pressure and volume for a given gas quantity.

Volume

Volume, in regard to gases, represents the amount of space that the gas occupies. It can change depending on external conditions, such as changes in pressure or temperature when speaking about compressible gases.
In our particular exercise, the piston allows the volume to change by mechanically altering the size of the container holding the gas. This change is slow and controlled to ensure that Boyle's Law applies, that is, constant temperature and energy contribution.
The inverse relationship between volume and pressure as dictated by Boyle's Law is important to note. When volume increases, as seen when the piston is pulled, pressure decreases provided temperature stays consistent. This principle is fundamental in applications ranging from breathing in biology to hydraulics in engineering.

Temperature

Temperature is an essential concept when discussing the behavior of gases. It is a measure of the average kinetic energy of the particles in a substance. For gases, temperature can influence the speed at which molecules move, thus affecting the gas's pressure and volume.
However, in our exercise, the temperature is assumed to be constant. In practical terms, the consistency of temperature ensures that all energy interchanges between pressure and volume are due to mechanical work only, without any added heat energy.
It's helpful to recall that keeping the temperature constant aligns with Boyle's Law application, and such situations are described as isothermal processes. Despite the temperature not changing, it's a key underlying factor because any increase would have influenced the pressure-volume relationship differently.