Problem 31 - An Introduction to Thermal Physics

Question

Imagine some helium in a cylinder with an initial volume of 1 liter and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 liters, in such a way that its pressure rises in direct proportion to its volume.

Step-by-Step Solution

Answer

The final pressure is 3 atm.

Step 1: Understand the Problem

We have helium gas starting at an initial volume of \( V_1 = 1 \) liter and initial pressure of \( P_1 = 1 \) atm. The gas expands such that the pressure is directly proportional to the volume, i.e., \( P \propto V \). This means if the volume becomes 2 liters, the pressure doubles, if it's 3 liters, the pressure triples, etc.

Step 2: Determine the Relationship

Since the pressure rises in direct proportion to volume, we can express this relationship as \( P = kV \), where \( k \) is a constant of proportionality. Initially, when \( V = 1 \) liter, \( P = 1 \) atm, so \( k = 1 \).

Step 3: Calculate the Final Pressure

Using the relationship \( P = kV \) and knowing \( k = 1 \), when the final volume \( V_2 = 3 \) liters, the pressure \( P_2 = kV_2 = 1 \times 3 = 3 \) atm.

Step 4: Review the Process

We found that the pressure is directly proportional to the volume change as given, so each increase in liter volume results in an equal increase in atmospheric pressure. This meant applying the proportionality constant throughout the given conditions.

Key Concepts

Helium Gas

Helium is one of the noble gases, characterized by its lightness and low reactivity. It's the second lightest element after hydrogen, and it exists as a monatomic gas under normal conditions. Helium is often used in applications where inertness is valuable, such as in balloons, airships, and cooling systems.
Helium is particularly interesting in thermodynamics due to its simple diatomic structure and predictable behavior under various conditions. This makes it an ideal candidate in theoretical studies and practical applications. Understanding how helium behaves when subjected to changes in pressure, volume, and temperature is crucial for engineers and scientists alike.
In our problem, we're looking at helium gas's behavior in an expansion scenario which is a typical exercise in thermodynamics to explore the physical principles governing gas behaviors.

Proportional Pressure-Volume Relationship

The relationship between pressure and volume is a fundamental concept in thermodynamics, especially relevant for ideal and real gases. In this exercise, the pressure of the helium gas increases directly proportional to its volume. This means that as the volume of the gas increases, its pressure increases by the same factor.
Mathematically, this relationship can be described by the equation \( P = kV \), where \( P \) is the pressure, \( V \) is the volume, and \( k \) is a constant of proportionality. This relationship is distinct from Boyle's Law, which states that pressure and volume are inversely proportional for ideal gases at constant temperature.
When applying this to a real-world problem, remembering that each change in volume results in a directly proportional change in pressure can simplify calculations and enhance predictions about gas behavior. The direct proportionality seen in this scenario is not commonly natural but can occur in controlled environments.

Gas Expansion Calculation

Calculating gas expansion involves understanding and applying the relationships between pressure, volume, and sometimes temperature. In the given problem, the helium gas starts with an initial volume of 1 liter and a pressure of 1 atm. It then expands to a volume of 3 liters, with its pressure increasing proportionally.
To determine the final pressure, we use the equation derived from the direct proportionality \( P = kV \). Given that initially \( P_1 = 1 \) atm at \( V_1 = 1 \) liter, the constant \( k = 1 \). This constant helps calculate the final pressure when the volume changes.
Thus, when the volume expands to 3 liters, applying \( P = 1 \times 3 \), we find the final pressure is 3 atm. Understanding this helps in practical applications where controlling or predicting changes in gaseous systems is required. Gas expansion calculations are essential in processes ranging from industrial applications to everyday technologies.