Ideal Gas Behavior with Volume Change

When an ideal gas is contained within a piston-cylinder arrangement, its physical properties like pressure, volume, and temperature are interconnected. Understanding how these properties change under specific conditions is fundamental to the study of thermodynamics and gas laws.

Gas Laws and Boyle's Law

The behavior of ideal gases is described by several gas laws, which are derived from the ideal gas law equation. One crucial law applicable to this scenario is Boyle's Law.

• Ideal Gas: An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle attractive or repulsive forces. Its behavior is accurately described by the ideal gas law.
• Boyle's Law: This law states that for a fixed amount of an ideal gas kept at a constant temperature, the pressure and volume are inversely proportional. In simpler terms, if the volume of the gas increases, its pressure decreases, and vice-versa, assuming the temperature remains unchanged.

Mathematically, Boyle's Law can be expressed as:

$$P \propto \frac{1}{V} \quad \text{or} \quad PV = k$$

Where:

• \(P\) is the pressure of the gas
• \(V\) is the volume of the gas
• \(k\) is a constant value, provided the temperature and the amount of gas remain constant.

For two different states of the same ideal gas at constant temperature, Boyle's Law can be written as:

$$P_1V_1 = P_2V_2$$

Where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume, respectively.

Pressure Calculation for Doubled Volume

Let's apply Boyle's Law to the given problem statement. We are given the following conditions:

• The gas is an ideal gas.
• The gas is in a piston-cylinder arrangement.
• The temperature of the gas is constant.
• The volume of the gas gets doubled.

Let the initial pressure be \(P_1\) and the initial volume be \(V_1\). According to the problem:

• Initial Volume = \(V_1\)
• Final Volume, \(V_2\) = \(2V_1\) (since the volume gets doubled)
• Initial Pressure = \(P_1\)
• Final Pressure = \(P_2\) (what we need to find)

Using Boyle's Law equation for constant temperature:

$$P_1V_1 = P_2V_2$$

Substitute the known values into the equation:

$$P_1V_1 = P_2(2V_1)$$

To find the final pressure \(P_2\), we can rearrange the equation:

$$P_2 = \frac{P_1V_1}{2V_1}$$

The term \(V_1\) cancels out from the numerator and denominator:

$$P_2 = \frac{P_1}{2}$$

This result shows that the final pressure \(P_2\) is half of the initial pressure \(P_1\).

Pressure Outcome for Ideal Gas

Based on Boyle's Law, when the volume of an ideal gas is doubled at a constant temperature, its pressure becomes half of its original value. This inverse relationship is a key characteristic of ideal gas behavior under isothermal (constant temperature) conditions.