To tackle this problem, we can apply the principles of the ideal gas law and the concept of pressure in a closed system. When we have a cylinder containing air at a certain pressure and we remove some of that air while keeping the temperature constant, we can predict how the pressure will change based on the amount of air left in the cylinder.

Understanding the Initial Conditions

Initially, we have a cylinder containing 2 kg of air at a pressure of P_a. According to the ideal gas law, pressure (P), volume (V), and temperature (T) are related to the number of moles (n) of gas in the system. For our purposes, we can simplify this by focusing on mass instead of moles since we are dealing with air in kilograms.

Applying the Ideal Gas Law

The ideal gas law can be expressed in a simplified form for our scenario as:

• P1 * V = n1 * R * T
• P2 * V = n2 * R * T

Where:

• P1 and P2 are the initial and final pressures, respectively.
• n1 and n2 are the initial and final amounts of air (in kg).
• R is the specific gas constant for air.
• T is the temperature, which remains constant in this case.

Calculating the Final Pressure

Initially, we have 2 kg of air (n1 = 2 kg). When we pump out 1 kg of air, we are left with 1 kg (n2 = 1 kg). Since the temperature and volume of the cylinder remain constant, we can set up a ratio of the pressures:

Using the relationship between the initial and final states:

P1 / P2 = n1 / n2

Substituting the known values:

P_a / P2 = 2 kg / 1 kg

This simplifies to:

P2 = P_a / 2

Final Result

Thus, the pressure in the cylinder after removing 1 kg of air, while keeping the temperature constant, will be half of the initial pressure:

P2 = P_a / 2

In summary, by removing half of the air from the cylinder, the pressure is reduced to half of its original value, demonstrating the direct relationship between the amount of gas and pressure in a closed system at constant temperature.