An ideal gas is contained within closed box behind a removable wall. The walls of the container are insulated, so that no heat may be transferred from the system. Initially, the gas has an initial pressure P and an initial temperature T. The gas is then allowed to freely expand to twice the initial volume by the removal of the inside wall. Which of the following statements regarding the final pressure is true?

The final pressure is twice the initial pressure.

The final pressure is one-half the initial pressure.

The final pressure is one-fourth the initial pressure.

The final pressure is equal to the initial pressure.

The final pressure is four times the initial pressure.

http://www.schoolphysics.co.uk/age16-19/Thermal%20physics/Gas%20laws/text/Adiabatic_change_equation/index.html

Note example problem with adiabatic expansion to twice volume.

Would it be twice the initial pressure?

To determine the final pressure, we can use the ideal gas law: PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.

Since the gas is expanding freely without any heat exchange, we can assume that the number of moles of gas remains constant (n is constant). Additionally, since the system is insulated, there is no heat exchange, so the temperature remains constant as well (T is constant).

Initially, the gas has a pressure P and a temperature T, and it occupies a volume V. After the removal of the inside wall, the gas freely expands to twice its initial volume. Therefore, the final volume is 2V.

Using the ideal gas law, we can write:

(P)(V) = (n)(R)(T) (Initial state)
(P)(2V) = (n)(R)(T) (Final state)

Canceling out the constant terms (n, R, and T) on both sides of the equations, we get:

P = (2P)

Simplifying it further, we find:

P = 2P

This equation tells us that the final pressure is equal to twice the initial pressure. Therefore, the correct statement is:

The final pressure is twice the initial pressure.