A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase if the gas is compressed to half its original volume?

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We have a cylinder with a movable piston containing 3 moles of hydrogen gas at standard temperature and pressure (STP). The walls of the cylinder are heat insulators, and the piston is insulated by a pile of sand.

To solve this problem, we can utilize the ideal gas law, which states:

PV=nRTPV=nRT

Where:

• PP is the pressure of the gas,
• VV is the volume of the gas,
• nn is the number of moles of gas,
• RR is the ideal gas constant, and
• TT is the temperature of the gas in Kelvin.

Since the temperature and the number of moles of gas remain constant, we can rewrite the ideal gas law as:

P1V1=P2V2P1V1=P2V2

Where:

• P1P1 and V1V1 are the initial pressure and volume, respectively, and
• P2P2 and V2V2 are the final pressure and volume, respectively.

Given that the gas is compressed to half its original volume, V2=12V1V2=21V1.

Now, let's solve for P2P2:

P1V1=P2V2P1V1=P2V2

P2=P1V1V2P2=V2P1V1

P2=P1V112V1P2=21V1P1V1

P2=2P1P2=2P1

So, the pressure of the gas increases by a factor of 2 when it is compressed to half its original volume.

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