If we continuously compress a real gas in a cylinder with a movable piston while keeping the container at a constant temperature in an ice bath: (a) What will eventually happen to the gas and why? (b) How well will the ideal gas law apply to the real gas shown when it is close to doing what you said it will do in part (a)?

As we continuously compress the real gas inside the cylinder with a movable piston, the pressure of the gas will increase, and the volume of the gas will decrease. The constant temperature inside an ice bath implies that the gas's internal energy will remain constant, as will the temperature of the gas. Eventually, due to the continuous compression, the gas molecules will be in very close proximity to each other, and the intermolecular forces will start to dominate. When intermolecular forces become dominant, the gas will eventually condense into a liquid. This occurs because the increased pressure forces the gas particles to come closer together, and when they are close enough, the attractive forces between the particles will cause the gas to change its phase and become a liquid. So, the answer to part (a) is that the real gas will eventually condense into a liquid due to the continuous compression and dominance of intermolecular forces.

The ideal gas law, given by PV = nRT, works well under the assumption that gas particles have negligible volume and negligible intermolecular forces between them. However, when the real gas is close to condensing into a liquid, as described in part (a), the volume of the gas particles becomes significant compared to the overall volume of the gas. Moreover, the intermolecular forces also become significant due to the close proximity of the gas particles. Under these conditions, the ideal gas law no longer holds true, and we need to consider a more realistic model like Van der Waals equation of state to incorporate these deviations from the ideal gas law. The Van der Waals equation is given by \[ \left( P + \frac{an^2}{V^2}\right) (V-nb) = nRT,\] where a and b are Van der Waals constants specific to the gas under consideration. So, the answer to part (b) is that the ideal gas law will not apply well to the real gas when it is close to condensing into a liquid, because the assumptions of negligible volume and intermolecular forces are no longer valid under these conditions.