Methane, CH4, is studied over a range of temperatures to investigate real-gas behavior. For each temperature, researchers collect a set of P versus Vm (molar volume) data at constant T.
The following qualitative observations are made:
• At 400 K, the P–Vm curve is nearly a hyperbola, closely matching the ideal-gas equation.
• At 250 K, the curve lies below the ideal prediction at moderate volumes but rises above it at very small volumes.
• At 120 K, the curve shows a pronounced minimum in pressure before rising sharply at small volumes.
(a) Describe, in words, the physical meaning of the region where the 250 K curve lies below the ideal-gas prediction.
(b) Explain why the 250 K curve eventually rises above the ideal prediction at very small molar volumes.
(c) For the 120 K curve, account for the presence of a pressure minimum. In your explanation, integrate molecular-level reasoning about attractive and repulsive interactions.
(d) Propose one modification to the ideal-gas model that would allow a single equation to reproduce all three qualitative behaviors described. Your proposal should identify what physical effect is being incorporated and how it alters the pressure–volume relationship.