An ideal gas is contained in a cylinder fitted with a frictionless piston. Which of the following are correct?

1. AIf the gas is heated at constant pressure, the work done by the gas equals $nR\Delta T$
2. BAt constant temperature, the product $PV$ remains constant as the piston moves
3. CIf the external pressure is doubled at constant temperature, the density of the gas doubles
4. DThe number of molecules per unit volume at constant $T$ is independent of pressure