Q4. The absolute temperature of a gas is doubled from 300 K to 600 K. By what factor does the average kinetic energy of its molecules change? And the rms speed?
The average kinetic energy is proportional to T, so it doubles. The rms speed is proportional to √T, so it rises by √2, about 1.41 times.
Q5. What is the total translational kinetic energy of the molecules in 1 m of a gas at 1 atm (101 325 Pa)?
PV = (2/3)E, so E = (3/2)PV = 1.5 × 101 325 × 1, about 1.52 × 10 J. You need only P and V: the kind of gas does not matter, and T is not needed separately (at the same P and V, a hotter gas simply has fewer molecules).
Q6. Explain, using kinetic theory, why a gas heats up when it is compressed quickly by a piston.
Molecules bouncing off a piston that is moving in rebound faster than they arrived, just as a ball rebounds faster off a massive bat moving towards it. If heat loss is negligible and work goes into the gas, its average random kinetic energy and temperature rise. Rapid compression alone does not justify the reversible PV^γ formula.