A player bounces a basketball on the floor, compressing it to $80.0\%$ of its original volume. The air (assume it is essentially N₂ gas) inside the ball is originally at $20.0$°C and $2.00$ atm. The ball's inside diameter is $23.9$ cm. What temperature does the air in the ball reach at its maximum compression? Assume the compression is adiabatic and treat the gas as ideal.

On a warm summer day, a large mass of air (atmospheric pressure $1.01\times {10}^5$ Pa) is heated by the ground to $26.0$°C and then begins to rise through the cooler surrounding air. (This can be treated approximately as an adiabatic process; why?) Calculate the temperature of the air mass when it has risen to a level at which atmospheric pressure is only $0.850\times {10}^5$ Pa. Assume that air is an ideal gas, with $g=1.40$. (This rate of cooling for dry, rising air, corresponding to roughly $1$ C° per $100$ m of altitude, is called the dry adiabatic lapse rate.)

A cylinder contains 0.100 mol of an ideal monatomic gas. Initially the gas is at $1.00\times {10}^5$ Pa and occupies a volume of $2.50\times {10}^{-3}$ m³. If the gas is allowed to expand to twice the initial volume, find the final temperature (in kelvins) and pressure of the gas if the expansion is (i) isothermal; (ii) isobaric; (iii) adiabatic.