Question

A massive piston traps a fixed amount of helium gas as shown. After being brought to point (a) the system equilibrates to room temperature. Weight is then added ADIABATICALLY compressing the gas to half of its original volume (b). Either greater than, less than, or equal to.

greater/thanless/thanequal to the pressure P_b _________________ P_a.
greater/thanless/thanequal to The entropy of the gas at "b" is __________________ the entropy of the gas at "a".
greater/thanless/thanequal to The temperature T_b _________________ T_a.
greater/thanless/thanequal to The internal energy U_b __________________

Answer 1

Adiabatic, reversible = isentropic.

Compressing the gas increases the pressure.
The temperature should increase, but the system remains at room temp through the compression, so clearly heat must be leaving the system.

So entropy must change.
dS = dQ / T; T constant, dQ < 0 (heat leaves system to keep temp const), dS < 0

Temperature remains the same.

Internal energy: dU = T dS - p dV
dS < 0, p increases and dV < 0 :  
Just guessing at this one, but it should increase, because work is being done on the gas in excess of the heat being allowed to escape. i.e. T dS decreasing, but -p dV > 0 increasing faster than T dS decreasing.

Pressure necessarily increases.