A researcher investigates how simultaneous cooling and compression affect the internal energy of an ideal monatomic gas. Starting from equilibrium state X, the gas is quasi-statically compressed to one-half of its initial volume while a thermal reservoir removes energy at a steady rate so that the final temperature is 0.60 times the initial temperature.
(1) Derive an algebraic expression for the change in internal energy ΔU of the gas in terms of the initial temperature Ti, the final temperature Tf, and any relevant constants. Show each reasoning step.
(2) Using your result from part (a), explain whether ΔU is positive, negative, or zero for this experiment and interpret the physical meaning of the sign.
(3) Without substituting numerical values, determine the sign of the work done on the gas and the sign of the energy transferred by heating Q. Provide a logical argument that references both the mathematical form of the first law and the thermodynamic changes stated.
(4) If the researcher repeated the experiment but removed energy more rapidly so that the final temperature were 0.40 times the initial temperature, would the magnitude of the work done on the gas change, remain the same, or be impossible to determine from the given information? Justify your answer thoroughly.