3.3 Figure 3.22 on the next page shows the initial state of an apparatus consisting of an ideal gas in a bulb, a stopcock, a porous plug, and a cylinder containing a frictionless piston. The walls are diathermal, and the surroundings are at a constant temperature of 300.0K and a constant pressure of 1.00 bar. When the stopcock is opened, the gas diffuses slowly through the porous plug, and the piston moves slowly to the right. The process ends when the pressures are equalized and the piston stops moving. The system is the gas. Assume that during the process the temperature through- out the system differs only infinitesimally from 300.0K and the pressure on both sides of the piston differs only infinitesimally from 1.00 bar. p = 3.00 bar porous V = 0.500 m³ plug piston T = 300.0 K . gas a Text = 300.0 K Pext = 1.00 bar Figure 3.22 (a) Which of these terms correctly describes the process: isothermal, isobaric, isochoric, reversible, irreversible? (b) Calculate q and W. 3.4 Consider a horizontal cylinder-and-piston device similar to the one shown in Fig. 3.4 on page 70. The piston has mass m. The cylinder wall is diathermal and is in thermal contact with a heat reservoir of temperature Text. The system is an amount n of an ideal gas confined in the cylinder by the piston. The initial state of the system is an equilibrium state described by pi and T = Text. There is a constant external pressure Pext> equal to twice P1, that supplies a constant external force on the piston. When the piston is released, it begins to move to the left to compress the gas. Make the idealized assumptions that (1) the piston moves with negligible friction; and (2) the gas remains practically uniform (because the piston is massive and its motion is slow) and has a practically constant temperature T = Text (because temperature equilibration is rapid). (a) Describe the resulting process. (b) Describe how you could calculate w and q during the period needed for the piston velocity to become zero again. (c) Calculate w and q during this period for 0.500 mol gas at 300 K. Ffric Fgas Fext x Figure 3.4 Forces acting on the piston (cross hatched) in a cylinder-and-piston device containing a gas (shaded). The direction of Ffric shown here is for expansion. 3.5 This problem is designed to test the assertion on page 59 that for typical thermodynamic pro- cesses in which the elevation of the center of mass changes, it is usually a good approximation to set w equal to Wlab- The cylinder shown in Fig. 3.23 on the next page has a vertical orienta- tion, so the elevation of the center of mass of the gas confined by the piston changes as the pis- ton slides up or down. The system is the gas. Assume the gas is nitrogen (M = 28.0 g at 300K, and initially the vertical length l of the gas column is one meter. Treat the nitro- gen as an ideal gas, use a center-of-mass local frame, and take the center of mass to be at the midpoint of the gas column. Find the difference between the values of w and Wlab, expressed as a percentage of w, when the gas is expanded reversibly and isothermally to twice its initial volume. gas Figure 3.23 3.6 Figure 3.24 on the next page shows an ideal gas confined by a frictionless piston in a vertical cylinder. The system is the gas, and the boundary is adiabatic. The downward force on the piston can be varied by changing the weight on top of it. (a) Show that when the system is in an equilibrium state, the gas pressure is given by p = mgh/V where m is the combined mass of the piston and weight, go is the acceleration of free fall, and h is the elevation of the piston shown in the figure. (c) It might seem that by making the weight placed on the piston sufficiently large, V2 could be made as close to zero as desired. Actually, however, this is not the case. Find ex- pressions for V2 and T2 in the limit as m2 approaches infinity, and evaluate V2/V1 in this limit if the heat capacity is Cy = (3/2)nR (the value for an ideal monatomic gas at room temperature). vacuum weight ideal h gas Figure 3.24 p = 3.00 bar p = 0 V = 0.500 m³ V = 1.001 m³ T = 300.0K gas o vacuum Text = 300.0K Figure 3.25 3.8 Figure 3.25 shows the initial state of an apparatus containing an ideal gas. When the stopcock is opened, gas passes into the evacuated vessel. The system is the gas. Find q, w, and AU under the following conditions. (a) The vessels have adiabatic walls. (b) The vessels have diathermal walls in thermal contact with a water bath maintained at 300. K, and the final temperature in both vessels is T = 300. K.