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
V = 0.500 m³
T = 300.0 K
Text = 300.0 K
Pext = 1.00 bar
(a) Which of these terms correctly describes the process: isothermal, isobaric, isochoric,
(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.
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
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
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
p = 3.00 bar
p = 0
V = 0.500 m³
V = 1.001 m³
T = 300.0K
Text = 300.0K
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.
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