Page 67 - Physical chemistry eng

44 CHAPTER 2 Heat, Work, Internal Energy, Enthalpy, and the First Law of Thermodynamics

P2.37 Calculate ¢H and ¢U for the transformation of P2.41 The Young’s modulus (see Problem P2.40) of muscle
7
2.50 mol of an ideal gas from 19.0°C and 1.00 atm to 550.°C and fiber is approximately 2.80 * 10 Pa . A muscle fiber 3.25 cm
T –1 –1 in length and 0.125 cm in diameter is suspended with a mass
19.5 atm if C P,m = 20.9 + 0.042 in units of J K mol .
K M hanging at its end. Calculate the mass required to extend
P2.38 A 1.75 mole sample of an ideal gas for which the length of the fiber by 10%.
-1
C V,m = 20.8 J K mol -1 is heated from an initial tempera- P2.42 DNA can be modeled as an elastic rod that can be
ture of 21.2°C to a final temperature of 380.°C at constant
twisted or bent. Suppose a DNA molecule of length L is bent
volume. Calculate q, w, ¢U , and ¢H for this process.
such that it lies on the arc of a circle of radius R c . The
P2.39 An ideal gas undergoes a single-stage expansion reversible work involved in bending DNA without twisting is
against a constant external pressure P external = P f at constant w = BL where B is the bending force constant. The
temperature from T, P , V , to T, P , V . bend 2R 2 c
f
f
i
i
a. What is the largest mass m that can be lifted through the DNA in a nucleosome particle is about 680. Å in length.
height h in this expansion? Nucleosomal DNA is bent around a protein complex called
the histone octamer into a circle of radius 55 Å. Calculate the
b. The system is restored to its initial state in a single-state
¿
compression. What is the smallest mass m that must fall reversible work involved in bending the DNA around the his-
-28
tone octamer if the force constant B = 2.00 * 10 J m .
through the height h to restore the system to its initial state?
6
6
c. If h = 15.5 cm , P = 1.75 * 10 Pa , P = 1.25 * 10 Pa , P2.43 A 1.75 mole sample of an ideal gas is compressed
i
f
T = 280. K , and n = 2.25 mol , calculate the values of the isothermally from 62.0 L to 19.0 L using a constant external
pressure of 2.80 atm. Calculate q, w, ¢U , and ¢H .
masses in parts (a) and (b).
P2.44 Assume the following simplified dependence of the
P2.40 The formalism of the Young’s modulus is sometimes
pressure in a ventricle of the human heart as a function of the
used to calculate the reversible work involved in extending or
volume of blood pumped.
compressing an elastic material. Assume a force F is applied
and length L . As a
to an elastic rod of cross-sectional area A 0 0
result of this force the rod changes in length by ¢L . The
Young’s modulus E is defined as
F > P s
tensile stress A 0 FL 0 Pressure
E = = = P
tensile strain ¢L > A ¢L d
0
L 0
a. Relate k in Hooke’s Law to the Young’s modulus expres-
0 75 150
sion just given. 3
V/cm
b. Using your result in part (a) show that the magnitude of
the reversible work involved in changing the length L of P , the systolic pressure, is 120. mm Hg, corresponding to
0
s
an elastic cylinder of cross-sectional area A by ¢L is 0.158 atm. P , the diastolic pressure, is 80.0 mm Hg, corre-
0
d
1 ¢L 2 sponding to 0.105 atm. If the volume of blood pumped in one
w = a b EA L .
0 0
3
2 L 0 heartbeat is 75.0 cm , calculate the work done in a heartbeat.
Web-Based Simulations, Animations, and Problems
W2.1 A simulation is carried out in which an ideal gas is W2.4 The isochoric heating and cooling of an ideal gas is
heated under constant pressure or constant volume conditions. simulated for different values of volume. The number of
The quantities ¢V (or ¢P ), w, ¢U , and ¢T are determined moles of gas and ¢U are calculated from the constant V value
as a function of the heat input. The heat taken up by the gas and from the T and P values obtained in the simulation.
under constant P or V is calculated and compared with ¢U W2.5 Reversible cyclic processes are simulated in which
and ¢H . the cycle is either rectangular or triangular on a P–V plot. For
W2.2 The reversible isothermal compression and expansion each segment and for the cycle, ¢U , q, and w are determined.
of an ideal gas is simulated for different values of T. The For a given cycle type, the ratio of work done on the sur-
work w is calculated from the T and V values obtained in the roundings to the heat absorbed from the surroundings is
simulation. The heat q and the number of moles of gas in the determined for different P and V values.
system are calculated from the results. W2.6 The reversible adiabatic heating and cooling of an
W2.3 The reversible isobaric compression and expansion of ideal gas is simulated for different values of the initial tem-
an ideal gas is simulated for different values of pressure gas perature. The quantity g = C P,m >C V,m as well as C P,m and
as heat flows to/from the surroundings. The quantities q, w, C V,m are determined from the P, V values of the simulation;
and ¢U are calculated from the ¢T and ¢V values obtained ¢U and ¢U are calculated from the V, T, and P values
in the simulation. obtained in the simulation.