Page 261 - Physical Chemistry

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7.53 Calculate the pressure inside a bubble of gas in water at 7.63 On the sea bottom at the Galápagos Rift, water heated to
20°C if the pressure of the water is 760 torr and the bubble 350°C gushes out of hydrothermal vents at a depth of 3000 m.
radius is 0.040 cm. See Prob. 7.48 for g. Will this water boil or remain liquid at this depth? The vapor
pressure of water is 163 atm at 350°C. (The heat of this water
7.54 At 20°C, the capillary rise at sea level for methanol
is used as an energy source by sulfide-oxidizing bacteria con-
in contact with air in a tube with inside diameter 0.350 mm is
tained in the tissues of tube worms living on the ocean floor.)
3.33 cm. The contact angle is zero. The densities of methanol
3
and air at 20°C are 0.7914 and 0.0012 g/cm . Find g for 7.64 In Prob. 4.28b, we found that G is 2.76 cal/g for the
CH OH at 20°C. conversion of supercooled water to ice, both at 10°C and
3
1 atm. The vapor pressure of ice is 1.950 torr at 10°C. Find
7.55 For the Hg–air interface on glass, u 140°. Find the the vapor pressure of supercooled water at 10°C. Neglect the
capillary depression of Hg in contact with air at 20°C in a glass effect of pressure changes on G of condensed phases.
tube with inside diameter 0.350 mm. For Hg at 20°C, r 13.59 m
3
2
g/cm and g 490 ergs/cm . (See Prob. 7.54). 7.65 The vapor pressure of liquid water at 0.01°C is 4.585
torr. Find the vapor pressure of ice at 0.01°C.
7.56 At 20°C, the interfacial tension for the liquids n-hexane
2
and water is 52.2 ergs/cm . The densities of n-hexane and water 7.66 (a) Consider a two-phase system, where one phase is pure
3
at 20°C are 0.6599 and 0.9982 g/cm . Assuming a zero contact liquid A and the second phase is an ideal gas mixture of A vapor
angle, calculate the capillary rise at 20°C in a 0.350-mm inside with inert gas B (assumed insoluble in liquid A). The presence of
l
diameter tube inserted into a two-phase n-hexane–water system. gas B changes m , the chemical potential of liquid A, because B
A
increases the total pressure on the liquid phase. However, since
7.57 (a) In Eq. (7.37), h is the height of the bottom of the the vapor is assumed ideal, the presence of B does not affect m ,
g
A
meniscus. Hence, Eq. (7.37) neglects the pressure due to the the chemical potential of A in the vapor phase [see Eq. (6.4)].
small amount of liquid b that is above the bottom of the menis- Because of its effect on m ,gas B affects the liquid–vapor equi-
l
cus. Show that, if this liquid is taken into account, then g A
1 1 librium position, and its presence changes the equilibrium vapor
b
a
2 (r r )gr(h r) for u 0. (b) Rework Prob. 7.54 using pressure of A. Imagine an isothermal infinitesimal change dP in
3
this more accurate equation.
the total pressure P of the system. Show that this causes a change
7.58 Two capillary tubes with inside radii 0.600 and 0.400 dP in the vapor pressure of A given by
A
3
mm are inserted into a liquid with density 0.901 g/cm in con- l l
3
tact with air of density 0.001 g/cm . The difference between the dP A V m,A V m,A P A const. T (7.38)
dP g RT
capillary rises in the tubes is 1.00 cm. Find g. Assume a zero V m,A
contact angle. Equation (7.38) is often called the Gibbs equation. Because V l
m,A
is much less than V g m,A , the presence of gas B at low or moder-
Section 7.9 ate pressures has only a small effect on the vapor pressure of A.
7.59 Let K° be the concentration-scale standard equilibrium (b) The vapor pressure of water at 25°C is 23.76 torr. Calculate
c
constant for the equilibrium nL ∆ L between monomers and the vapor pressure of water at 25°C in the presence of 1 atm of
n
micelles in solution, where L is an uncharged species (for ex- inert ideal gas insoluble in water.
ample, a polyoxyethylene). (a) Let c be the stoichiometric con-
7.67 The vapor pressure of water at 25°C is 23.766 torr.
centration of the solute (that is, the number of moles of
Calculate G° for the process H O(l) → H O(g). Assume the
monomer used to prepare a liter of solution) and let x be the 298 2 2
vapor is ideal. Compare with the value found from data in the
concentration of micelles at equilibrium: x [L ], where the
n
1/n
brackets denote concentration. Show that c nx (x/K ) . Appendix.
c
Assume all activity coefficients are 1. (b) Let f be the fraction 7.68 Benzene obeys Trouton’s rule, and its normal boiling
of L present as monomer. Show that f 1 nx/c. (c) For n point is 80.1°C. (a) Use (7.22) to derive an equation for the vapor
50 and K° 10 200 , calculate and graph [L], n[L ], and f as func-
c n pressure of benzene as a function of T.(b) Find the vapor pressure
tions of c. (Hint: Calculate c for various assumed values of x, of C H at 25°C. (c) Find the boiling point of C H at 620 torr.
6
6
6
6
rather than the reverse.) (d) If the cmc is taken as the value of
7.69 Some vapor pressures for H O(l) are 4.258 torr at
c for which f 0.5, give the value of the cmc. 2
1.00°C, 4.926 torr at 1.00°C, 733.24 torr at 99.00°C, and
787.57 torr at 101.00°C. (a) Calculate H , S , and G for
General m m m
the equilibrium vaporization of H O(l) at 0°C and at 100°C.
7.60 Which has the higher vapor pressure at 20°C, ice or 2
Explain why the calculated 100°C H value differs slightly
supercooled liquid water? Explain. vap m
from the true value. (b) Calculate H°and S° for vaporization
7.61 At the solid–liquid–vapor triple point of a pure sub- of H O(l) at 0°C; make reasonable approximations. The vapor
2
stance, which has the greater slope, the solid–vapor or the liquid– pressure of water at 0°C is 4.58 torr.
vapor line? Explain.
s
7.70 A solution is prepared by mixing n moles of the solvent
A
s
7.62 A beaker at sea level contains pure water. Calculate the A with n moles of the solute B. The s stands for stoichiometric
B
difference between the freezing point of water at the surface and indicates that these mole numbers need not be the number
and water 10 cm below the surface. of moles of A and B actually present in the solution, since A and