Page 54 - Physical Chemistry

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1.28 Two evacuated bulbs of equal volume are connected by 1.41 Find / y of: (a) 5x y sin(axy) 3; (b) cos (by z);
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2
a tube of negligible volume. One bulb is placed in a 200-K (c) xe ; (d) tan (3x 1); (e) 1/(e a/y 1); (f ) f (x)g(y)h(z).
x/y
constant-temperature bath and the other in a 300-K bath, and
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then 1.00 mol of an ideal gas is injected into the system. Find 1.42 Take ( / T) P,n of (a) nRT/P; (b) P/nRT (where R is a
the final number of moles of gas in each bulb. constant).
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2 3
1.29 An oil-diffusion pump aided by a mechanical forepump 1.43 (a) If y 4x 6x, find dy. (b) If z 3x y , find dz.
can readily produce a “vacuum” with pressure 10 6 torr. (c) If P nRT/V, where R is a constant and all other quantities
Various special vacuum pumps can reduce P to 10 11 torr. At are variables, find dP.
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25°C, calculate the number of molecules per cm in a gas at 1.44 If c is a constant and all other letters are variables, find
(a)1 atm; (b) 10 6 torr; (c) 10 11 torr.
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(a) d(PV); (b) d(1 T); (c) d(cT ); (d) d(U PV).
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1.30 A certain mixture of He and Ne in a 356-cm bulb 1.45 Let z x /y . Evaluate the four second partial deriva-
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5
weighs 0.1480 g and is at 20.0°C and 748 torr. Find the mass tives of z; check that z/( x y) z/( y x).
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2
and mole fraction of He present.
1.46 (a) For an ideal gas, use an equation like (1.30) to show
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1.31 The earth’s radius is 6.37 10 m. Find the mass of the
that dP P(n 1 dn T 1 dT V 1 dV) (which can be written
earth’s atmosphere. (Neglect the dependence of g on altitude.)
as d ln P d ln n d ln T d ln V). (b) Suppose 1.0000 mol
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1.32 (a)If10 P/bar 9.4, what is P?(b)If10 T/K 4.60, of ideal gas at 300.00 K in a 30.000-L vessel has its tempera-
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3
what is T?(c)If P/(10 bar) 1.2, what is P?(d)If10 (K/T) ture increased by 1.00 K and its volume increased by 0.050 L.
3.20, what is T? Use the result of (a) to estimate the change in pressure, P.
(c) Calculate P exactly for the change in (b) and compare with
1.33 A certain mixture of N and O has a density of 1.185 g/L
2 2 the estimate given by dP.
at 25°C and 101.3 kPa. Find the mole fraction of O in the mix-
2
ture. (Hint: The given data and the unknown are all intensive
properties, so the problem can be solved by considering any Section 1.7
convenient fixed amount of mixture.) 1.47 Find the molar volume of an ideal gas at 20.0°C and
1.000 bar.
1.34 The mole fractions of the main components of dry air at
sea level are x 0.78, x 0.21, x 0.0093, x 1.48 (a) Write the van der Waals equation (1.39) using the
N 2 O 2 Ar CO 2
0.0004. (a) Find the partial pressure of each of these gases in molar volume instead of V and n. (b) If one uses bars, cubic
dry air at 1.00 atm and 20°C. (b) Find the mass of each of these centimeters, moles, and kelvins as the units of P, V, n, and T,
gases in a 15 ft 20 ft 10 ft room at 20°C if the barometer give the units of a and of b in the van der Waals equation.
reads 740 torr and the relative humidity is zero. Also, find the 1.49 For a liquid obeying the equation of state (1.40), find
density of the air in the room. Which has a greater mass, you or expressions for a and k.
the air in the room of this problem?
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1.50 For H O at 50°C and 1 atm, r 0.98804 g/cm and k
2
1
Section 1.6 4.4 10 10 Pa . (a) Find the molar volume of water at 50°C
and 1 atm. (b) Find the molar volume of water at 50°C and
1.35 On Fig. 1.15, mark all points where df/dx is zero and cir-
100 atm. Neglect the pressure dependence of k.
cle each portion of the curve where df/dx is negative.
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m
1.36 Let y x x 1. Find the slope of the y-versus-x 1.51 For an ideal gas: (a) sketch some isobars on a V -T dia-
curve at x 1 by drawing the tangent line to the graph at x gram; (b) sketch some isochores on a P-T diagram.
1 and finding its slope. Compare your result with the exact 1.52 A hypothetical gas obeys the equation of state PV
slope found by calculus. nRT(1 aP), where a is a constant. For this gas: (a) show that
a 1/T and k 1/P(1 aP); (b) verify that ( P/ T) a/k.
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3 3x
1.37 Find d/dx of (a) 2x e ; (b) 4e 3x 12; (c) ln 2x; V
2
(d) 1/(1 x); (e) x/(x 1); (f ) ln (1 e 2x ); (g) sin 3x. 1.53 Use the following densities of water as a function of T
for water at 25°C and
2
2 3x
2
1.38 (a) Find dy/dx if xy y 2. (b) Find d (x e )/dx . and P to estimate a, k, and 10P>0T2 V m 3
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1 atm: 0.997044 g/cm at 25°C and 1 atm; 0.996783 g/cm at
2
(c) Find dy if y 5x 3x 2/x 1. 26°C and 1 atm; 0.997092 g/cm at 25°C and 2 atm.
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x
1.39 Use a calculator to find: (a) lim x→0 x for x 0; 1.54 By drawing tangent lines and measuring their slopes, use
1/x
(b) lim x→0 (1 x) . Fig. 1.14 to estimate for water: (a) a at 100°C and 500 bar;
(b) k at 300°C and 2000 bar.
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x
1.40 (a) Evaluate the first derivative of the function y e at
x 2 by using a calculator to evaluate y/ x for x 0.1, 1.55 For H O at 17°C and 1 atm, a 1.7 10 4 K 1 and
2
0.01, 0.001, etc. Note the loss of significant figures in y as x k 4.7 10 5 atm . A closed, rigid container is completely
1
decreases. If you have a programmable calculator, you might filled with liquid water at 14°C and 1 atm. If the temperature is
try programming this problem. (b) Compare your result in raised to 20°C, estimate the pressure in the container. Neglect
(a) with the exact answer. the pressure and temperature dependences of a and k.