Page 61 - Essentials of physical chemistry

Ideal and Real Gas Behavior 23

terms of the reduced parameters: P R ¼ (P=P C ), V R ¼ (V=V C ), and T R ¼ (T=T C ), which yields a
universal van der Waals equation

3 1 8
¼ T R :
P R þ 2 V R
V 3 3
R
Although the van der Waals gas equation is generally more accurate than the ideal gas law, the
equation is cubic in V which leads to a practical difficulty in solving for V. An iterative Newton–
Raphson technique is suggested for use with some sort of computer-aided way to evaluate V. The
idea of the critical point is applied to introduce an analytical technique termed supercritical fluid
chromatography, which is shown to be able to separate and clearly resolve high-molecular weight
materials. Discussion followed as to potential applications of SCF chromatography for nondestruc-
tive analysis of forensic evidence. The SCF behavior of carbon dioxide was described as a good
‘‘solvent’’ near its critical point. The exercise in calculus to find the van der Waals parameters offers
a chance to motivate improvement in mathematical skills when it is seen that the formulas can be
used to find the (a, b) parameters for a number of real gases.

PROBLEMS
1.1 Calculate the volume in liters of 2 mol of He gas at 658F and 740 mmHg pressure.
1.2 Calculate the pressure in a tire inflated to 30 psi in winter at a temperature of 108F if the tire
has the same volume in July when the temperature is 908F.
1.3 Calculate the pressure of 2 mol of H 2 contained in a 10 L container at 308C using the van der
Waals equation.
1.4 Calculate the volume of 1 mol of CO 2 gas at 308C when the pressure is 20 bars using the van
der Waals equation. (Hint: Estimate V from the ideal gas equation and use the Newton–
Raphson method for the van der Waals gas starting from that value.)
1.5 Calculate P c , V c , and T c for NH 3 gas using the van der Waals parameters (a, b) and compare to
the values in a handbook or the Internet.
1.6 A glass light bulb shell is sealed by a glassblower and one small tip is pulled out into a long
narrow point with an entrance hole of about 1=8 in. diameter. The open bulb is weighed and
found to weigh 46.345 g. Then the bulb is completely filled with water using an eye dropper
with a thin tip and the filled bulb is found to have a mass of 237.93 g. Use the density of water
3
at 208C of 0.99821 g=cm to find the internal volume of the bulb. The bulb is then aspirated to
remove the water and dried in an oven. When dry and cool the bulb is reweighed and found to
still be 46.345 g prior to filling it with about 5 mL of a volatile liquid. The bulb with this small
amount of liquid is placed carefully into a 250 mL beaker of water, which is heated to a boil at
exactly 1008C. After reaching equilibrium at this temperature and all evidence of liquid is
gone the glass tip is sealed with a tiny drop of glue, the bulb is removed from the boiling
water, carefully dried, and found to weigh 47.309 g. Assuming the pressure was 1 atm when
the bulb was sealed, what is the estimated molecular weight and a possible compound that has
a volatile liquid close to this molecular weight?
1.7 Calculate the moles of gas collected over water in a 600 mL container at 756 mmHg pressure
at 208C given that the vapor pressure of water is 2.3388 kPa at 208C.
1.8 Calculate the uncertainty in the mole answer in Problem 1.7 if the uncertainty is 5 mL in
volume, 1 mmHg in pressure, and 0.38C in temperature. Give the value in % and in moles.
1.9 Calculate the volume of a 9.39 in. diameter basketball in liters. (See Introduction: Math and
Physics Review.)
1 qV 1 qV
.(Introduction:
1.10 Using PV ¼ nRT, n ¼ 1, calculate a ¼ and b ¼
V qT V qP
p T
Math and Physics Review.)