Chapter 14 Practice Problems In problems that require the atomic masses (atomic weights) of atomic hydrogen, oxygen, nitrogen, and carbon, we will use the rounded values, 1, 16, 14, and 12, respectively. The atomic masses of molecular hydrogen, oxygen, and nitrogen are 2, 32, and 28, respectively. 1. (a) How many atoms are there in one mole of hydrogen gas (H2)? (b) Nitrogen gas (N2 )? 2. (a) How many hydrogen atoms are there one mole of methane, CH4? (b) What is the mass (in grams) of one molecule of methane? 3. A cylindrical glass of water (H2O) has a radius of 4.50 cm and a height of 12.0 cm. The density of water is 1.0 g/cm 3. (a) How many moles of water are in the glass? (a) How many molecules? 4. It takes 0.16 grams of helium (He, atomic mass 4)) to fill a balloon. How many grams of nitrogen gas would be required to fill the balloon to the same pressure, volume, and temperature? 5. An empty oven with a volume of 0.150 m 3 and a temperature of 300 K is vented so that the air pressure inside is always the same as the air pressure of the environment. Initially, the air pressure is 1000 millibars, but after the oven has warmed up to a final temperature of 460 K, the atmospheric air pressure has decreased by 50 millibars. How many moles of air leave the oven while it is heating up? 6. An ideal gas at 15.5 o C and a pressure of 1.72 x 10 5 Pa occupies a volume of 2.81 m 3. (a) How many moles of gas are present? (b) If the volume is raised to 4.16 m 3 and the temperature raised to 28.2 o C, what will be the pressure of the gas? 7. Oxygen (O2 ) pressurized in a tank at 65.0 atm for hospital patients is allowed to expand and is delivered to patients lungs at 1.0 atm. Suppose the pressurized gas is initially at temperature of 288 o K, but arrives at the lungs of the patients at 297 o K. For each cubic meter of oxygen that is drawn from the tank, how many cubic meters of oxygen are received by the patients? 8. A clown at a birthday party has brought along a helium cylinder, with which he intends to fill balloons. When full, each balloon contains 0.034 m 3 of helium at an absolute pressure of 1.2 x 10 5 Pa. The cylinder contains helium at an absolute pressure of 1.6 x 10 7 Pa and has a volume of 0.0031 m 3. The temperature of the helium in the cylinder and in the balloons is the same and remains constant. What is the maximum number of balloons that can be filled? 9. In a diesel engine, the piston compresses air at 305 o C to a volume that is one-sixteenth of the original volume and a pressure that is 48.5 times the original pressure. What is the temperature of the air after the compression?
10. A 0.030-m 3 container is initially evacuated. Then, 4.0 grams of water is placed in the container, and after some time, all the water evaporates. If the temperature of the water vapor is 388 K, what is its pressure? Physics 23 Chapter 14 Supplement to Practice Problems 1. A cylinder contains oxygen gas at a temperature of 20 o C and a pressure of 15 atm in a volume of 100 liters. A fitted piston is lowered into the cylinder, decreasing the volume occupied by the gas to 80 liters and raising the temperature to 25 o C. What then is the gas pressure? 2. An air bubble of 20 cm 3 volume is at the bottom of a lake 40 meters deep where the temperature is 4 o C. The bubble rises to the surface which is at a temperature 20 o C. Take the temperature to be the same as that of the surrounding water and find its volume just before it reaches the surface. 3. An automobile tire has a volume of 1000 in 3 and contains air at a gauge pressure of 24 lbs/in 2 when the temperature is 0 o C. What is the gauge pressure of the air in the tires when its temperature rises to 27 o C and its volume increases to 1020 in 3? 4. Compute the number of molecules in a gas contained in a volume of 1 cm 3 at a pressure of 0.001 atm and a temperature of 200 K. 5. Oxygen gas at 273 K at 1-atm pressure is confined to a cubical container 10 cm on a side. How long does it take a typical molecule to cross the container? 6. (a) What is the internal energy of 20 grams of helium (atomic mass = 4) gas at 50 o C? (b) What is the average energy per atom? (c) What is the average speed? 7. (a) How many calories of heat energy must flow into a gas containing two moles of argon molecules at 20 o C to raise its temperature to 60 o C? (b) Calculate the specific heat capacity of argon gas.
Chapter 14 Problem Solutions 1. H2 and N2 have two atoms per molecule (a) 2 x 6.02 x 10 23 = 1.204 x 10 24 (b) same as above 3. Atomic mass of water = 16 +2(1) = 18 m = ρv = 1.0 [π (4.50) 2 12.0] = 763 g (a) n = 763/18 = 42.4 moles (b) N = 42.4 x NA = 42.4 (6.02 x 10 23 ) = 2.55 x 10 25 2. (a) 4 x 6.02 x 10 23 (b) Atomic mass of CH4 = 12 + 4 x 1 = 16 m =16/6.02 x 10 23 = 2.66 x 10-23 g 4. n = PV/RT P, V, and T are the same for both gases, so n is also the same: n = 0.16/4 = 0.04 moles of helium = 0.04 moles of nitrogen Atomic weight of N2 = 28 0.04 x 28 = 1.12 g 5. P1 = 100,000 Pa P2 = 95,000 Pa V1 = V2 = 0.150 m 3 = V T1 = 300 o K T2 = 460 o K n = PV/RT Δn = n2 - n1 = P2V2/RT2 - P1V1/RT1 = (V/R) (P2/T2 - P1/T1) = (0.150/8.31) (95,000/460-100,000/300) = - 2.29 moles
6. T = 273.0 + 15.5 = 288.5 o K P = 1.72 x 10 5 Pa V = 2.81 m 3 (a) n = PV/RT = 201.6 mol 7. P1 = 65 atm P2 = 1 atm T1 = 288 o K T2 = 297 o K V1 = 1.00 m 3 V2 = nrt2 /P2 (b) P = nrt/v =121,298 Pa 8. T1 = T2 P1 = 1.6 x 10 7 Pa P2 = 1.2 x 10 5 Pa V1 = 0.0031 m 3 V2 =? As in Problem 7: V2/V1 = (T2/T1)(P1/P2) = (1)(1.6 x 10 7 / 1.2 x 10 5 ) = 133 V2 = 133 V1 = 133(0.0031) = 0.412 m 3 Number of balloons: 0.412/0.034 = 12 V1 = nrt1 /P1 V2/V1 = (T2/T1)(P1/P2) V2 = (297/288)(65/1) = 67 m 3 9. n1 = n2 = n V2 = V1/16 P2 = 48.5 P1 T1 = 305 o C = 578 o K T2 =? T = PV/nR: T2 = P2V2/nR T1 = P1V1/nR T2/T1 = (P2/P1) (V2/V1) = (48.5)(1/16) = 3.03 T2 = 3.03 T1 = 3.03 (578) = 1751 o K 7. P1 = 65 atm P2 = 1 atm T1 = 288 o K T2 = 297 o K V1 = 1.00 m 3 V2 = nrt2 /P2 V1 = nrt1 /P1 V2/V1 = (T2/T1) (P1/P2) V2 = (297/288) (65/1) = 67 m 3 10. Atomic mass of water: 18.0 g n = 4.0/18.0 = 0.22 mol T = 388 K V = 0.03 m 3 P = nrt/v = 0.22 (8.31)(388)/0.03 = 2.36 x 10 4 Pa
Chapter 14 Supplemental Practice Problems Solutions 1. T1 = 273 + 20 = 293 o K T2 = 273 + 25 = 298 o K V1 = 100 L V2 = 80 L P1= 15 atm P2/P1 = (V1/V2)(T2/T1) P2 = (100/80)(298/293)15 = 19 atm 3. V1 = 1000 in 3 : P1 = 24.0 + 14.7 = 38.7 lbs/in 2 (psi) T1 = 273 o K T2 = 300 o K: V2 = 1020 in 3 P2 = (V1/V2)(T2/T1) P1 = 41.7 psi Gauge Pressure = 41.7-14.7 = 27 psi 2. V2 = (T2/T1)(P1/P2) V1 T1 = 273 + 4 = 277 o K T2 = 273 + 20 = 293 o K P1 =101,000 + 1000 (9.8)40 = 4.93 x 10 5 Pa P2 = 101,000 Pa V1 = 20 cm 3 V2 = 103 cm 3 4. k = 1.38 x 10-23 J/K o P = 0.001 atm = 1.01 x 10 2 Pa V = (1.0 x 10-2 m) 3 = 1.0 x 10-6 m 3 T = 200 o K PV = NkT N = PV/kT = 3.7 x 10 16
5. O2 atomic mass = 32 g m = 32/6.02 x 10 23 = 5.3 x 10-23 g = 5.3 x 10-26 kg ½ mv 2 = 3/2 kt T = 273 o K v = 462 m/s t = d/v = 0.10/462 = 2.2 x 10-4 s 6. T = 323 o K n = 20/4 = 5 moles N = 5(6.02 x 10 23 ) = 3.01 x 10 24 (a) U = 3/2 NkT =2.0 x 10 4 J (b) K = U/N = 6.6 x 10-21 J (c) m = 4/NA =6.6 x 10-24 g = 6.6 x 10-27 kg ½ mv 2 = 6.6 x 10-21 J v = 1414 m/s 7. (a) U = 3/2 nrt Q = ΔU = U2 - U1 Q = 3/2 (2)(8.31)(333-293) = 997 J = 997 J/(4.19 J/cal) = 238 cal (b) ΔT = 40 C o Atomic mass of argon = 39.9 g m = 2 (39.9) = 79.8 g Q = mcδt c = Q/mΔT = 0.075 cal/g-c o