Atomic Mass and Atomic Mass Number The mass of an atom is determined primarily by its most massive constituents: protons and neutrons in its nucleus. The sum of the number of protons and neutrons is called the atomic mass number: A = number of protons + number of neutrons The atomic mass, in u, is close, but not exactly, equal, to the atomic mass number. u is the atomic mass unit: 1 u = 1.66 10 27 kg 2017 Pearson Education, Inc. Slide 18-1 Moles and Molar Mass By definition, one mole of matter, be it solid, liquid, or gas, is the amount of substance containing Avogadro s number N A of particles. N A = 6.02 10 23 mol 1 The number of moles in a substance containing N basic particles is One mole of helium, sulfur, copper, and mercury. 2017 Pearson Education, Inc. Slide 18-2 Moles and Molar Mass The molar mass of a substance is the mass of 1 mol of substance. The molar mass, which we ll designate M, has units g/mol. kg/mol should be used in SI calculations. The number of moles in a system of mass m consisting of atoms or molecules with molar mass M is n = m M For example, water has a molar mass of 18 g/mol. Therefore, 63 grams of water would contain 3.5 moles of water or 2.1 x 10 24 molecules of water. 2017 Pearson Education, Inc. Slide 18-3 1
Which contains more molecules, a mole of hydrogen gas (H 2 ) or a mole of oxygen gas (O 2 )? A. The hydrogen B. The oxygen C. They each contain the same number of molecules. D. Can t tell without knowing their temperatures. 2017 Pearson Education, Inc. Slide 18-4 Example 1 How many moles of oxygen is in 100 g of oxygen gas? Also, how many molecules are there in 100 g of oxygen gas? 2017 Pearson Education, Inc. Slide 18-5 Ideal Gases The ideal-gas model is one in which we model atoms in a gas as being hard spheres. Such hard spheres fly through space and occasionally interact by bouncing off each other in perfectly elastic collisions. Experiments show that the ideal-gas model is quite good for gases if two conditions are met: 1.The density is low (i.e., the atoms occupy a volume much smaller than that of the container), and 2.The temperature is well above the condensation point. 2017 Pearson Education, Inc. Slide 18-6 2
The Ideal-Gas Law The pressure p, the volume V, the number of moles n and the temperature T of an ideal gas are related by the ideal-gas law as follows: where R is the universal gas constant: R = 8.31 J/mol K Or: where N is the number of molecules and k B is Boltzman s constant: k B = 1.38 10 23 J/K 2017 Pearson Education, Inc. Slide 18-7 If the volume of a sealed container of gas is decreased, the gas temperature A. Increases. B. Stays the same. C. Decreases. D. Not enough information to tell. 2017 Pearson Education, Inc. Slide 18-8 Two identical cylinders, A and B, contain the same type of gas at the same pressure. Cylinder A has twice as much gas as cylinder B. Which is true? A. T A < T B B. T A = T B C. T A > T B D. Not enough information to make a comparison. 2017 Pearson Education, Inc. Slide 18-9 3
Two cylinders, A and B, contain the same type of gas at the same temperature. Cylinder A has twice the volume as cylinder B and contains half as many molecules as cylinder B. Which is true? A. p A = 4p B B. p A = 2p B 1 C. p A = 2 p B 1 D. p A = p B 4 2017 Pearson Education, Inc. Slide 18-10 Example 2 100 g of oxygen gas is distilled into an evacuated 600 cm 3 container. What is the gas pressure at a temperature of 150ºC? 2017 Pearson Education, Inc. Slide 18-11 Example 3 An empty aluminum scuba tank contains 11.0 L of air at 21ºC and 1 atm. When the tank is filled rapidly from a compressor, the air temperature is 42ºC and the gauge pressure is 2.10 x 10 7 Pa. What mass of air was added? 2017 Pearson Education, Inc. Slide 18-12 4
Ideal Gases All ideal-gas problems involve a gas in a sealed container. The number of moles (and number of molecules) will not change during a problem. In that case, If the gas is initially in state i, characterized by the state variables p i, V i, and T i, and at some later time in a final state f, the state variables for these two states are related by 2017 Pearson Education, Inc. Slide 18-13 The temperature of a rigid (i.e., constant-volume), sealed container of gas increases from 100ºC to 200ºC. The gas pressure increases by a factor of A. 2 B. 1.3 C. 0.8 D. 0.5 2017 Pearson Education, Inc. Slide 18-14 A quantity of an ideal gas is contained in a balloon. Initially the gas temperature is 27ºC. You double the pressure on the balloon and change the temperature so that the balloon shrinks to one-quarter of its original volume. What is the new temperature of the gas? A. 54ºC B. 27ºC C. 13.5ºC D. 123ºC 2017 Pearson Education, Inc. Slide 18-15 5
Example 4 In an automobile engine, a mixture of air and vaporized gasoline is compressed in the cylinders before being ignited. A typical engine has a compression ratio of 9.00 to 1. If the quantity of gas is constant, what is the final temperature of the compressed gas if its initial temperature is 27ºC and the initial and final pressures are 1.00 atm and 21.7 atm, respectively? 2017 Pearson Education, Inc. Slide 18-16 In-class Activity #1 What is the volume of a container that holds exactly 1 mole of an ideal gas at STP (standard temperature and pressure, defined to be 0ºC and 1 atm)? 2017 Pearson Education, Inc. Slide 18-17 The van der Waals equation The model used for the ideal-gas equation ignores the volumes of molecules and the attractive forces between them. The van der Waals equation is a more realistic model: 2016 Pearson Education Inc. 6
pv-diagrams These show isotherms, or constant-temperature curves, for a constant amount of an ideal gas. 2016 Pearson Education Inc. This p-v diagram shows three possible states of a certain amount of an ideal gas. Which state is at the highest temperature? A. state #1 B. state #2 C. state #3 D. Two of these are tied for highest temperature. E. All three of these are at the same temperature. O p 1 3 2 V 2016 Pearson Education, Inc. pv-diagrams A pv-diagram for a nonideal gas shows isotherms for temperatures above and below the critical temperature T c. At still lower temperatures the material might undergo phase transitions from liquid to solid or from gas to solid. 2016 Pearson Education Inc. 7
Pressure in a Gas Why does a gas have pressure? The pressure in a gas is due to collisions of the molecules with the walls of its container. The steady rain of a vast number of molecules striking a wall each second exerts a measurable macroscopic force. The gas pressure is the force per unit area (p = F/A) resulting from these molecular collisions. The figure shows a molecule that collides with a wall, exerting an impulse on it.the average force on the wall is 2017 Pearson Education, Inc. Slide 20-22 Pressure in a Gas The pressure is the average force on the walls of the container per unit area: (N/V ) is the number density of the gas in m 3. Note that the average velocity of many molecules traveling in random directions is zero. v rms is the root-mean-square speed of the molecules, which is the square root of the average value of the squares of the speeds of the molecules: 2017 Pearson Education, Inc. Slide 20-23 Molecular speeds and kinetic energies The root-mean-square speed of the molecules in a gas is: Thus the average kinetic energy of the molecules is given by Total kinetic energy of n moles of a gas is given by 2016 Pearson Education Inc. 8
A rigid container holds both hydrogen gas (H 2 ) and nitrogen gas (N 2 ) at 100ºC. Which statement describes their rms speeds? A. v rms of H 2 < v rms of N 2 B. v rms of H 2 = v rms of N 2 C. v rms of H 2 > v rms of N 2 2017 Pearson Education, Inc. Slide 20-25 A rigid container holds both hydrogen gas (H 2 ) and nitrogen gas (N 2 ) at 100ºC. Which statement describes the average translational kinetic energies of the molecules? A. K tr of H 2 < K tr of N 2 B. K tr of H 2 = K tr of N 2 C. K tr of H 2 > K tr of N 2 2017 Pearson Education, Inc. Slide 20-26 Example 5 A container holds helium at a pressure of 200 kpa and a temperature of 60.0ºC. What is the rms speed of the helium atoms? 2017 Pearson Education, Inc. Slide 20-27 9
Consider two specimens of ideal gas at the same temperature. Specimen #1 has the same total mass as specimen #2, but the molecules in specimen #1 have greater molar mass than the molecules in specimen #2. In which specimen is the total translational kinetic energy of the entire gas greater? A. specimen #1 B. specimen #2 C. The answer depends on the particular mass of gas. D. The answer depends on the particular molar masses. E. Both C and D are correct. 2017 Pearson Education, Inc. Slide 20-28 Example 6 What is the average translational kinetic energy of an idealgas molecule at 27ºC? What is the total random translational kinetic energy of the molecules in 1 mole of this gas? 2017 Pearson Education, Inc. Slide 20-29 In-class Activity #2 What is the total translational kinetic energy of the molecules in 1.0 mol of gas at STP? 2017 Pearson Education, Inc. Slide 20-30 10